I’ll demonstrate how careful analysis of group relations yields unexpected constructions, addressing several central questions in group theory. These include a question by Elliott, Jonusas, Mesyan, Mitchell, Morayne, and Peresse regarding Zariski topologies on groups and semigroups, a series of questions by Amir, Blachar, Gerasimova, and Kozma concerning algebraic group laws, and a longstanding question by Wiegold on invariably generated groups.

In 1970, Frisch and Bourret introduced a q-deformation of independent Gaussian random variables (say "q-Gaussian system"). In one-variable case, q-Gaussian is the distribution whose orthogonal polynomials are q-Hermite polynomials, and this distribution interpolates between Rademacher (q=-1), semicircle (q=0), Gaussian (q=1) distribution. In multivariable case, q-Gaussian system is represented as a tuple of operators (which are non-commutative in general) on the q-deformed Fock space introduced by Bożejko and Speicher. 

In this talk, I will explain related combinatorics (pair partitions and number of crossings) and analysis for q-Gaussian system. 

Initially studied by Markov around 1880, the Lagrange spectrum, $L$, and the Markov spectrum, $M$, are complicated subsets of the real line that play a crucial role in the study of Diophantine approximation and binary quadratic forms. Perron's 1920s description of the spectra in terms of continued fractions allowed powerful dynamical machinery to come to bear on many problems. In this talk, I will discuss recent work with Harold Erazo and Carlos Gustavo Moreira investigating the function $d_\textrm{loc}(t)$ that determines the local Hausdorff dimension at a point $t$ in $L'$.

Biholomorphic mapping problems for domains in complex Euclidean space and in complex manifolds will be discussed.

Placing a two-dimensional lattice on another with a small rotation gives rise to periodic “moiré” patterns on a superlattice scale much larger than the original lattice.  The Bistritzer-MacDonald (BM) model attempts to capture the electronic properties of twisted bilayer graphene (TBG) by an effective periodic continuum model over the bilayer moiré pattern. We use the mathematical techniques developed to study waves in inhomogeneous media to identify a regime where the BM model emerges as the effective dynamics for electrons modeled as wave-packets spectrally concentrated at the monolayer Dirac points of linear dispersion, up to error that we rigorously estimate. Using measured values of relevant physical constants, we argue that this regime is realized in TBG at the first “magic" angle where the group velocity of the wave packet is zero and strongly correlated electronic phases (superconductivity, Mott insulators, etc.) are observed. 

We are working to develop models of TBG which account for the effects of mechanical relaxation and to couple our relaxed BM model with interacting TBG models.  We are also extending our approach to essentially arbitrary moirématerials such as twisted multilayer transition metal dichalcogenides (TMDs) or even twisted heterostructures consisting of layers of distinct 2D materials.

Ultrafilters show up in many places, including logic, topology, and analysis. Despite this, the concept does not seem to be well-known among mathematicians (indeed, the speaker completed several years of graduate school without learning about them). The goal of this talk is to present a friendly introduction to ultrafilters and to highlight a few of their various manifestations. If all goes well, the talk will include a topological proof of Arrow’s Impossibility Theorem, a classic result from political science.

We will discuss some ongoing work on the subject, specifically in the rank $2$, genus $2$ case. The talk will start with a quick review of existing literature on $M_2$ and some of its étale covers, along with results and constructions involving moduli of rank $2$ bundles. We will go over their generalizations to the universal setting and outline the usage of these tools for computing the Chow ring. Lastly, we shall go over some ideas to relate the generators to tautological classes.

The symbol length problem is a longstanding question concerning the Brauer group of a field. In the case of fields of positive characteristic, every Brauer class is split by a finite extension of height 1. This observation suggests a connection between the symbol length problem and the Galois theory of purely inseparable extensions, where the restricted Lie algebra naturally arise. In this talk, we will explore how various symbol length problems in Brauer groups relate to restricted Lie algebras and introduce a moduli-theoretic description of restricted subspaces in a restricted Lie algebra.

I will introduce the notion of a new equivalence relation on the class of countable discrete groups, called von Neumann orbit equivalence (vNOE). I will also discuss the stability of vNOE under the operations of taking free products and graph products of groups. This is based on a joint work with Aoran Wu.

Classification of the steady states for 2D Euler equation is a classical topic in fluid mechanics. In this talk, we consider the rigidity of the analytic steady states in  bounded simply-connected domains. By studying an over-determined elliptic problem in Serrin type, we show the stream functions for the steady state are either radial functions or solutions to semi-linear elliptic equations.  This is the joint work with Tarek Elgindi, Ayman Said and Chunjing Xie.

Let $G_{n,p}$ denote the random $n$-vertex graph obtained by including each edge independently and with probability $p$. Given a graph $F$, let $\mathrm{ex}(G_{n,p},F)$ denote the size of a largest $F$-free subgraph of $G_{n,p}$. When $F$ is non-bipartite, the asymptotic behavior of $\mathrm{ex}(G_{n,p},F)$ is determined by breakthrough work done independently by Conlon-Gowers and by Schacht, but the behavior for bipartite $F$ remains largely unknown.

We will discuss some recent developments that have been made for bipartite $F$, with a particular emphasis on the case of theta graphs.  Based on joint work with Gwen McKinley.

Optimization models that incorporate uncertainty and hierarchical structures have attracted much attention in data science. Recent advances in polynomial optimization offer promising methods to certify global optimality for these complex models. In this talk, I will use two-stage stochastic optimization as a major model to demonstrate how polynomial optimization can be efficiently applied to data science optimization under uncertainty.

One of the central problems in Arithmetic Statistics is counting number field extensions of a fixed degree with a given Galois group, parameterized by discriminants. This talk focuses on C2 ≀ H extensions over an arbitrary base field. While Jürgen Klüners has established the main term in this setting, we present an alternative approach that provides improved power-saving error terms for the counting function.

Drugs of abuse, such as opiates, have been widely associated with enhancing susceptibility to HIV infection, intensifying HIV replication, accelerating disease progression, diminishing host-immune responses, and expediting neuropathogenesis. In this talk, I will present a variety of mathematical models to study the effects of the drugs of abuse on several aspects of HIV infection and replication dynamics. The models are parameterized using data collected from simian immunodeficiency virus infection in morphine-addicted macaques. I will demonstrate how mathematical modeling can help answer critical questions related to the HIV infection altered due to the presence of drugs of abuse. Our models, related theories, and simulation results provide new insights into the HIV dynamics under drugs of abuse. These results help develop strategies to prevent and control HIV infections in drug abusers.

Many data-driven problems in the modern world involve solving nonconvex optimization problems. The large-scale nature of many of these problems also necessitates the use of first-order optimization methods, i.e., methods that rely only on the gradient information, for computational purposes. But the first-order optimization methods, which include the prototypical gradient descent algorithm, face a major hurdle for nonconvex optimization: since the gradient of a function vanishes at a saddle point, first-order methods can potentially get stuck at the saddle points of the objective function. And while recent works have established that the gradient descent algorithm almost surely escapes the saddle points under some mild conditions, there remains a concern that first-order methods can spend an inordinate amount of time in the saddle neighborhoods. It is in this regard that we revisit the behavior of the gradient descent trajectories within the saddle neighborhoods and ask whether it is possible to provide necessary and sufficient conditions under which these trajectories escape the saddle neighborhoods in linear time. The ensuing analysis relies on precise approximations of the discrete gradient descent trajectories in terms of the spectrum of the Hessian at the saddle point, and it leads to a simple boundary condition that can be readily checked to ensure the gradient descent method escapes the saddle neighborhoods of a class of nonconvex functions in linear time. The boundary condition check also leads to the development of a simple variant of the vanilla gradient descent method, termed Curvature Conditioned Regularized Gradient Descent (CCRGD). The talk concludes with a convergence analysis of the CCRGD algorithm, which includes its rate of convergence to a local minimum of a class of nonconvex optimization problems.

As we know, Kaehler-Ricci flow can be reduced to solve a class of  parabolic   complex Monge-Amp\`ere equations for Kaehler potentials and  the solutions usually depend on the Kaehler class of initial metric.   Thus there  gives a  degeneration of Kaehler metrics arising from the Kaehler-Ricci flow.  For a class of $G$-spherical manifolds,   we can  use  the local estimate  of  Monge-Amp\`ere equations as well as  the H-invariant for $C^*$-degeneration  to determine the limit of  Kaehler-Ricci flow after resales.  In particular,  on such manifolds,  the flow will develop the singularities of  type II.  

We present a highly effective algorithmic approach, PMM, for generating differentially private synthetic data in a bounded metric space with near-optimal utility guarantees under the 1-Wasserstein distance. In particular, for a dataset in the hypercube [0,1]^d, our algorithm generates synthetic dataset such that the expected 1-Wasserstein distance between the empirical measure of true and synthetic dataset is O(n^{-1/d}) for d>1. Our accuracy guarantee is optimal up to a constant factor for d>1, and up to a logarithmic factor for d=1. Also, PMM is time-efficient with a fast running time of O(\epsilon d n). Derived from the PMM algorithm, more variations of synthetic data publishing problems can be studied under different settings.

Riemannian geometry, the study of smooth manifolds and how to define distances and angles on them, can be viewed either intrinsically or extrinsically. In this talk, we discuss how Nash unified these views starting with his 1954 paper “C1 Isometric Imbeddings,” where the isometric embedding and the solution to the corresponding system of partial differential equations is constructed as the limit of iteratively defined subsolutions. This technique is cited as one of the first instances of what is now known as convex integration, and is used to construct solutions to many problems in geometry and PDE.

In this talk we will discuss extensions of the 4 fundamental products of graphs (cartesian, categorical, lexicographical, and strong products) to quantum graphs, and provide bounds on the resulting graphs akin to those for products of classical graphs. We will pay particular attention to the lexicographical product, discussing our notion of a quantum b-fold chromatic number as a tool for computing the quantum chromatic number of the lexicographical products.

This is joint work with A. Meenakshi McNamara.

We make explicit the relationship between Hilbert modular forms and orthogonal modular forms arising from positive definite quaternary lattices over the ring of integers of a totally real number field.  Our work uses the Clifford algebra, and it generalizes that of Ponomarev, Bocherer--Schulze-Pillot, and others by allowing for general discriminant, weight, and class group of the base ring.  This is joint work with Eran Assaf, Dan Fretwell, Colin Ingalls, Adam Logan, and Spencer Secord.

[pre-talk at 3:00PM]

Epigenetic cell memory, the inheritance of gene expression patterns across subsequent cell divisions, is a critical property of multi-cellular organisms. It was previously found via simulations of stochastic models that the time scale separation between establishment (fast) and erasure (slow) of chromatin modifications (such as DNA methylation and histone modifications) extends the duration of cell memory, and that different asymmetries between erasure rates of chromatin modifications can lead to different gene expression patterns. We provide a mathematical framework to rigorously validate these computational findings using stochastic models of chemical reaction networks. For our study of epigenetic cell memory, these are singularly perturbed, finite state, continuous time Markov chains. We exploit special structure in our models and extend beyond existing theory to study these singularly perturbed Markov chains when the perturbation parameter is small. We also develop comparison theorems to study how different erasure rates affect the behavior of our chromatin modification circuit. The theoretical tools developed in our work not only allow us to set a rigorous mathematical basis for highlighting the effect of chromatin modification dynamics on epigenetic cell memory, but they can also be applied to other singularly perturbed Markov chains beyond the applications in this work, especially those associated with chemical reaction networks.

The widely-used k-means procedure returns k clusters that have arbitrary convex shapes. In high dimension, such a clustering might not be easy to understand. A more interpretable alternative is to constraint the clusters to be the leaves of a decision tree with axis-parallel splits; then each cluster is a hyperrectangle given by a small number of features.

Is it always possible to find clusterings that are intepretable in this sense and yet have k-means cost that is close to the unconstrained optimum? A recent line of work has answered this in the affirmative and moreover shown that these interpretable clusterings are easy to construct.

I will give a survey of these results: algorithms, methods of analysis, and open problems.

Is every finite group isomorphic to the Galois group of some Galois extension of the rational numbers? Although this question remains open in general, powerful methods have led to an affirmative answer in some cases, including that of solvable groups, symmetric and alternating groups, and most of the sporadic groups. In this talk, we call upon seemingly disconnected areas of algebra, topology, and complex analysis to describe the rigidity method of inverse Galois theory.

We discuss an analogue of the Goldbach conjecture for polynomials with coefficients in semidomains (i.e., subsemirings of an integral domain).

Empirical evidence suggests that for a variety of overparameterized nonlinear models, most notably in neural network training, the growth of the loss around a minimizer strongly impacts its performance. Flat minima -- those around which the loss grows slowly -- appear to generalize well. This work takes a step towards understanding this phenomenon by focusing on the simplest class of overparameterized nonlinear models: those arising in low-rank matrix recovery. We analyze overparameterized matrix and bilinear sensing, robust PCA, covariance matrix estimation, and single hidden layer neural networks with quadratic activation functions. In all cases, we show that flat minima, measured by the trace of the Hessian, exactly recover the ground truth under standard statistical assumptions. For matrix completion, we establish weak recovery, although empirical evidence suggests exact recovery holds here as well.

The q-circular system is a tuple of non-commutative random variables (operators with some state) which interpolate independent standard complex Gaussian random variables (q=1) in classical probability and freely independent circular random variables (q=0) in free probability. One of the interesting results on q-deformed probability is that -1<q<1 case has similar properties to free case (q=0). Haagerup inequality is one of such properties, which was originally proved for generators of free groups with respect to the left regular representation.    

