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.

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.
 

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.

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.
 

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.

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

Please register here:

https://forms.gle/yDcUa9LAmpY1F2178.

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.