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.

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.

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.

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.

TBA

TBA

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.