 In this talk, I will explain the strong version of Haagerup inequality for the q-circular system, which was originally proved by Kemp and Speicher for q=0. This talk is based on a joint project with T. Kemp.

This talk discusses the matrix Moment-SOS hierarchy for solving polynomial matrix optimization problems. We first establish the finite convergence of this hierarchy under the Archimedean property, provided the nondegeneracy condition, strict complementarity condition, and second-order sufficient condition hold at every minimizer. Furthermore, we also prove that every minimizer of the moment relaxation must exhibit a flat truncation when the relaxation order is sufficiently large.

Quantum interaction models discussed here are the (asymmetric) quantum Rabi model (QRM) and non-commutative harmonic oscillator (NCHO). The QRM is the most fundamental model describing the interaction between a photon and two-level atoms. The NCHO can be considered as a covering model of the QRM, and recently, the eigenvalue problems of NCHO and two-photon QRM (2pQRM) are shown to be equivalent. Spectral degeneracy can occur in models, but correspondingly there is a hidden symmetry relates geometrical nature described by hyperelliptic curves. In addition, the analytical formula for the heat kernel (propagator)/partition function of the QRM is described as a discrete path integral and gives the meromorphic continuation of its spectral zeta function (SZF). This discrete path integral can be interpreted to the irreducible decomposition of the infinite symmetric group $\mathfrak{S}_\infty$ naturally acting on $\mathbb{F}_2^\infty$, $\mathbb{F}_2$ being the binary field. Moreover, from the special values of the SZF of NCHO, an analogue of the Apéry numbers is naturally appearing, and their generating functions are, e.g., given by modular forms, Eichler integrals of a congruence subgroup. The talk overviews those above and present questions which are open.

[pre-talk at 3:00PM]

There is a special entropy quantity associated to the Gauss curvature flow which plays an important rule for the convergence of the flow. Similar entropy can also be defined for a class of generalized Gauss curvature flows, in particular for anisotropic flows. One crucial property is monotonicity of the associated entropy along the flow. Another is the fact that critical point of entropy associated to the anisotropic flow under volume constraint is a solution to the $L^p$-Minkowski problem. This provides a flow approach to the $L^p$-Minkowski problem. The main question is under what condition entropy can control the diameter, as to obtain non-collapsing estimate for the flow.  We will discuss the main steps of the approach, and open problems related to inhomogeneous type flows.

I will prove strict comparison of C* algebras associated to free groups and then use it to solve the C* version of Tarski's problem from 1945 in the negative. It is joint work with Amrutam, Gao and Patchell and another joint work with Schafhauser.

The key property of linear dispersive flows is that waves with different frequencies travel with different group velocities, which leads to the phenomena of dispersive decay. Nonlinear dispersive flows also allow for interactions of linear waves, and their long time behavior is determined by the balance of linear dispersion on one hand, and nonlinear effects on the other hand. 

The first goal of this talk will be to present a new set of conjectures which aim to describe the global well-posedness and the dispersive properties of solutions in the most difficult case when the nonlinear effects are dominant, assuming only small initial data. This covers many interesting physical models, yet, as recently as a few years ago, there was no clue even as to what one might reasonably expect. The second objective of the talk will be to describe some very recent results in this direction.  This is joint work with Mihaela Ifrim.

Deep learning has been wildly successful in practice and most state-of-the-art artificial intelligence systems are based on neural networks. Lacking, however, is a rigorous mathematical theory that adequately explains the amazing performance of deep neural networks. In this talk, I present a new mathematical framework that provides the beginning of a deeper understanding of deep learning. This framework precisely characterizes the functional properties of trained neural networks. The key mathematical tools which support this framework include transform-domain sparse regularization, the Radon transform of computed tomography, and approximation theory. This framework explains the effect of weight decay regularization in neural network training, the importance of skip connections and low-rank weight matrices in network architectures, the role of sparsity in neural networks, and explains why neural networks can perform well in high-dimensional problems. At the end of the talk we shall conclude with a number of open problems and interesting research directions.

This talk is based on work done in collaboration with Rob Nowak, Ron DeVore, Jonathan Siegel, Joe Shenouda, and Michael Unser.

Coq is a programming language and "proof assistant", where one can state and prove theorems which are checked for soundness by Coq itself. Looking at an example formalization of the hypernatural numbers, we'll explore what makes such a tool useful, interesting, and even fun! At the end of this talk attendees will hopefully have reasons to consider using Coq or similar tools themselves, and incidentally be able to construct a non-standard model of arithmetic (whenever the need arises).

Many data science problems require solving a least-squares problem min_x || A x - b ||^2. Efficiently solving this problem becomes a challenge when A has millions of rows, or even higher. I am developing solutions based on randomized numerical linear algebra:

1. If A is small enough to fit in working memory, an efficient solution is conjugate gradient with randomized preconditioning.

2. If A is too large to fit in working memory but x fits in memory, an intriguing possibility is randomized Kaczmarz.

3. If x is too large to fit in working memory, the final possibility is randomly sparsified Richardson iteration.

Two tracial von Neumann algebras are elementarily equivalent if they cannot be distinguished by first-order sentences or, more algebraically, if they have isomorphic ultrapowers. The same definition can be made for (countable, discrete) groups, and it is natural to wonder whether or not there is a connection between two groups being elementarily equivalent and their corresponding group von Neumann algebras being elementarily equivalent.  In the first part of the talk, I will give examples to show that, in general, there is no connection in either direction.  In the second part of the talk, I will introduce a strengthening of elementary equivalence, called back-and-forth equivalence (in the sense of computability theory) and show that back-and-forth equivalent groups have back-and-forth equivalent group von Neumann algebras.  I will also discuss how the same is true for the group measure space von Neumann algebra associated to the Bernoulli action of a group on an arbitrary tracial von Neumann algebra.  The latter half of the talk represents joint work with Matthew Harrison-Trainor.

The superspace ring of rank $n$ is defined as the tensor product of the polynomial ring over $n$ variables and the exterior product of $n$ additional variables. This carries an action of the symmetric group, as well as the hyperoctahedral group (the group of signed permutations). For each of these actions, we define the coinvariant ideal as the ideal generated by invariants under the action with vanishing constant term. We explore some results on bases and Hilbert series of the quotient rings cut out by these ideals.

There is an idea in number theory that if two objects are congruent modulo a prime p, then the congruence can also be seen for the special values of L functions attached to the objects. Here is a context explicating this idea: Suppose f and f' are holomorphic cuspidal eigenforms of weight k and level N, and suppose f is congruent to f' modulo p; suppose g is another cuspidal eigenform of weight l; if the difference k - l is large then the Rankin-Selberg L function L(s, f x g) has enough critical points; same for L(s, f' x g); one expects then that there is a congruence modulo p between the algebraic parts of L(m, f x g) and L(m, f' x g) for any critical point m. In this talk, after elaborating on this idea, I will describe the results of some computational experiments where one sees such congruences for ratios of critical values for Rankin-Selberg L-functions. Towards the end of my talk, time-permitting, I will sketch a framework involving Eisenstein cohomology for GL(4) over Q which will permit us to prove such congruences. This is joint work with my student P. Narayanan.

In this talk, we present a unified framework for accelerated gradient methods through the variable and operator splitting. The operator splitting decouples the optimization process into simpler subproblems, and more importantly, the variable splitting leads to acceleration. The key contributions include the development of strong Lyapunov functions to analyze stability and convergence rates, as well as advanced discretization techniques like Accelerated Over-Relaxation (AOR) and extrapolation by the predictor-corrector (EPC) methods. The framework effectively handles a wide range of optimization problems, including convex problems, composite convex optimization, and saddle point systems with bilinear coupling. A dynamic updating parameter, which serves as a rescaling of time, is introduced to handle the weak convex cases.

Many stochastic processes (such as the full family of Lévy processes) can be linearly and deterministically transformed into a white noise process. Consequently these processes can be viewed as the deterministic "mixing" of a white noise process. This formulation is the so-called "innovation model" of Bode, Shannon, and Kailath (ca. 1950-1970), where the white noise represents the stochastic part of the process, called its innovation. This allows for a conceptual decoupling between the correlation properties of a process (which are imposed by the whitening operator L) and its underlying randomness, which is determined by its innovation. This reduces the study of a stochastic process to the study of its underlying innovation. In this talk, we will introduce the innovations approach to studying stochastic processes and adopt the beautiful formalism of generalized stochastic processes of Gelfand (ca. 1955), where stochastic processes are viewed as random tempered distributions (more generally, random variables that take values in the dual of a nuclear space). This formulation uses the so-called characteristic functional (infinite-dimensional analog of the characteristic function of a random variable) of a stochastic process in lieu of more traditional concepts such as random measures and Itô integrals. A by-product of this formulation is that the characteristic functional of any stochastic process that satisfies the innovation model can be readily derived, providing a complete description of its law. We will then discuss some of my recent work where we have derived the characteristic functional of random neural networks to study their properties. This setting will reveal the true power of the characteristic functional: Any property of a stochastic process can be derived with short and simple proofs. For example, we will show that, as the "width" of these random neural networks tends to infinity, these processes can converge in law not only to Gaussian processes, but also to non-Gaussian processes depending on the law of the parameters. Our asymptotic results provide a new take on several classical results that have appeared in the machine learning community (wide networks converge to Gaussian processes) as well as some new ones (wide networks can converge to non-Gaussian processes). This talk is based on joint work with Pakshal Bohra, Ayoub El Biari, Mehrsa Pourya, and Michael Unser from our recent preprint arxiv:2405.10229.

To a smooth projective curve over a finite field, we associate rigid-analytic moduli stacks of isocrystals together with the Verschiebung endomorphism. We develop relevant foundations of rigid-analytic stacks, and discuss the examples and properties of such moduli stacks. We also illustrate how such moduli can be used to count p-adic coefficient objects on the curve of rank one.

The main talk will be given by Oh. In the pre-talk, Shimizu will introduce integrable connections and isocrystals, which will be the key objects in the main talk.

[pre-talk at 1:00PM]

In the current stage of numerical methods for PDE, the primary challenge lies in addressing the complexities of high dimensionality while maintaining physical fidelity in our solvers. In this presentation, I will introduce deep learning assisted particle methods aimed at addressing some of these challenges.  These methods combine the benefits of traditional structure-preserving techniques with the approximation power of neural networks, aiming to handle high dimensional problems with minimal training. I will begin with a discussion of general Wasserstein-type gradient flows and then extend the concept to the Landau equation in plasma physics.

No, it's not; that question doesn't even make sense. But pretending it is for a minute lets us construct a special class of bijections involving sets of binary trees (known in the literature as "particularly elementary" bijections, or sometimes "very explicit" bijections), and even deduce nice equivalent conditions for when such a bijection exists. Based on the paper "Seven Trees in One" by Andreas Blass, this talk explores whether we can ever "solve for" the set of binary trees, and whether we should.
 

An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. In this talk, we will restrict to the case of log Calabi-Yau pairs (X,B). We prove that if (X,B) admits a polarized endomorphism that preserves the boundary structure, then (X,B) is a finite quotient of a toric log Calabi-Yau fibration over an abelian variety. This is joint work with Joaquin Moraga and Wern Yeong.

Hierarchically hyperbolic spaces provide a uniform framework for working with many important examples, including mapping class groups, right angled Artin groups, Teichmuller space, and others.  In this talk I'll provide an introduction to studying groups and spaces from this point of view. I'll focus on a few of my favorite "hyperbolic features" and how they manifest in many examples. This talk will include joint work with M. Hagen and A. Sisto, as well as with C. Abbott and M. Durham. 

We present new objects called quilts of alternating sign matrices with respect to two given posets. Quilts generalize several commonly used concepts in mathematics. For example, the rank function on submatrices of a matrix gives rise to a quilt with respect to two Boolean lattices. When the two posets are chains, a quilt is equivalent to an alternating sign matrix and its corresponding corner sum matrix. Such rank functions are used in the definition of Schubert varieties in both the Grassmannian and the complete flag manifold. Quilts also generalize the monotone Boolean functions counted by the Dedekind numbers, which is known to be a #P-complete problem. Quilts form a distributive lattice with many beautiful properties and contain many classical and well known sublattices, such as the lattice of matroids of a given rank and ground set. While enumerating quilts is hard in general, we prove two major enumerative results, when one of the posets is an antichain and when one of them is a chain. We also give some bounds for the number of quilts when one poset is the Boolean lattice. Several open problems will be given for future development. This talk is based on joint work with Matjaz Konvalinka in arxiv:2412.03236.

In collaboration with Han, Padurariu, and Park, we show that the number of $5$-isogenies of elliptic curves defined over $\mathbb{Q}$ with naive height bounded by $H > 0$ is asymptotic to $C_5\cdot H^{1/6} (\log H)^2$ for some explicitly computable constant $C_5 > 0$. This settles the asymptotic count of rational points on the genus zero modular curves $X_0(m)$. We leverage an explicit $\mathbb{Q}$-isomorphism between the stack $\mathscr{X}_0(5)$ and the generalized Fermat equation $x^2 + y^2 = z^4$ with $\mathbb{G}_m$ action of weights $(4, 4, 2)$.

Pretalk: I will explain how to count isomorphism classes of elliptic curves over the rationals. On the way, I will introduce some basic stacky notions: torsors, quotient stacks, weighted projective stacks, and canonical rings.

[pre-talk at 3:00PM]

The study of orbit dynamics for the upper triangular subgroup $P \subset \mathrm{SL}(2, \mathbb R)$ holds fundamental significance in homogeneous and Teichmüller dynamics. In this talk, we shall discuss the quantitative density properties of $P$-orbits for translation surfaces near Teichmüller curves. In particular, we discuss the Teichmüller space $H(2)$ of genus two Riemann surfaces with a single zero of order two, and its corresponding absolute period coordinates, and examine the asymptotic dynamics of $P$-orbits in these spaces.

Given a random walk on a countable group, the Poisson boundary is a measure space which captures all asymptotic events of the markov chain. The Poisson boundary can sometimes be identified with a concrete geometric "boundary at infinity", but almost all previous results relied strongly on moment conditions of the random walk. I will discuss a technique which allows us to identify the Poisson boundary on any group with hyperbolic properties without moment conditions, new even in the free group case, making progress on a question of Kaimanovich and Vershik. This is joint work with Behrang Forghani, Joshua Frisch, and Giulio Tiozzo.

Growth of bacterial colonies on solid surfaces is commonplace; yet, what occurs inside a growing colony is complex even in the simplest cases. Robust colony expansion kinetics featuring linear radial growth and saturating vertical growth in diverse bacteria indicates a common developmental program, which will be elucidated in this talk using a combination of findings based on modeling and experiments.  Agent-based simulations reveal the crucial role of emergent mechanical constraints and spatiotemporal dynamics of nutrient gradients which govern observed expansion kinetics. The consequences of such emergent features will also be examined in the context of pattern formation in multi-species bacterial communities. Future directions and opportunities in theoretical modeling of such pattern formation systems will be discussed.

In this talk, we discuss the semiclassical asymptotics for Bergman kernels in exponentially weighted spaces of holomorphic functions. We will first review various approaches to the construction of asymptotic Bergman projections, for smooth weights and for real analytic weights. We shall then explore the case of Gevrey weights, which can be thought of as the interpolating case between the real analytic and smooth weights. In the case of Gevrey weights, we show that Bergman kernel can be approximated in certain Gevrey symbol class up to a Gevrey type small error, in the semiclassical limit. We will also introduce some microlocal analysis tools in the Gevrey setting, including Borel's lemma for symbols and complex stationary phase lemma. This talk is based on joint work with Hang Xu.

In this talk, I will present a full resolution of the question of whether the Regularity problem for the parabolic PDE $-\partial_tu + {\rm div}(A\nabla u)=0$ on a Lipschitz cylinder $\mathcal O\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients satisfy a very natural Carleson condition (a parabolic analog of the so-called DKP-condition).

We prove that for some $p_0>1$ the Regularity problem is solvable in the range $(1,p_0)$. We note that answer to this question was not known previously even in the  "small Carleson case", that is, when the Carleson norm of coefficients is sufficiently small.

In the elliptic case the analogous question was only fully resolved recently (2022) independently by two groups using two very different methods; one involving S. Hofmann, J. Pipher and the presenter, the second by M. Mourgoglou, B. Poggi and X. Tolsa. Our approach in the parabolic case is motivated by that of the first group, but in the parabolic setting there are significant new challenges. The result is a joint work with L. Li and J. Pipher.

Consider minx≥ 0 f(x), where the objective function f: ℝ→ ℝ is smooth, and the variable is required to be nonnegative. A naive "squared variable" technique reformulates the problem to minv∈ ℝ f(v2). Note that the new problem is now unconstrained, and many algorithms, e.g., gradient descent, can be applied. In this talk, we discuss the disadvantages of this approach known for decades and the possible surprising fact of equivalence for the two problems in terms of (i) local minimizers and (ii) points satisfying the so-called second-order optimality conditions, which are keys for designing optimization algorithms. We further discuss extensions to the vector case (where the vector variable is required to have all entries nonnegative) and the matrix case (where the matrix variable is required to be a positive semidefinite) and demonstrate such an equivalence continues to hold.

In this talk, various aspects of the game of chess will be explored.  Like true philosophers, we will make random observations, pose rhetorical questions and draw strange parallels, all without claim to the truth, while touching on various topics from the nature of randomness to ways to maximize cognitive performance. Familiarity with the game is not required but will be helpful.
 

In this talk, we consider operator-valued polynomials in Gaussian Unitary Ensemble random matrices. I will explain a new strategy to bound its $L^p$-norm, up to an asymptotically small error, by the operator norm of the same polynomial evaluated in free semicircular variables as long as $p=o(N^{2/3})$. As a consequence, if the coefficients are $M$-dimensional matrices with $M=exp(o(N^{2/3}))$, then the operator norm of this polynomial converges towards the one of its free counterpart. In particular this provides another proof of the Peterson-Thom conjecture thanks to the result of Ben Hayes. The approach that we take in this paper is based on an asymptotic expansion obtained in a previous paper combined with a new result of independent interest on the norm of the composition of the multiplication operator and a permutation operator acting on a tensor of $C^*$-algebras.

By the way, can I assume that the people attending the seminar will be familiar with notions of free probability?

Binary search trees (BSTs) are a fundamental data structure, optimized for data retrieval, entry, and deletion as needed for priority queue tasks. A fundamental statistic that controls each of these operations is the height of the tree with $n$ nodes, $h_n$, which returns the maximal depth of a node within the tree. Devroye established the height of a random BST, generated using a uniform permutation of length $n$, has height that limits to $c^*\approx 4.311$ when scaled by $\log n$. We are interested in studying the  heights of random block BSTs, $h_{n,m}$, which correspond to uniform permutations combined using Kronecker or wreath products of permutations of lengths $n$ and $m$, that thus arise naturally in the setting of a BST data structure implemented using parallel architecture. We show using one such product of two permutations suffices to increase the asymptotic height of a random BST, while maintaining a logarithmic scaling with respect to the length of the generated permutation. We then explore the question of how much can the height increase when repeatedly using such products. These butterfly trees correspond to block BSTs formed using uniform butterfly permutations, that include a particular 2-Sylow subgroup of the symmetric group of $N = 2^n$ objects formed by taking $n$-iterated wreath products of $S_2$. In this setting, we show the expected heights for the corresponding block BSTs are now polynomial, with a lower bound of $N^\alpha$ for $\alpha \approx 0.585$. We provide exact nonasymptotic and asymptotic distributional descriptions for the case of simple butterfly permutations, which also include connections to other well-studied permutations statistics (e.g., the longest increasing subsequence, number of cycles, left-to-right maxima), while for nonsimple butterfly permutations we provide power-law bounds on the expected heights. This project is joint with Chenyang Zhong.

Nowadays prismatic crystals are gathering an increasing interest as they unify various coefficients in $p$-adic cohomology theories. Recently, attached to any $p$-adic formal scheme $X$, Drinfeld and Bhatt-Lurie constructed certain ring stacks, including the prismatization of $X$, on which quasi-coherent complexes correspond to various crystals on the prismatic site of $X$. While such a stacky approach sheds some new light on studying prismatic crystals, little is known outside of the Hodge-Tate locus. In this talk, we will introduce our recent work on studying quasi-coherent complexes on the prismatization of $X$ via various charts.

[pre-talk at 3:00PM]

A dilation surface is roughly a surface made up of polygons in the complex plane with parallel sides glued together by complex linear maps. The action of $\mathrm{SL}(2, \mathbb R)$ on the plane induces an action of $\mathrm{SL}(2, \mathbb R)$ on the collection of all dilation surfaces, aka the moduli space, which possesses a natural manifold structure. As with translation surfaces, which can be identified with holomorphic $1$-forms on Riemann surfaces, dilation surfaces can be thought of as “twisted” holomorphic $1$-forms.

The first result that I will present, joint with Nick Salter, produces an $\mathrm{SL}(2, \mathbb R)$-invariant measure on the moduli space of dilation surfaces that is mutually absolutely continuous with respect to Lebesgue measure. The construction fundamentally uses the group cohomology of the mapping class group with coefficients in the homology of the surface. It also relies on joint work with Matt Bainbridge and Jane Wang, showing that the moduli space of dilation surfaces is a $K(\pi,1)$ where $\pi$ is the framed mapping class group. 

The second result that I will present, joint with Bainbridge and Wang, is that an open and dense set of dilation surfaces have Morse-Smale dynamics, i.e. the horizontal straight lines spiral towards a finite set of simple closed curves in their forward and backward direction. A consequence is that any fully supported $\mathrm{SL}(2, \mathbb R)$ invariant measure on the moduli space of dilation surfaces cannot be a finite measure.

In 2003, P. Biane characterized a free semi-circular system in terms of free Poincaré inequality, which is an inequality related to the non-commutative L^2-norm of free difference quotients. In this talk, we will generalize his result to $B$-valued semi-circular system using a “natural” $B$-valued free Poincaré inequality. If time permits, we will also give a counterexample to Voiculescu’s conjecture related to $B$-valued free Poincaré inequality.

This talk will focus on systems-theoretic and control theory tools that help characterize the responses of nonlinear systems to external inputs, with an emphasis on how network structure “motifs” introduce constraints on finite-time, transient behaviors.  Of interest are qualitative features that are unique to nonlinear systems, such as non-harmonic responses to periodic inputs or the invariance to input symmetries. These properties play a key role as tools for model discrimination and reverse engineering in systems biology, as well as in characterizing robustness to disturbances. Our research has been largely motivated by biological problems at all scales, from the molecular (e.g., extracellular ligands affecting signaling and gene networks), to cell populations (e.g., resistance to chemotherapy due to systemic interactions between the immune system and tumors; drug-induced mutations; sensed external molecules triggering activations of specific neurons in worms), to interactions of individuals (e.g., periodic or single-shot non-pharmaceutical “social distancing'” interventions for epidemic control). Subject to time constraints, we'll briefly discuss some of these applications.

Many data-driven problems in the modern world involve solving nonconvex optimization problems. The large-scale nature of many of these problems also necessitates the use of first-order optimization methods, i.e., methods that rely only on the gradient information, for computational purposes. But the first-order optimization methods, which include the prototypical gradient descent algorithm, face a major hurdle for nonconvex optimization: since the gradient of a function vanishes at a saddle point, first-order methods can potentially get stuck at the saddle points of the objective function. And while recent works have established that the gradient descent algorithm almost surely escapes the saddle points under some mild conditions, there remains a concern that first-order methods can spend an inordinate amount of time in the saddle neighborhoods. It is in this regard that we revisit the behavior of the gradient descent trajectories within the saddle neighborhoods and ask whether it is possible to provide necessary and sufficient conditions under which these trajectories escape the saddle neighborhoods in linear time. The ensuing analysis relies on precise approximations of the discrete gradient descent trajectories in terms of the spectrum of the Hessian at the saddle point, and it leads to a simple boundary condition that can be readily checked to ensure the gradient descent method escapes the saddle neighborhoods of a class of nonconvex functions in linear time. The boundary condition check also leads to the development of a simple variant of the vanilla gradient descent method, termed Curvature Conditioned Regularized Gradient Descent (CCRGD). The talk concludes with a convergence analysis of the CCRGD algorithm, which includes its rate of convergence to a local minimum of a class of nonconvex optimization problems.

For a smooth complex projective variety, the Lefschetz standard conjectures of Grothendieck predict the existence of algebraic self-correspondences that provide inverses to the hard Lefschetz isomorphisms. These conjectures have broad implications for Hodge theory and the theory of motives. In this talk, we describe recent progress on the Lefschetz standard conjectures for hyper-Kähler varieties of generalized Kummer deformation type. 

I will introduce a new matroid (graph) invariant: The Arboricity Polynomial.   Arboricity is a numerical invariant first introduced by Nash-Williams, Tutte and Edmonds.  It captures the minimum number of independent sets (forests) needed to decompose the ground set of a matroid (edges of a graph).    The arboricity polynomial enumerates the number of such decompositions.  We examine this counting function in terms of scheduling, Ehrhart theory, quasisymmetric functions, matroid polytopes and the permutohedral fan. 

Consider the conjugation action of \({\rm GL}_2(K)\) on the polynomial ring \(K[X_{2\times 2}]\). When \(K\) is an infinite field, the ring of invariants is a polynomial ring generated by the trace and the determinant. We describe the ring of invariants when \(K\) is a finite field, and show that it is a hypersurface.

Let $X$ be a proper geodesic Gromov hyperbolic space whose isometry group contains a uniform lattice $\Gamma$. For instance, $X$ could be a negatively curved contractible manifold or a Cayley graph of a hyperbolic group. Let $H$ be a discrete subgroup of isometries of $X$ with critical exponent (exponential growth rate) strictly less than half of the growth rate of $\Gamma$. We show that the injectivity radius of $X/H$ grows linearly along almost every geodesic in $X$ (with respect to the Patterson-Sullivan measure on the Gromov boundary of $X$). The proof will involve an elementary analysis of a novel concept called the "sublinearly horospherical limit set" of $H$ which is a generalization of the classical concept of "horospherical limit set" for Kleinian groups. This talk is based on joint work with Inhyeok Choi and Keivan Mallahi-Kerai.

Khintchine's theorem is a key result in Diophantine approximation. Given a positive non-increasing function f defined over the integers, it states that the set of real numbers that are f-approximable has zero or full Lebesgue measure depending on whether the series of terms (f(n))_n converges or diverges. I will present a recent work in collaboration with Weikun He and Han Zhang in which we extend Khintchine's theorem to any self-similar probability measure on the real line. The argument involves the quantitative equidistribution of upper triangular random walks on SL_2(R)/SL_2(Z).

In classical algebra, the Picard group of a commutative ring R is invariant under quotient by nilpotent elements. In joint work in progress with Ishan Levy and Guchuan Li, we study Picard groups of some quotient ring spectra. Under a vanishing condition, we prove that Pic(R/v^{n+1}) --> Pic(R/v^n) is injective for a ring spectrum R such that R/v is an E_1-R-algebra. This allows us to show Picard groups of quotients of Morava E-theory by a regular sequence in its π_0 are always ℤ/2. Running the profinite descent spectral sequence from there, we prove the Picard group of any K(n)-local generalized Moore spectrum of type n is finite. At height 1 and all primes p, we compute the Picard group of K(1)-local S^0/p^k when k is not too small.

The Ramsey number R(s,t) is the smallest integer n such that every red-blue coloring of the edges of the complete graph Kn contains a red clique of size s or a blue clique of size t. Ramsey numbers and their variants are some of the most famous numbers in combinatorics, yet computing even small exact values is notoriously difficult. Indeed, Erdős quipped that it would be more difficult for humans to compute the Ramsey number R(6,6) than to fend off an alien invasion. In this talk we highlight recent successes of Boolean satisfiability (SAT) solvers in Ramsey theory in both the arithmetic and graph theoretic settings.

Super-resolution algorithms to learn fine-scale features of a signal beyond the resolution at which the signal was measured. This talk will provide an overview of the mathematical theory of super-resolution, including new results by the presenter and collaborators, and show how these mathematical techniques can also be used to design quantum algorithms for problems in scientific computing. This talk is designed for a broad mathematical audience and assumes no prior knowledge in super-resolution or quantum computation.

The Banach-Tarski paradox states that a ball can be disassembled into finitely many disjoint pieces and reassembled via translations and rotations into two copies of the original ball. It turns out that this "paradoxical decomposition" is precisely characterized by the group theoretic property known as (non)-amenability. Along the way, we will encounter various equivalent definitions of amenable group (of which there are many) and some applications. This talk will be accessible to anyone who knows what a group is.

In this talk, we describe different compactifications of the moduli space of n distinct weighted labeled points in a flag of affine spaces. These spaces are constructed via generalizations of the Fulton-MacPherson compactification. For specific weight choices, we show that our moduli problem admits toric compactifications that coincide with the polypermutohedral variety of Crowly-Huh-Larson-Simpson-Wang and with the polystellahedral variety of Eur-Larson. This is joint work with J. Gonzalez-Anaya and J.L. Gonzalez.

In 2010, Dabrowski showed that a von Neumann algebra generated by self-adjoint operators is a factor when they admit a conjugate system. We extend this to the operator-valued case by defining an operator valued partial derivative and conjugate systems with respect to completely positive maps. We show that the center of the von Neumann algebra generated by B and its relative commutant is the center of B.

Given a representation $W$ of a group $G$, polarization is a technique to obtain polynomial invariants for the diagonal action of $G$ on $W^{\oplus r+1}$ from invariants of $W^{\oplus r}$. Weyl's theorem on polarization tells us when one can obtain all polynomial invariants of $W^{\oplus r+1}$ via this process. I will survey some results on polarization in the positive characteristic setting from the last three decades and explain how this can be used to obtain negative answers to some noetherian problems in infinite-dimensional/noncommutative algebra.

This work concerns the zeroth-order global minimization of continuous nonconvex functions with a unique global minimizer and possibly multiple local minimizers. We formulate a theoretical framework for inexact proximal point (IPP) methods for global optimization, establishing convergence guarantees under mild assumptions when either deterministic or stochastic estimates of proximal operators are used. The quadratic regularization in the proximal operator and the scaling effect of a positive parameter create a concentrated landscape of an associated Gibbs measure that is practically effective for sampling. The convergence of the expectation under the Gibbs measure is established, providing a theoretical foundation for evaluating proximal operators inexactly using sampling-based methods such as Monte Carlo (MC) integration. In addition, we propose a new approach based on tensor train (TT) approximation. This approach employs a randomized TT cross algorithm to efficiently construct a low-rank TT approximation of a discretized function using a small number of function evaluations, and we provide an error analysis for the TT-based estimation. We then propose two practical IPP algorithms, TT-IPP and MC-IPP. The TT-IPP algorithm leverages TT estimates of the proximal operators, while the MC-IPP algorithm employs MC integration to estimate the proximal operators. Both algorithms are designed to adaptively balance efficiency and accuracy in inexact evaluations of proximal operators. The effectiveness of the two algorithms is demonstrated through experiments on diverse benchmark functions and various applications.

Connes Rigidity Conjecture (1980) states that any ICC (infinite conjugacy class) property (T) group is W*-superrigid, meaning the group can be completely recognized from its group von Neumann algebra. The first examples of groups satisfying the conjecture, wreath-like product groups, were constructed in the work of Chifan-Ioana-Osin-Sun (2021). Building on these groups, we investigate the reconstruction of groups with infinite center from their group von Neumann algebras. We introduce the first examples of groups with infinite center whose direct product structure and ICC part are completely recognizable, and the first examples of property (T) W*-superrigid groups with infinite center.  This is based on joint work with Ionuţ Chifan and Adriana Fernández Quero, and upcoming joint work with Ionuţ Chifan, Adriana Fernández Quero and Denis Osin.

This talk will concern the three dimensional half-wave maps equation (HWM), a nonlocal geometric equation arising in the study of integrable spin systems. In high dimensions, n≥4, the equation is known to admit global solutions for suitably small initial data, however the extension of these results to three dimensions presents significant difficulties due to the loss of a key Strichartz estimate. In this talk I will introduce the half-wave maps equation and discuss a global wellposedness result for the three dimensional problem under the assumption that the initial data has angular regularity. The proof combines techniques from the study of wave maps with new microlocal arguments involving commuting vector fields and improved Strichartz estimates.

Consider a variety $X$ in the space of matrices, stable under the action of a product of general linear groups by row and column operations. How does its coordinate ring decompose as a direct sum of irreducible representations? We argue that this question is effectively studied by imposing a crystal graph structure on the standard monomials of the defining ideal of $X$ (with respect to some term order). For the standard monomials of "bicrystalline" ideals, we obtain such a crystal structure from the crystal graph on monomials introduced by Danilov–Koshevoi and van Leeuwen. This yields an explicit combinatorial rule we call "filtered RSK" for their irreducible representation multiplicities. In this talk, we will explain our rule and show that Schubert determinantal ideals (among others) are bicrystalline. Based on joint work with Abigail Price and Alexander Yong, https://arxiv.org/abs/2403.09938.

A central problem in cryptanalysis involves computing the set of solutions within a bounded region to systems of modular multivariate polynomials. Typical approaches to this problem involve identifying shift polynomials, or polynomial combinations of input polynomials, with good computational properties. In particular, we care about the size of the support of the shift polynomials, the degree of each monomial in the support, and the magnitude of coefficients. While shift polynomials for systems of a single modular univariate polynomial have been well understood since Coppersmith's original 1996 work, multivariate systems have been more difficult to analyze. Most analyses of shift polynomials only apply to specific problem instances, and it has long been a goal to find a general method for constructing shift polynomials for any system of modular multivariate polynomials. In recent work, we have made progress toward this goal by applying Groebner bases, graph optimization algorithms, and Ehrhart's theory of polytopes to this problem. This talk focuses on these mathematical aspects as they relate to our work, as well as open conjectures about the asymptotic performance of our strategies.

[pre-talk at 3:00PM]

TBA

We prove a homotopy formula which yields almost sharp estimates in all (positive-indexed) Sobolev and Hölder-Zygmund spaces for the $\overline \partial$ equation on finite type domains in $\mathbb C^2$, extending the earlier results of Fefferman-Kohn (1988), Range (1990), and Chang-Nagel-Stein (1992). The main novelty of our proof is the construction of holomorphic support functions that admit precise estimates when the parameter variable lies in a thin shell outside the domain, which generalizes Range's method.

I will review a general framework for developing numerical methods working with non-parametrically defined surfaces for various problems involving. The main idea is to formulate appropriate extensions of a given problem defined on a surface to ones in the narrow band of the surface in the embedding space. The extensions are arranged so that the solutions to the extended problems are equivalent, in a strong sense, to the surface problems that we set out to solve. Such extension approaches allow us to analyze the well-posedness of the resulting system, develop, systematically and in a unified fashion, numerical schemes for treating a wide range of problems involving differential and integral operators, and deal with similar problems in which only point clouds sampling the surfaces are given.

Nonstandard analysis involves the direction manipulation of infinite and infinitesimal quantities to circumvent many uses of epsilons and deltas in analytical arguments. These techniques provide a new perspective on analysis, and they can make rigorous many intuitively appealing arguments that are difficult to formalize with the standard approach. In this talk, we will build the logical foundation for nonstandard analysis and the ever-important transfer principle, and discuss some applications. For instance, we’ll reinterpret the df/dx notation for derivatives as a genuine quotient of infinitesimals.

Motivic degree 0 Donaldson-Thomas theory on a variety \(X\) is a point counting theory on the Hilbert scheme of points on \(X\) parametrizing zero-dimensionally supported quotient sheaves of \(\mathcal{O}_X\). On the other hand, the high rank DT theory is about the so called punctual Quot scheme parametrizing zero-dimensional quotient sheaves of the vector bundle \(\mathcal{O}_X^{\oplus r}\), while the unframed DT theory is about the stack of zero-dimensional coherent sheaves on \(X\). I will talk about some recent progresses on explicit computations of these theories for singular curves \(X\). For example, we found the exact count of \(n\times n\) matrix solutions \(AB=BA, A^2=B^3\) over a finite field (a problem corresponding to the motivic unframed DT theory for the curve \(y^2=x^3\)), and its generating function is a series appearing in the Rogers-Ramanujan identities. In other families of examples, it turns out that such computations discover new Rogers-Ramanujan type identities.

The theory of countable Borel equivalence relations analyzes the actions of countable groups on Polish spaces. The main question studied is how much information is encoded by the corresponding orbit space. The amount of encoded information reflects the extent to which the action is rigid.

In this talk we will discuss rigidity results in the theory of countable Borel equivalence relations. While the first rigidity results by Adams and Kechris use Zimmer's work, more recent results are based on newer cocycle superrigidity theorems, hinting at a deeper interplay than what we currently know. We will also discuss open questions and new directions in set theoretic rigidity.

Direct policy search has achieved great empirical success in reinforcement learning. Many recent studies have revisited its theoretical foundation for continuous control, which reveals elegant nonconvex geometry in various benchmark problems. In this talk, we introduce a new and unified Extended Convex Lifting (ECL) framework to reveal hidden convexity in classical optimal and robust control problems from a modern optimization perspective. Our ECL offers a bridge between nonconvex policy optimization and convex reformulations, enabling convex analysis for nonconvex problems. Despite non-convexity and non-smoothness, the existence of an ECL not only reveals that minimizing the original function is equivalent to a convex problem but also certifies a class of first-order non-degenerate stationary points to be globally optimal. Therefore, no spurious stationarity exists in the set of non-degenerate policies. We believe that the new ECL framework may be of independent interest for analyzing nonconvex problems beyond control. This talk is based on our recent work: arxiv.org/abs/2312.15332, and arxiv.org/abs/2406.04001.

Yang Zheng is an assistant professor in the ECE department at UC San Diego. Yang Zheng received his DPhil (Ph.D.) degree in Engineering Science from the University of Oxford in 2019. He received the B.E. and M.S. degrees from Tsinghua University in 2013 and 2015, respectively.  He was a research associate at Imperial College London and was a postdoctoral scholar in SEAS and CGBC at Harvard University. His research interests include learning, optimization, and control of network systems, and their applications to autonomous vehicles and traffic systems. Dr. Zheng received the 2019 European Ph.D. Award on Control for Complex and Heterogeneous Systems, and the 2022 Best Paper Award in the IEEE Transactions on Control of Network Systems. He was also a recipient of the National Scholarship, Outstanding Graduate at Tsinghua University, and the Clarendon Scholarship at the University of Oxford. Dr. Zheng also won an NSF CAREER Award in 2024, and the 2023 Best Graduate Teacher Award from the ECE department at UC San Diego.

TBA

Let $p$ be a prime number and $ Q_p$  the field  of $p$-adic numbers.

The representations of  a cousin of the Galois group of an algebraic closure of $ Q_p$ are related (the {\bf Langlands's bridge}) to the representations of reductive $p$-adic groups, for instance $SL_2(Q_p),  GL_n(Q_p) $.   The irreducible representations $\pi$ of reductive $p$-adic groups are  easier  to study than those of the Galois groups but they are rarely finite dimensional. Their classification is very involved but their behaviour  around the identity, that we call the ``asymptotics'' of $\pi$, are expected to be more uniform. We shall survey what is known  (joint work with Guy Henniart), and what it suggests.

In this talk I will describe some properties of Meinhardt's model of sidebranching. This is a four-species reaction-diffusion model dating from 1976 describing the interaction of four fields, the concentrations of an activator, an inhibitor, the substrate, and a marker for differentiation. The model exhibits rich dynamics that are absent from simpler RD systems. I will describe two of these: propagation failure of differentiation fronts and, in a different parameter regime, the presence of intermittent spiking. The former is traced to the presence of so-called T-points in parameter space. The latter is characterized by a Poisson probability distribution function of interspike intervals, indicating that the spiking process is memoryless. The role of a (subcritical) Turing instability in generating (unstable) spikes will be emphasized.

This is joint work with Arik Yochelis, Ben-Gurion University, Be'er Sheva, Israel.

Gaussian elimination (GE) remains one of the most used dense linear solvers. In the error analysis of GE with selected pivoting strategies on well-conditioned systems, the analysis can be reduced to studying the behavior of the growth factor, which represents the largest entry encountered at each step of the elimination process. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided very recently by Huang and Tikhomirov’s average-case analysis of GEPP, which showed GEPP growth stays at most polynomial with very high probability when using Gaussian matrices. Research on GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman, and Urschel in the last year. I am interested in studying how GEPP and GECP behave on the same linear systems, with a focus on large growth systems and orthogonal matrices. One direction will explore when GECP is less stable than GEPP, which will lead to new empirical lower bounds on how much worse GECP can behave compared to GEPP in terms of growth. Another direction will include an empirical study on a family of exponential GEPP growth matrices whose polynomial behavior in small neighborhoods limits to the initial GECP growth factor.

I will give a combination math-history-show-and-tell talk in which I explain the theory and background of subtractive synthesis, an approach to making sounds via electronics. Although not technically the first type of synthesis implemented successfully in an electronic instrument, subtractive synthesis is by far the most popular approach to the electronic creation of sounds for music. We will go over the rudiments of subtractive synthesis, briefly cover the history of the original Moog synthesizer, and along the way I will showcase these ideas using the Moog Mavis, a modern, smaller instrument from the same company based on the same principles.

In this talk, I will describe how to evaluate the virtual count of maps from a fixed-domain smooth curve to a hypersurface in a Grassmannian. We use the Quot scheme to compactify the space of maps and perform a virtual intersection-theoretic calculation. I will also discuss the conditions under which the virtual count is enumerative. This talk is based on joint work with Alina Marian.

Pipe dreams are combinatorial objects that compute Grothendieck polynomials. We introduce a new combinatorial object that naturally recasts the pipe dream formula. From this, we obtain the first direct combinatorial formula for the top degree components of Grothendieck polynomials, also known as the Castelnuovo-Mumford polynomials. We also prove the inverse fireworks case of a conjecture of Meszaros, Setiabrata, and St. Dizier on the support of Grothendieck polynomials.

The dual Cauchy identity for Schur polynomials is a fundamental result in symmetric function theory and representation theory. It states that the sum of products of two Schur polynomials indexed by conjugate partitions, in two sets of variables, equals the generating function of binary matrices. Combinatorially, this identity is realized through the dual RSK insertion, which provides a bijection between such matrices and pairs of tableaux. 

Schubert polynomials, often seen as non-symmetric generalizations of Schur polynomials, satisfy a Cauchy-type formula involving triangular binary matrices. We present an explicit insertion algorithm that establishes a bijection realizing this identity using the Pieri rule. Remarkably, our algorithm retains key features of the classical RSK and naturally involves traversals of increasing binary trees. This talk is based on ongoing joint work with Johnny Gao and Sylvester Zhang.

In the talk, we will discuss the irreducibility and the Galois group of random polynomials over the integers. After giving motivation (coming from work of Breuillard--Varju, Eberhard, Ferber--Jain--Sah--Sawhney, and others), I will present a result, conditional on the extended Riemann hypothesis, showing that the characteristic polynomial of certain random tridiagonal matrices is irreducible, with probability tending to 1 as the size of the matrices tends to infinity. 

The proof involves random walks in direct products of \({\rm SL}_2(\mathbb{F}_p)\), where we use results of Breuillard--Gamburd and Golsefidy--Srinivas. 

Joint work with Lior Bary-Soroker and Sasha Sodin.

Some Shimura varieties are moduli spaces of Abelian varieties with extra structure.

The Tate module of a universal Abelian variety is a natural source of $\ell$-adic local systems on such Shimura varieties. Remarkably, the theory allows one to build these local systems intrinsically from the Shimura variety in an essentially tautological way, and this construction can be carried out in exactly the same way for Shimura varieties whose moduli interpretation remains conjectural.

This suggests the following program: Show that these tautological local systems "look as if" they were arising from the cohomology of geometric objects. In this talk, I will describe some recent progress. It is based on joint work with Kiran Kedlaya, Christian Klevdal, and Stefan Patrikis, as well as joint work with Yifei Zhang.

[pre-talk at 3pm]

Kinetoplast DNA, often described as molecular chainmail, is found in the mitochondria of trypanosome parasites and consists of thousands of topologically interlocked circular molecules. In addition to its biological role in gene editing, it has been explored recently as a model system for materials science, due to its unique topological connectivity and its two-dimensional structure. In this talk, I will discuss some mathematical investigations that have emerged out of materials-based research of kinetoplast DNA, including the relationship between the link topology of the network and the Gaussian curvature of chainmail membranes, as well as methods to detect Borromean linking within densely linked networks.

In recent years, tensor networks have emerged as a powerful low-rank approximation framework for addressing exponentially large data science problems without requiring exponential computational resources. In this talk, we demonstrate how tensor networks, when combined with accelerations from randomized numerical linear algebra (rNLA), can enable the efficient representation and manipulation of large-scale, complex datasets originating from quantum physics, high-dimensional function approximation, and neural network compression. We will start by describing how to construct a tensor network directly from input data. Building on this foundation, we then describe a new randomized algorithm called Successive Randomized Compression (SRC) that asymptotically accelerates the tensor network analog of matrix-vector multiplication using the randomized singular value decomposition. As a demonstration, we present examples showing how tensor network based simulations of quantum dynamics in 2^100 dimensions can be performed on a personal laptop.

Beginning in the 1970s, Mazur's "Program B" kicked off efforts to classify the rational points on all modular curves $X_H$, as $H$ ranges through open subgroups of $\text{GL}_2(\hat{\mathbb Z})$. Fifty years later, it remains a very active field of research in arithmetic geometry: even as late as 2017, the determination of the rational points on a single "cursed curve" was heralded a breakthrough in the subject. In this talk, we will outline a possible approach to settle Mazur's Program B in full generality, i.e. for any number field. The inputs required are (1) a resolution to Serre's uniformity question, and (2) an algorithm to obtain rational points on any modular curve of genus at least 2. For (1), we discuss a possible approach via Borcherds products, and for (2), we discuss equationless approaches to quadratic and motivic Chabauty algorithms, following the respective recent work of Balakrishnan-Dogra-Muller-Tuitman-Vonk and Corwin.

We will give a taste of the flavors of math that constitute the study of random walks on compact groups, followed by which we will describe the author's work with Prof. Golsefidy in solving a question of Lindenstrauss and Varju. Namely, can the spectral gap of a random walk on a product of groups be related to those of the projections onto its factors.

I will talk about recent joint work with Nansen Petrosyan where we studied the behavior of $L^2$-Betti Numbers under group-theoretic Dehn filling, a quotienting process of groups motivated by 3-manifold theory. As applications, we verified the Singer Conjecture for Einstein manifolds constructed from arithmetic lattices of $SO(n, 1)$. Another application appears in my collaboration with Francesco Fournier-Facio where we constructed the first uncountable family of finitely generated torsion-free groups which are mutually non-measure equivalent.

The goal of quantized compressed sensing (QCS) is to recover structured signals from quantized measurements. The performance bounds of hamming distance minimization (HDM) were well established and known to be optimal in recovering sparse signals, but HDM is in general computationally infeasible. In this talk, we propose an efficient projected gradient descent (PGD) algorithm for QCS which generalizes normalized binary iterative hard thresholding (NBIHT) in one-bit compressed sensing for sparse vectors.  Under sub-Gaussian design, we identify the conditions under which PGD achieves essentially the same error rates as HDM, up to logarithmic factors. These conditions are easy to validate and include estimates of the separation probability, a small-ball probability and some moments. We specialize the general framework to several popular memoryless QCS models and show that PGD achieves the optimal rate O(k/m) in recovering sparse vectors, and the best-known rate O((k/m)^{1/3}) in recovering effectively sparse signals. This is joint work with Ming Yuan. An initial version is available in https://arxiv.org/abs/2407.04951. 

The work deals with the existence of solutions of an integro-differential equation in the case of the normal diffusion and the influx/efflux term proportional to the Dirac delta function in the presence of the drift term. The proof of the existence of solutions relies on a fixed point technique. We use the solvability conditions for the non-Fredholm elliptic operators in unbounded domains and discuss how the introduction of the transport term influences the regularity of the solutions.

The probability that every vertex in a random orientation of the edges of a given graph has the same in-degree and out-degree is equivalent to counting Eulerian orientations, a problem that is known to be ♯P-hard in general. This count also appears under the name residual entropy in physical applications, most famously in the study of the behaviour of ice. Using a new tail bound for the cumulant expansion series, we derive an asymptotic formula for graphs of sufficient density. The formula contains the inverse square root of the number of spanning trees, for which we do not have a heuristic explanation. We will also show a strong experimental correlation between the number of spanning trees and the number of Eulerian orientations even for graphs of bounded degree. This leads us to propose a new heuristic for the number of Eulerian orientations which performs much better than previous heuristics for graphs of chemical interest. The talk is based on two recent papers arXiv:2309.15473 and arXiv:2409.04989 joint with B.D.McKay and R.-R. Zhang.

My research is mostly about exploring how symmetries can be used to generate randomness or unveil structural insights. In this talk, I will focus on random walks on compact groups, and give you a glimpse of some of the tools that I use to study such a random process:

• Connection with expander graphs,
• Property (T),
• Growth within algebraic structures: sum-product and product results,
• Entropy and the Bourgain-Gamburd technique.

At the end, I will mention more recent results of Srinivas and mine on random walks on group extensions.

The genealogy process is typically the most time-consuming part of -- and a limiting factor in the success of -- forensic investigative genetic genealogy, which is a new approach to solving violent crimes and identifying human remains. We formulate a stochastic dynamic program that -- given the list of matches and their genetic distances to the unknown target -- chooses the best decision at each point in time: which match to investigate (i.e., find its ancestors), which ancestors of these matches to descend from (i.e., find its descendants), or whether to terminate the investigation. The objective is to maximize the probability of finding the target minus a cost on the expected size of the final family tree. We estimate the  parameters of our model using data from 17 cases (eight solved, nine unsolved) from the DNA Doe Project. We assess the Proposed Strategy using simulated versions of the 17 DNA Doe Project cases, and compare it to a Benchmark Strategy that ranks matches by their genetic distance to the target and only descends from known common ancestors between a pair of matches. The Proposed Strategy solves cases 25-fold faster than the Benchmark Strategy, and does so by aggressively descending from a set of potential most recent common ancestors between the target and a match even when this set has a low probability of containing the correct most recent common ancestor.

This lecture is jointly sponsored by the UCSD Rady School of Management and the UCSD Mathematics Department.

The MPR2 conference room is just off Ridge Walk. It is on the same level as Ridge Walk. You will see the glass-walled MPR2 conference room on your left as you come into the Rady School area. 

FREE REGISTRATION REQUIRED: https://forms.gle/jv8nVFajV9mZ6U3v6 

The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem. The results are relevant for the construction of curved folding devices.

Robust statistics answers the question of how to build statistical estimators that behave well even when a small fraction of the input data is badly corrupted. While the information-theoretic underpinnings have been understood for decades, until recently all reasonably accurate estimators in high dimensions were computationally intractable. Recently however, a new class of algorithms has arisen that overcome these difficulties providing efficient and nearly-optimal estimates. Furthermore, many of these techniques can be adapted to cover the case where the majority of the data has been corrupted. These algorithms have surprising applications to clustering problems even in the case where there are no errors.

In this talk, we will present a new software application for publishing interactive math content online. It works like Overleaf, where you type text, LaTeX, and more in your browser, but instead of a PDF it produces a live, interactive website.  This app has now been tested in several math courses at UCSD, and we hope it can support your teaching as well!

The study of enumerative invariants dates back at least as far as Euclid’s work circa 300 BC, who observed that through two distinct points in the plane there is a unique line. In 1849, Cayley-Salmon found that there are 27 lines on a nonsingular cubic surface. In 1879, Schubert found that there are 2875 lines on a generic non-singular quintic threefold; Katz correctly counted 609250 conics in a generic nonsingular quintic threefold in 1986. In 1991, physicists Candelas-de la Ossa-Green-Parkes gave a generating function for genus 0 Gromov-Witten invariants of a generic non-singular quintic threefold by studying the mirror space. This observation represented a change in our approach to enumerative problems by counting rational degree d curves inside of the quintic threefold “all at once;” other landmark achievements in modern enumerative geometry include Kontsevich-Manin’s recursive formula for the number of rational plane curves. From the perspective of homological mirror symmetry, enumerative invariants come from the Hochschild cohomology of the Fukaya category. I’m interested in a different question, which asks what enumerative data can be gleaned from the bounded derived category of coherent sheaves. I’ll share results on giving presentations of derived categories, and if time allows, will describe Kalashnikov’s method to recover Givental’s small J-function and the genus 0 Gromov-Witten potential for CP^1 by viewing it as a toric quiver variety associated to the Kronecker quiver; i.e., from a presentation of the bounded derived category of coherent sheaves.  

Support varieties for Hopf algebras (and more general tensor categories) give a way of associating geometry to finite-dimensional modules. The support variety of a module is empty if and only if the module is projective. We give a method for extending a support variety theory from the finite-dimensional modules to the infinite-dimensional ones, and give conditions under which the theory still detects projectivity. This talk will include joint work with Nakano—Yakimov and with Cai.

A countable discrete group is called Choquet-Deny if for every non-degenerate probability measure on the group, the corresponding space of bounded harmonic functions is trivial. Recently a complete characterization of Choquet-Deny groups was obtained by Frisch, Hartman, Tamuz and Ferdowsi. In this talk, we will look at the extension of the Choquet-Deny property to the framework of discrete measured groupoids. Our main result gives a complete characterisation of this property in terms of the associated measured equivalence relation and the isotropy groups of the groupoid. This talk is based on a joint work with Tey Berendschot, Milan Donvil, Mario Klisse and Se-Jin Kim.

Nonlinearly Partitioned Runge--Kutta (NPRK) methods are a newly proposed class of time integration schemes which target differential equations in which different scales, stiffnesses or physics are coupled in a nonlinear way. In this talk, I will provide a broad overview of this new class of methods. First, I will motivate these methods as a nonlinear generalization of classical Runge--Kutta (RK) and Additive Runge--Kutta (ARK) methods. Subsequently, I will discuss order conditions for NPRK methods; we obtain the complete order conditions using an edge-colored rooted tree framework. Interestingly, NPRK methods have nonlinear order conditions which have no classical additive counterpart. We will show how these nonlinear order conditions can be used to obtain embedded estimates of state-dependent nonlinear coupling strength and present a numerical example to demonstrate these embedded estimates. I will then discuss how these methods yield efficient semi-implicit time integration of numerical partial differential equations; numerical examples from radiation hydrodynamics will be presented. Finally, I will discuss our recent work on multirate NPRK methods, which target problems with nonlinearly coupled processes occurring on different timescales. We will discuss properties of these multirate methods such as timescale coupling, stability and efficiency, and conclude with several numerical examples, such as a fast-reaction viscous Burgers’ equation and the thermal radiation diffusion equations.

The Riemann Mapping Theorem is a fundamental result in classical complex analysis in one variable: If $\Omega\subset \mathbb C$ is a simply connected domain, $\Omega\neq \mathbb C$, then there is a biholomorphic map $F\colon \Omega\to\mathbb D:=\{|z|<1\}$. One of the first things we teach students in several complex variables is that the analogous fails miserably for domains in $\mathbb C^n$ for $n\geq 2$, as was already discovered by Poincaré; There is no biholomorphic map from the bidisk $\mathbb D^2:=\{(z_1,z_2)\colon |z_1|<1, |z_2|<1\}$ to the unit ball $\mathbb B^2=\{|z_1|^2+|z_2|^2<1\}$. There are clearly no topological obstructions to the existence, which is essentially the only obstruction to a Riemann map in one variable (but what about $\Omega\neq \mathbb C$?). As a first reaction, one might then give up and exclaim "if this example doesn't work, there is no hope for a reasonable Riemann Mapping Theorem in higher dimensions". Well, I intend to convince the audience that one would be wrong, and one would then miss an extremely rich theory that blends real and complex geometry, partial differential equations, and, of course, real and complex analysis.

The continuous random energy model (CREM) is a Gaussian process indexed by a binary tree of depth T, introduced by Derrida and Spohn (1988) and Bovier and Kurkova (2004) as a toy model of a spin glass. In this talk, I will present recent results on hardness thresholds for algorithms that search for low-energy states. I will first discuss the existence of an algorithmic hardness threshold x_*: finding a state of energy lower than -x T is possible in polynomial time if x < x_*, and takes exponential time if x > x_*, with high probability. I will then focus on the transition from polynomial to exponential complexity near the algorithmic hardness threshold, by studying the performance of a certain beam-search algorithm of beam width N depending on T — we believe this algorithm to be natural and asymptotically optimal. The algorithm turns out to be essentially equivalent to the time-inhomogeneous version of the so-called N-particle branching Brownian motion (N-BBM), which has seen a lot of interest in the last two decades. Studying the performance of the algorithm then amounts to investigating the maximal displacement at time T of the time-inhomogeneous N-BBM. In doing so, we are able to quantify precisely the nature of the transition from polynomial to exponential complexity, proving that the transition happens when the log-complexity is of the order of the cube root of T. This result appears to be the first of its kind and we believe this phenomenon to be universal in a certain sense.

We show that the Bergman metric on ball quotients $\mathbb{B}^2/\Gamma$ is Kähler-Einstein if and only if $\Gamma$ is trivial, leading to a characterization of the unit ball among certain two-dimensional Stein spaces, confirming a version of Cheng’s conjecture. We also relate the boundary type of two-dimensional Stein spaces to the local algebraic degree of their Bergman kernel, characterizing ball quotients via the local rationality of the Bergman kernel. Finally, we derive the rotational symmetry properties of certain domains in $\mathbb{C}^n$ from the orthogonality of holomorphic monomials in their Bergman spaces.

Let  M  be a smooth real codimension two compact submanifold in a Stein manifold. We will prove the following theorem: Suppose that  M  has two elliptic complex tangents and that CR points are non-minimal. Assume further that  M  is contained in a bounded strongly pseudoconvex domain. Then  M  bounds a unique smoothly up to  M  Levi-flat hypersurface  $\widehat{M}$  that is foliated by Stein hyper-surfaces diffeomorphic to the ball. Moreover,  $\widehat{M}$  is the hull of holomorphy of M . This subject has a long history of investigation dating back to E. Bishop and Harvey-Lawson. I will discuss both the historical context and the techniques used in the proof of the aforementioned theorem.

The regulation and maintenance of a tissue’s shape and structure is a major outstanding question in developmental biology and plant biology. In this talk, through iterations between experiments and multiscale model simulations that include a mechanistic description of interkinetic nuclear migration, we will show that the local curvature, height, and nuclear positioning of cells in the Drosophila wing imaginal disc are defined by the concurrent patterning of actomyosin contractility, cell-ECM adhesion, ECM stiffness, and interfacial membrane tension. The biologically calibrated model describing both tissue growth and morphogenesis incorporates the spatial patterning of fundamental subcellular properties. Additionally, the model implements for the first time the dynamics of interkinetic nuclear migration within the simulated pseudostratified epithelium. This includes the basal to apical motion of the nucleus, mitotic rounding, and cell division dynamics. Key characteristics of global tissue architecture, such as the local curvature of the basal wing disc epithelium, cell height, and nuclear positioning, serve as metrics for model calibration. The experiments have shown how these physical features are jointly regulated through spatiotemporal dynamics in the localization of pMyoII, β-Integrin, and ECM stiffness. As the disc grows, there are progressive changes in the patterning of key subcellular features such as actomyosin contractility. The predictions made by the model simulations agree with the observed changes in contractility and cell-ECM adhesion during wing disc morphogenesis. Multiscale modeling approach combined with experiments was also applied to studying stem cell maintenance in multilayered shoot apical meristems (SAMs) of plants which requires strict regulation of cell growth and division. In this talk, the combined approach will be demonstrated through testing three hypothesized mechanisms for the regulation of cell division plane orientation and the direction of anisotropic cell expansion in the corpus.

A fundamental task in linear algebra is that of trace estimation: Suppose we have a PSD matrix A that can be accessed only by matrix-vector products. Then, with as few matrix-vector products as possible, estimate the trace of A to relative error with high probability. This is an essential subroutine in all sorts of applications, for instance in efficiently estimating the log-determinant of a matrix.

In the first part of the talk, I'll rigorously introduce this problem, the prior state-of-the-art algorithm (the Girard-Hutchinson Estimator), and our improvement upon it (the Hutch++ Estimator), which we show to have asymptotically optimal matrix-vector complexity. In the second part of the talk, I'll introduce a Kronecker-structured variant of this problem with applications for tensorized data, alongside the only known algorithm that solves this problem.

However, we'll see that this algorithm converges very slowly. We will show this is a result of this Kronecker-structured computational model, which elicits strange computational properties. We will see that good design decisions in the non-Kronecker case can cause catastrophic failure in the Kronecker case, that using complex random variables leads to exponential speedups over reals, and that subgaussianity does not suffice to understand the performance of randomized algorithms here.

Joint work with Haim Avron, David Woodruff, and William Swartworth.

I will discuss some recent results about relative entropies in QFT, with particular emphasis on the singular limits of such entropies.

We give an explicit geometric computation of the quantum K rings of symplectic Grassmannians of lines, which are deformations of their Grothendieck rings of vector bundles and refinements of their quantum cohomology rings. We prove that their Schubert structure constants have signs that alternate with codimension (just like in the Grothendieck ring) and vanish for degrees at least 3. We also give closed formulas that characterize the multiplicative structure of these rings. This is based on joint work with V. Benedetti and N. Perrin.

We survey some recent observations and ongoing work motivated by a hope to better understand p-adic K-theory. More specifically, we discuss arithmetic problems—and potential approaches—related to syntomic cohomology in positive and mixed characteristics. At the level of the structure sheaf, syntomic cohomology is an 'intelligent version' of p-adic étale Tate twists at the characteristic and (among other things) provides a motivic filtration on p-adic étale K-theory via the theory of trace invariants.

Let R be a polynomial ring with m x n many indeterminate over the complex numbers. We can think of the indeterminates as a matrix X of size m x n.  

Consider the group G = Gl(m) x Gl(n). Then G acts on R via the group action (A,B)X =AXB^{-1}. In 1980, DeConcini, Eisenbud, and Procesi introduced the ideals that are invariant under this group action.

In the same paper, they described various properties of those ideals, e.g., associated primes, primary decomposition, and integral closures.  In recent work with Sudipta Das, Tài Huy Hà, and Jonathan Montaño, we described their rational powers and proved that they satisfy the binomial summation formula. In an ongoing work, Alexandra Seceleanu and I are formulating symbolic properties of these ideals. In this talk, I will describe these ideals and the properties we are interested in. I will also showcase some results from my collaborations.

In his seminal paper, Ozawa demonstrated the solidity property for ${\rm II}_1$ factor arising from Biexact groups. In this talk, I will discuss a relative version of the solidity property for biexact groups in the setting of measure equivalence and its applications to measure equivalence rigidity. This is a joint work with Daniel Drimbe.

We introduce BiLO (Bilevel Local Operator Learning), a novel neural network-based approach for solving inverse problems in partial differential equations (PDEs). BiLO formulates the PDE inverse problem as a bilevel optimization problem: at the upper level, we optimize PDE parameters by minimizing data loss, while at the lower level, we train a neural network to locally approximate the PDE solution operator near given PDE parameters. This localized approximation enables accurate descent direction estimation for the upper-level optimization. We apply gradient descent simultaneously on both the upper and lower level optimization problems, leading to an effective and fast algorithm. Additionally, BiLO can infer unknown functions within PDEs by introducing an auxiliary variable. Extensive experiments across various PDE systems demonstrate that BiLO enforces strong PDE constraints, is robust to sparse and noisy data, and eliminates the need for manually balancing residual and data loss, a common challenge in soft PDE constraints. We also discuss how to apply the BILO for uncertainty quantification in a Bayesian framework.

In 1995, Kuperberg introduced a remarkable collection of trivalent web bases which encode tensor invariants of $U_q(\mathfrak{sl}_3)$. Extending these bases to general $\mathfrak{sl}_r$ has been an open problem ever since. We present a solution to the $r=4$ case by introducing hourglass plabic graphs - a new generalization of Postnikov's plabic graphs. Joint work with Christian Gaetz, Oliver Pechenik, Jessica Striker and Joshua Swanson.

For q a prime power, let F_q be the finite field of order q. There are a finite number of isomorphism classes of (smooth, projective, geometrically irreducible) curves of genus g over F_q. Can one give a closed form expression for this number? We discuss how to correctly interpret this question; how to generalize it by also counting marked points; what is known for small g; and what information can be gained by making complete tables of curves of a given genus.

Solving linear systems is a crucial subroutine and challenge in the large-scale data setting. In this presentation, we introduce an iterative method for approximating the solution of large-scale multi-linear systems, represented in the form A*X=B under the tensor t-product. Unlike previously proposed randomized iterative strategies, such as the tensor randomized Kaczmarz method (row slice sketching) or the tensor Gauss-Seidel method (column slice sketching), which are natural extensions of their matrix counterparts, our approach delves into a distinct scenario utilizing frontal slice sketching. In particular, we explore a context where frontal slices, such as video frames, arrive sequentially over time, and access to only one frontal slice at any given moment is available. This talk will present our novel approach, shedding light on its applicability and potential benefits in approximating solutions to large-scale multi-linear systems.
 

A celebrated theorem of Eskin–Mozes–Shah gives an asymptotic counting formula for the number of integral (n x n)-matrices with a prescribed irreducible (over the integers/rationals) integral characteristic polynomial. We obtain a power saving error term for the counting problem for (3 x 3)-matrices. We do this by using the connection to homogeneous dynamics and proving effective equidistribution of translates of tori in SL_3(R)/SL_3(Z). A key tool is that the limiting Lie algebra corresponding to the translates of tori is a certain nilpotent Lie algebra. This allows us to use the recent breakthrough work of Lindenstrauss–Mohammadi–Wang–Yang on effective versions of Shah's/Ratner's theorems. We actually study the phenomenon more generally for any semisimple Lie group which we may discuss if time permits.

We consider scalar field theories on the line with Ginzburg-Landau  (double-well) self-interaction potentials. Prime examples include the  $\phi^4$ model and the sine-Gordon model. These models feature simple  examples of topological solitons called kinks. The study of their = asymptotic stability leads to a rich class of problems owing to the  combination of weak dispersion in one space dimension, low power  nonlinearities, and intriguing spectral features of the linearized  operators such as threshold resonances or internal modes.

We present a perturbative proof of the full asymptotic stability of the sine-Gordon kink outside symmetry under small perturbations in weighted Sobolev norms. The strategy of our proof combines a space-time resonances approach based on the distorted Fourier transform to capture modified scattering effects with modulation techniques to take into account the invariance under Lorentz transformations and under spatial translations. A major difficulty is the slow local decay of the radiation term caused by the threshold resonances of the non-selfadjoint linearized matrix operator around the modulated kink. Our analysis hinges on two remarkable null structures that we uncover in the quadratic nonlinearities of the evolution equation for the radiation term as well as of the modulation equations.

The entire framework of our proof, including the systematic development of the distorted Fourier theory, is general and not specific to the sine-Gordon model. We conclude with a discussion of potential applications in the generic setting (no threshold resonances) and with a discussion of the outstanding challenges posed by internal modes such as in the well-known $\phi^4$ model.

This is joint work with Gong Chen (GeorgiaTech).

In this talk, I will present two recent pieces of research that are connected by the common theme of multiscale models for motile microorganisms. 

In the first part, I will discuss the orientational dynamics of microscopic organisms, such as bacteria, swimming in biofluids with properties that differ from those of isotropic Newtonian fluids, instead exhibiting characteristics of liquid crystals. These environments have a preferred direction, which forces the swimmers to align with it. However, certain types of bacteria can overcome this external torque and swim across the preferred direction. I will present a nonlinear PDE system that couples liquid crystal hydrodynamics with a model of a prototypical microswimmer. This model identifies the conditions for non-trivial reorientation dynamics and allows for deriving the homogenized limit, effectively describing the dynamics of the microswimmer colony. This is the joint work with I. Aronson (PSU), L. Berlyand (PSU), H. Chi (PSU), A. Yip (Purdue U.), and L. Zhang (SJTU). 

In the second part of the talk, I will focus on a computational model that describes how motile cancer cells interact with the extracellular matrix (ECM) during the initial invasion phase, including ECM degradation and mechanical remodeling. The model highlights the role of elastic interactions in the dynamics of cell clusters, including their shapes, sizes, and orientations. These results are joint work with O. Kim (Virginia Tech), Y. Klymenko (Indiana U.), M. Alber (UCR), and I. Aranson (PSU).

If G is a finite group then the collection of all finite dimensional complex representations of G carries two important operations: direct sum and tensor product. A tensor category is an abstraction of this situation. Finding new examples of tensor categories is a very difficult problem. In recent work with Harman, we gave a general construction of tensor categories based on oligomorphic groups, a class of infinite permutation groups best known in model theory. I will give an overview of our work.

In machine learning, classification is often approached as a function approximation problem.  In this talk, we propose a  active learning framework inspired by signal separation and super-resolution theory.  Our approach enables efficient identification of class supports, even in the presence of overlapping distributions. This allows efficient clustering and label propagation from very few labeled points.

The Chern filtration is a natural filtration that can be defined on the cohomology of moduli spaces of sheaves. Its definition was originally made for the moduli of Higgs bundles, motivated by a comparison with the perverse and weight filtrations, but it also makes sense for the very classical moduli spaces of bundles on curves. A vanishing result conjectured by Newstead and proved by Earl-Kirwan in the 90s is secretly a statement about the Chern filtration. I will explain a new approach to this vanishing which is based on parabolic bundles: it turns out that enriching the problem with a parabolic structure gives access to powerful tools, such as wall-crossing, Hecke transforms and Weyl symmetry — together, these give a new proof of the Newstead-Jefrey-Kirwan vanishing and a related "d independence" statement. Part of the talk is based on work with W. Lim and W. Pi.
 

It is well-known that if a tensor category has an abelian algebra object A, one obtains a new category, essentially by tensoring over A. An important class of such algebra objects come from conformal inclusions for loop groups. While these algebra objects have been known for a long time, an explicit description of the corresponding categories was only recently found.

They are somewhat surprisingly closely related to representation categories of the isomeric quantum Lie super algebras. This talk is based on joint work with Edie-Michell and a paper by Edie-Michell and Snyder.

Recently, members of our group proved impressive results on the reduced C*-algebras of free groups—and, more generally, hyperbolic groups. Following the same general strategy, but using quite different methods, I obtain analogous results for higher-rank lattices (e.g., cocompact discrete subgroups of SL3(ℝ)). In the talk I’ll survey the structural properties of interest and outline the main ideas of the proofs.

Recently, a class of algorithms combining classical fixed point iterations with repeated random sparsification of approximate solution vectors has been successfully applied to eigenproblems with matrices as large as $10^{108} \times 10^{108}$. So far, a complete mathematical explanation for this success has proven elusive. The family of methods has not yet been extended to the important case of linear system solves. In this work we propose a new scheme based on repeated random sparsification that is capable of solving sparse linear systems in arbitrarily high dimensions. We provide a complete mathematical analysis of this new algorithm. Our analysis establishes a faster-than-Monte Carlo convergence rate and justifies use of the scheme even when the solution vector itself is too large to store.

AI is taking off and we could say we are living in “the AI Era”.  Progress in AI today is based on mathematics and statistics under the covers of machine learning models.  This talk will explain recent work on AI4Crypto, where we train AI models to attack Post Quantum Cryptography (PQC) schemes based on lattices. I will use this work as a case study in training ML models to solve hard math problems in practice.  Our AI4Crypto project has developed AI models capable of recovering secrets in post-quantum cryptosystems (PQC).  The standardized PQC systems were designed to be secure against a quantum computer, but are not necessarily safe against advanced AI!  

Understanding the concrete security of these standardized PQC schemes is important for the future of e-commerce and internet security.  So instead of saying that we are living in a “Post-Quantum” era, we should say that we are living in a “Post-AI” era!

The Lévy-Khintchine theorem is a classical result in Diophantine approximation that describes the growth rate of denominators of convergents in the continued fraction expansion of a typical real number. We make this theorem effective by establishing a quantitative rate of convergence. More recently, Cheung and Chevallier (Annales scientifiques de l'ENS, 2024) established a higher-dimensional analogue of the Lévy-Khintchine theorem in the setting of simultaneous Diophantine approximation, providing a limiting distribution for the denominators of best approximations. We also make their result effective by proving a convergence rate, and in addition, we establish a central limit theorem in this context. Our approach is entirely different and relies on techniques from homogeneous dynamics.

Starting  with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$ (for example, absorbing Brownian motion in a snowflake domain), one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of  reflected Dirichlet spaces. The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form. In this talk, I will address the following two fundamental questions:

1) How to characterize the boundary trace Dirichlet space in a concrete way?

2) How does the boundary trace process behave? 

Based on a joint work with Shiping Cao.

The study of singularities of minimal submanifolds has a long history, with isolated singularities being the best understood case. The next simplest case is that of minimal submanifolds with families with singularities locally modeled on the product of an isolated conical singularity and a Euclidean space — such submanifolds are said to have cylindrical tangent cones at these singularities. Despite work in many contexts on minimal submanifolds with such singularities, the only known explicit examples at present are global products or involve extra structure (e.g. Kahler subvarieties). In this talk, I will describe a method for constructing infinite-dimensional families of non-product minimal submanifolds in arbitrary codimension whose singular set is itself an analytic submanifold. The construction uses techniques from the analysis of singular elliptic operators and Nash-Moser theory. This talk is based on joint work with Rafe Mazzeo.

The Heilbronn triangle problem is a classical problem in discrete geometry with several old and new close connections to various topics in extremal and additive combinatorics, graph theory, incidence geometry, harmonic analysis, and number theory. In this talk, we will survey a few of these stories, and discuss some recent developments. Based on joint works with Alex Cohen and Dmitrii Zakharov. 

Deep neural networks have been widely adopted for climate prediction tasks and have achieved high prediction accuracy across many problems. However, their decision-making processes remain opaque, and the complexity of these models poses significant challenges for interpretation. A recent theoretical breakthrough, "Recursive Feature Machine" (RFM), provides an alternative methodology for climate prediction that is interpretable and data efficient. Applying RFM to El Niño–Southern Oscillation (ENSO) prediction yields promising interpretability results and offers insights into the most influential geographical features that the model learns from training data. The method is clean, easy to implement, and can be generalized to a broad range of scientific fields.

How in the world did they get those crazy pictures of electron orbitals? Those chemists had to have talked to somebody about it! It turns out they talked to math people (probably physicists, but physicists talk to math people, and so on). These orbitals can actually be derived in not-too-bad a way using representation theory. We'll go over what electron orbitals are, how they show up in the periodic table, how representation theory gets involved, and how to derive the electron orbitals ourselves. We will even find orbitals that are bigger than the highest electron on Oganesson! We'll hopefully also understand what physicists and engineers mean when they say they have a "tensor." I've also been studying the periodic table using mnemonic devices lately, so you'll be sure to hear about that.

Elliptic surfaces are complex surfaces with two discrete invariants, $g$ and $d>0$. We will discuss the moduli and Hodge theory of these surfaces for small values of $(g,d)$. The case $(g,d)=(1,1)$ is particularly interesting, in view of a new conjectural Fourier-Mukai type correspondence. It also provides a test case of the Hodge Conjecture in dimension 4.

The symmetric group $\mathfrak{S}_n$ acts naturally on the polynomial ring of rank $n$ by variable permutation. The classical coinvariant ring $R_n$ is the quotient of this action by the ideal generated by invariant polynomials with vanishing constant term. The ring $R_n$ has deep ties to the combinatorics of permutations and the geometry of the flag variety. The superspace coinvariant ring $SR_n$ is obtained by an analogous construction where one considers the action of $\mathfrak{S}_n$ on the algebra $\Omega_n$ of polynomial-valued differential forms on $n$-space. We describe the Macaulay-inverse system associated to $SR_n$, give a formula for its bigraded Hilbert series, and give an explicit basis of $SR_n$. The basis of $SR_n$ will be derived using Solomon-Terao algebras associated to free hyperplane arrangements. Joint with Robert Angarone, Patty Commins, Trevor Karn, Satoshi Murai, and Andy Wilson.

The Hilbert transform is a cornerstone of  the classical analysis. A key approach to establishing its Lp-boundedness is through Cotlar's identity, a powerful equation that not only yields optimal constants for the Lp bounds of the Hilbert transform but also generalizes to broader settings where the notion of "analytic functions" is meaningful. In this talk, I will revisit Cotlar’s identity and explore how modified versions  extend to branches of free groups and hyperbolic groups.

Given $r$-uniform hypergraphs $G$ and $F$ and an integer $n$, let $f_{F,G}(n)$ be the maximum $m$ such that every $n$-vertex $G$-free $r$-graph has an $F$-free induced subgraph on $m$ vertices. We show that $f_{F,G}(n)$ is polynomial in $n$ when $G$ is a subgraph of an iterated blowup of $F$. As a partial converse, we show that if $G$ is not a subgraph of an $F$-iterated blowup and is $2$-tightly connected, then $f_{F,G}(n)$ is at most polylogarithmic in $n$. Our bounds generalize previous results of Dudek and Mubayi for the case when $F$ and $G$ are complete. Joint work with Xiaoyu He.

Given $r$-uniform hypergraphs $G$ and $F$ and an integer $n$, let $f_{F,G}(n)$ be the maximum $m$ such that every $n$-vertex $G$-free $r$-graph has an $F$-free induced subgraph on $m$ vertices. We show that $f_{F,G}(n)$ is polynomial in $n$ when $G$ is a subgraph of an iterated blowup of $F$. As a partial converse, we show that if $G$ is not a subgraph of an $F$-iterated blowup and is $2$-tightly connected, then $f_{F,G}(n)$ is at most polylogarithmic in $n$. Our bounds generalize previous results of Dudek and Mubayi for the case when $F$ and $G$ are complete. Joint work with Xiaoyu He.

The explosion of real-time data in the physical world requires new generations of tools to model complex dynamical systems. Deep learning, the foundation of modern AI, offers highly scalable models for spatiotemporal data. On the other hand, deep learning is opaque and complex. Dynamical system theory plays a key role in describing the emerging behavior of deep neural networks. It provides new paths towards understanding the hidden structures in these complex systems. In this talk, I will give an overview of our research to explore the interplay between the two. I will showcase the applications of these approaches to different science and engineering tasks. 

Please register at https://forms.gle/yDcUa9LAmpY1F2178.

Please register here:

https://forms.gle/yDcUa9LAmpY1F2178.

The moduli space of genus g Riemann surfaces equipped with the Teichmuller metric exhibits rich geometric, analytic, and dynamical properties. A major challenge is to understand the totally geodesic submanifolds -- these share many properties with the moduli space itself. For many decades, research focused on the one (complex) dimensional case, i.e. the fascinating Teichmuller cuves. The discovery of interesting higher-dimensional examples in recent years has led to new questions. In this talk, I will discuss joint work with Benirschke and Rached in which we study the boundary of a totally geodesic subvariety in the Deligne-Mumford compactification, showing that the boundary is itself totally geodesic.

Consider the random bipartite Erdos-Renyi graph $G(n, m, p)$, where each edge with one vertex in $V_{1}=[n]$ and the other vertex in $V_{2}=[m]$ is connected with probability $p$ with $n \geq m$. For the centered and normalized adjacency matrix $H$, it is well known that the empirical spectral measure will converge to the Marchenko-Pastur (MP) distribution. However, this does not necessarily imply that the largest (resp. smallest) singular values will converge to the right (resp. left) edge when $p = o(1)$, due to the sparsity assumption. In Dumitriu and Zhu 2024, it was proved that almost surely there are no outliers outside the compact support of the MP law when $np = \omega(\log(n))$. In this paper, we consider the critical sparsity regime with $np =O(\log(n))$, where we denote $p = b\log(n)/\sqrt{mn}$, $\gamma = n/m$ for some positive constants $b$ and $\gamma$. For the first time in the literature, we quantitatively characterize the emergence of outlier singular values. When $b > b_{\star}$, there is no outlier outside the bulk; when $b^{\star}< b < b_{\star}$, outlier singular values only appear outside the right edge of the MP law; when $b < b^{\star}$, outliers appear on both sides. Meanwhile, the locations of those outliers are precisely characterized by some function depending on the largest and smallest degrees of the sampled random graph. The thresholds $b^{\star}$ and $b_{\star}$ purely depend on $\gamma$. Our results can be extended to sparse random rectangular matrices with bounded entries.

Stationary varifolds generalize minimal surfaces and can exhibit singularities. The most general regularity theorem in this context is the celebrated Allard's Regularity Theorem, which asserts that the set of singular points has empty interior. However, it is believed that the set of singular points should have codimension (at least) one. Despite more than 50 years having passed since Allard's breakthrough, stronger results have remained elusive. In this talk, after a brief discussion about the regularity theory for stationary varifolds, I will discuss the principle of unique continuation and the topic of rectifiability, both of which are linked to understanding the structure of singularities. This discussion is based on joint works with Stefano Decio, Camillo De Lellis, and Federico Franceschini.

The concept of ultraproducts in the context of tracial von Neumann algebras was effectively introduced by Wright in 1954. Since then, it has been used as a central technique in several important works on the classification and structure theory of von Neumann algebras, including works of McDuff and Connes. Developments beginning in the 2010s also connected the concept to ultraproducts in model theory. In this talk, I will be presenting a general overview of the technique and relevant results, both from a von Neumann algebra and from a continuous model theory perspective. I will also present several of my works, with various collaborators, that apply the technique and related techniques in C*-algebras and group theory.

The properties of a commutative scheme are strongly reflected in its category of quasi-coherent sheaves.  One approach to noncommutative geometry is to consider arbitrary categories with similar properties (e.g. Grothendieck categories) as geometric objects in their own right.  We discuss how one might to define an analog of closed subscheme in this context and give lots of examples of how the definition behaves in both reasonable and non-intuitive ways.

We propose a novel framework for solving nonlinear PDEs using sparse radial basis function (RBF) networks. Sparsity-promoting regularization is employed to prevent over-parameterization and reduce redundant features. This work is motivated by longstanding challenges in traditional RBF collocation methods, along with the limitations of physics-informed neural networks (PINNs) and Gaussian process (GP) approaches, aiming to blend their respective strengths in a unified framework. The theoretical foundation of our approach lies in the function space of Reproducing Kernel Banach Spaces (RKBS) induced by one-hidden-layer neural networks of possibly infinite width. We prove a representer theorem showing that the solution to the sparse optimization problem in the RKBS admits a finite solution and establishes error bounds that offer a foundation for generalizing classical numerical analysis. The algorithmic framework is based on a three-phase algorithm to maintain computational efficiency through adaptive feature selection, second-order optimization, and pruning of inactive neurons. Numerical experiments demonstrate the effectiveness of our method and highlight cases where it offers notable advantages over GP approaches. This work opens new directions for adaptive PDE solvers grounded in rigorous analysis with efficient, learning-inspired implementation.

I will discuss the role of regularity properties in the structure and classification of C*-algebras, singling out the property of strict comparison of positive elements by traces. A well-understood source of strict comparison is through tensorial absorption of the Jiang-Su C*-algebra. This property is, however, absent from naturally occurring examples such as the reduced group C*-algebras of free groups. Thus, for some time this notion was hindered by a lack of concrete examples (particularly non-nuclear ones). This situation changed after the recent breakthrough work of Amrutam, Gao, Kunnawalkam Elayavalli, and Patchel. This work exploited the connection between Voiculescu's free independence and strict comparison, encapsulated in the concept of "selfless C*-algebra", to confirm that large classes of reduced group C*-algebras indeed obey strict comparison. I will discuss ongoing joint work with Hayes and Kunnawalkam Elayavalli, where we continue the program of verifying selflessness, and thus strict comparison, for new classes of C*-algebras, this time those arising as reduced free products.

We study the algorithmic problem of finding a large independent set in an Erdős–Rényi random graph $G(n,p)$. For constant $p$ and $b=1/(1-p)$, the largest independent set has size $2\log_b n$, while a simple greedy algorithm -- where vertices are revealed sequentially and the decision at any step depends only on previously seen vertices -- finds an independent set of size $\log_b n$. In his seminal 1976 paper, Karp challenged to either improve this guarantee or establish its hardness. Decades later, this problem remains one of the most prominent algorithmic problems in the theory of random graphs.

In this talk we provide some evidence for the algorithmic hardness of Karp's problem. More concretely, we establish that a broad class of online algorithms, which we shall define, fails to find an independent set of size $(1+\epsilon)\log_b n$ for any constant $\epsilon>0$, with high probability. This class includes Karp’s algorithm as a special case, and extends it by allowing the algorithm to also query additional `exceptional' edges not yet `seen' by the algorithm. For constant~$p$ we also prove that our result is asymptotically tight with respect to the number of exceptional edges queried, by  designing an online algorithm which beats the half-optimality threshold when the number of exceptional edges is slightly larger than our bound.

Our proof relies on a refined analysis of the geometric structure of tuples of large independent sets, establishing a variant of the Overlap Gap Property (OGP) commonly used as a barrier for classes of algorithms. While OGP has predominantly served as a barrier to stable algorithms, online algorithms are not stable, i.e., our application of OGP-based techniques to the online setting is novel.

Based on joint work with D. Gamarnik and E. Kızıldağ; see arXiv:2504.11450.

Evolutionary dynamics permeates life and life-like systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in life sciences. In this talk I will discuss how spatial interactions may change the laws of evolution, giving rise to a system of scaling laws that describe the growth of disadvantageous, neutral, and advantageous mutants in growing populations. Applications of these laws to bacterial growth and carcinogenesis will be discussed.

We will discuss the following result. For every geometrically finite Kleinian group $\Gamma < SL_2(\mathbb C)$ there is $\epsilon_\Gamma$ such that for every $g \in SL_2(\mathbb C)$ the intersection $g \Gamma g^{-1} \cap SL_2(\mathbb R)$ is either a lattice or has critical exponent $\delta(g \Gamma g^{-1} \cap SL_2(\mathbb R)) \leq 1 - \epsilon_\Gamma$. This result extends Margulis-Mohammadi and Bader-Fisher-Milier-Strover. We will discuss some ideas of the proof. We will focus on the applications of a new ergodic component: the preservation of entropy in a direction.

I will begin by reviewing results about the evolution of the roots of real-rooted polynomials under repeated differentiation. In this case, the limiting evolution of the (real) roots can be described in terms of the concept of fractional free convolution, which in turn is equivalent to the operation of taking corners of Hermitian random matrices. 

Then I will present new results about the evolution of the complex roots of random polynomials under repeated differentiation—and more generally under repeated applications of differential operators. In this case, the limiting evolution of the roots has an explicit form that is closely connected to free probability and random matrix theory. 

The talk will be self-contained and will have lots of pictures and animations.

The replicator and the Generalized Lotka-Volterra equations are closely-related, foundational models in evolutionary game theory and community ecology, respectively. The concept of evolutionary stability and its relationship with dynamic stability has received significant attention: in the replicator equation, a mixed evolutionary stable strategy is also dynamically globally stable—i.e., will be reached by any trajectory originating from positive conditions. Intriguingly, the converse is not true: there are replicator equations yielding dynamically stable mixed strategies that are not evolutionary stable. Here we consider two classes of equivalence (i.e., transformations that do not alter the qualitative dynamics) for the replicator equation, to determine whether a globally-stable, but not evolutionary stable strategy maps into an equivalent state that is evolutionary stable—and show that this is the case for the examples that have been put forward so far. We derive the same two classes of equivalence for the Generalized Lotka-Volterra model, obtaining the same conditions for stability as for the replicator equation, and show that in this way we can characterize stability when other methods fail. By unifying the approach to proving stability for the replicator equation and Lotka-Volterra models, we bring these foundational equations even closer together.

In this talk, we will address areas of recent work centered around the themes of fairness and foundations in machine learning as well as highlight the challenges in this area. We will discuss recent results involving linear algebraic tools for learning, such as methods in non-negative matrix factorization that include tailored approaches for fairness. We will showcase our approach as well as practical applications of those methods.  Then, we will discuss new foundational results that theoretically justify phenomena like benign overfitting in neural networks.  Throughout the talk, we will include example applications from collaborations with community partners, using machine learning to help organizations with fairness and justice goals. This talk includes work joint with Erin George, Kedar Karhadkar, Lara Kassab, and Guido Montufar.

^{Prof. Deanna Needell earned her PhD from UC Davis before working as a postdoctoral fellow at Stanford University. She is currently a full professor of mathematics at UCLA, the Dunn Family Endowed Chair in Data Theory, and the Executive Director for UCLA's Institute for Digital Research and Education. She has earned many awards including the Alfred P. Sloan fellowship, an NSF CAREER and other awards, the IMA prize in Applied Mathematics, is a 2022 American Mathematical Society (AMS) Fellow and a 2024 Society for industrial and applied mathematics (SIAM) Fellow. She has been a research professor fellow at several top research institutes including the SLMath (formerly MSRI) and Simons Institute in Berkeley. She also serves as associate editor for several journals including Linear Algebra and its Applications and the SIAM Journal on Imaging Sciences, as well as on the organizing committee for SIAM sessions and the Association for Women in Mathematics.}

I will discuss join work with Flores, Kunnawalkam Elayavalli and Patchell where we introduce the property of subexponential decay, generalizing Haagerup-Jolissaint's property RD. I will provide examples of interest and also various applications. 

In this defense, I will motivate von Neumann algebras and give several examples of constructions inspired by group theory, highlighting the similarities and differences between the study of tracial von Neumann algebras and countable discrete groups. I will state recent results about how various combinations of these group-inspired constructions yield structural results, including: absence of tensor decomposition, sequential commutation, single generation, and the existence of exotic non-separable algebras. 

How many isomorphism classes of genus g curves are there over a finite field $\mathbb{F}_q$? In joint work with Samir Canning, Sam Payne, and Thomas Willwacher, we prove that the answer is a polynomial in q if and only if g is at most 8. One of the key ingredients is our recent progress on understanding low-degree odd cohomology of moduli spaces of stable curves with marked points.

We will discuss recent advances in the compression of pre-trained neural networks using both novel and existing computationally efficient   algorithms. The approaches we consider leverage sparsity, low-rank approximations of weight matrices, and weight quantization to achieve significant reductions in model size, while maintaining performance. We provide rigorous theoretical error guarantees as well as numerical experiments.

Diagonals of multivariate rational functions are an important class of functions arising in number theory, algebraic geometry, combinatorics, and physics. For instance, many hypergeometric functions are diagonals as well as the generating function for Apery's sequence. A natural question is to determine the diagonal grade of a function, i.e., the minimum number of variables one needs to express a given function as a diagonal. The diagonal grade gives the ring of diagonals a filtration. In this talk we study the notion of diagonal grade and the related notion of Hadamard grade (writing functions as the Hadamard product of algebraic functions), resolving questions of Allouche-Mendes France, Melczer, and proving half of a conjecture recently posed by a group of physicists. This work is joint with Andrew Harder.

[pre-talk at 3:00PM]

Introduced by Mallows in statistical ranking theory, the Mallows permutation model is a class of non-uniform probability measures on permutations. The general model depends on a distance metric on the symmetric group. This talk focuses on Mallows permutation models with L1 and L2 distances, which possess spatial structure and are also known as “spatial random permutations” in the mathematical physics literature.

A natural question from probabilistic combinatorics is: Picking a random permutation from either of the models, what does it “look like”? This may involve various features of the permutation, such as the length of the longest increasing subsequence and the cycle structure. In this talk, I will explain how multi-scale analysis and the hit and run algorithm—a Markov chain for sampling from both models—can be used to establish limit theorems for these features.

A complex manifold X is said to be Brody hyperbolic if it admits no entire curves, which are non-constant holomorphic maps from the complex numbers. When X is a smooth complex projective variety, Demailly introduced an algebraic analogue of this property known as algebraic hyperbolicity. We propose a conjecture on the algebraic hyperbolicity of generic sections of adjoint bundles on X, motivated by Fujita’s freeness conjecture and recent results by Bangere and Lacini on syzygies of adjoint bundles. We present some old and new evidence supporting this conjecture, including when X is any smooth projective toric variety or Gorenstein toric threefold. This is based on joint work with Joaquín Moraga.