A fundamental problem in geometry is to classify geometric structures. In one complex variable, for example, the Riemann Mapping Theorem asserts that any simply connected region of the plane, other than the plane itself, is biholomorphically equivalent to the unit disk. This is far from true in higher dimensions, where the local CR geometry of the boundary obstructs the existence of biholomorphisms. In this talk, we shall survey some results and open problems characterizing the unit ball and ball quotients, up to biholomorphism, by properties of the Bergman kernel (e.g., the Ramadanov Conjecture and one concerning algebraicity of the kernel) and the Bergman metric (Cheng’s Conjecture).  Particular focus will be on generalizing some of the results to algebraic surfaces, weakly pseudoconvex domains and solving Cheng's conjecture for Stein spaces in dimension 2.

Most mathematicians spend the minimal amount of time on understanding the ins and outs of LaTeX. However, LaTeX can be…. finicky, and sometimes this can come back to bite us. In this chill, shorter-than-normal (one might even say lazy) talk, we give a few ideas on how to minimize later pain without having to read hundreds of pages on the innards of LaTeX.

In this talk I will present examples of property (T) type II1 factors with trivial fundamental group, thus, providing progress towards the well-known open questions of Connes'94 and Popa'06. We will show that the semidirect product feature is an algebraic feature that survive passage to group von Neumann algebras for a class of inductive limit of property (T) groups arising from geometric group theory. Using Popa's deformation/rigidity in conjunction with group theoretic methods we proved that the acting group can be completely recoverable from the von Neumann algebra as well as the limit action of the acting group. In addition, the fundamental group of the group von Neumann algebras associated to these limit groups are trivial, which contrasts the McDuff case.  This is based on a joint work with S. Das. 

We study a generalization of the classical Multidimensional Scaling procedure (cMDS) to the setting of general metric measure spaces which can be seen as natural 'continuous limits' of finite data sets. We identify certain crucial spectral properties of the generalized cMDS operator thus providing a natural and rigorous formulation of cMDS in this setting. Furthermore, we characterize the cMDS output of several continuous exemplar metric measures spaces such as high dimensional spheres and tori (both with their geodesic distance). In particular, the case of spheres (with geodesic distance) requires that we establish that its cMDS operator is trace class, a condition which is natural in the context when the cMDS operator has infinite rank. Finally, we establish the stability of the generalized cMDS method with respect to the Gromov-Wasserstein distance.

Quasi-Newton methods form the basis of many effective methods for unconstrained and constrained optimization.  Quasi-Newton methods require only the first-derivatives of the problem to be provided and update an estimate of the Hessian matrix of second derivatives to reflect new approximate curvature information found during each iteration.  In the years following the publication of the Davidon-Fletcher-Powell (DFP) method in 1963 the Broyden-Fletcher-Goldfarb-Shanno (BFGS) update emerged as the best update formula for use in unconstrained minimization.  More recently, a number of quasi-Newton methods have been proposed that are intended to improve on the efficiency and reliability of the BFGS method.  Unfortunately, there is no known analytical means of determining the relative performance of these methods on a general nonlinear function, and there is no accepted standard set of test problems that may be used to verify that results reported in the literature are comparable. In this talk we will discuss ongoing work to provide a thorough derivation, implementation, and numerical comparison of these methods in a systematic and consistent way. We will look in detail at several modifications, discuss their relative benefits, and review relevant numerical results.

Consider now the family of solutions $U_\nu$ to the associated Navier-Stokes equation with the no-slip condition on the flat boundaries, for small viscosities $\nu=1/ Re$, and initial values close in $L^2$ to $Ae_1$. Under a conditional assumption on the energy dissipation close to the boundary, Kato showed in 1984 that $U_\nu$ converges to $Ae_1$ when the viscosity converges to 0 and the initial value converges to $A e_1$. It is still unknown whether this inviscid is unconditionally valid. Actually, the convex integration method predicts the possibility of layer separation. It produces solutions to the Euler equation with initial values $Ae_1 $, but with layer separation energy at time T up to:

 $$\|U(T)-Ae_1\|^2_{L^2}\equiv A^3T.$$

In this work, we prove that at the double limit for the inviscid asymptotic $\bar{U}$, where both the Reynolds number $Re$ converges to infinity and the initial value $U_{\nu}$ converges to $Ae_1$ in $L^2$, the energy of layer separation cannot be more than:

$$\| \bar{U}(T)-Ae_1\|^2_{L^2}\lesssim A^3T.$$

Especially, it shows that, even if the limit is not unique, the shear flow pattern is observable up to time $1/A$. This provides a notion of stability despite the possible non-uniqueness of the limit predicted by the convex integration theory.

The result relies on a new boundary vorticity estimate for the Navier-Stokes equation. This new estimate, inspired by previous work on higher regularity estimates for Navier-Stokes, provides a non-linear control scalable through the inviscid limit.

This dissertation lies at the intersection of extremal combinatorics and probabilistic combinatorics.  Roughly speaking, extremal combinatorics studies how large a combinatorial object can be.  For example, a classical result of Mantel's says that every $n$-vertex triangle-free graph has at most $\frac{1}{4} n^2$ edges.  The area of probabilistic combinatorics encompasses both the application of probability to combinatorial problems, as well as the study of random combinatorial objects such as random graphs and random permutations.  In this dissertation we study problems related to extremal properties of random objects.  In particular we study a certain card guessing game, $F$-free subgraphs of random hypergraphs, and thresholds of random hypergraphs.  Minimal prerequisites will be assumed.

We give a comprehensive structural analysis of amenable subrelations of a treed quasi-measure preserving equivalence relation. The main philosophy is to understand the behavior of the Radon-Nikodym cocycle in terms of the geometry of the amenable subrelation within the tree. This allows us to extend structural results that were previously only known in the measure-preserving setting, e.g., we show that every nowhere smooth amenable subrelation is contained in a unique maximal amenable subrelation. The two main ingredients are an extension of Carrière and Ghys's criterion for nonamenability, along with a new Ping-Pong-style argument we call the "Paddle-ball lemma" that we use to apply this criterion in our setting. This is joint work with Anush Tserunyan.

We establish sufficient conditions for a locally-warped product structure to propagate backward in time under the Ricci flow. As an application, we show that if a gradient shrinking soliton is asymptotic to a cone whose cross-section is a locally warped product of Einstein manifolds, the soliton must itself be a warped product over the same manifolds.

The p-widths of a closed Riemannian manifold are a nonlinear analog of the spectrum of its Laplace--Beltrami operator, which was defined by Gromov in the 1980s and correspond to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich--Marques--Neves, the p-widths obey a Weyl law, just like eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any ≥ 2-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich--Marques--Neves Weyl law constant in dimension 2.

In classical real and complex dynamics, one studies topological and analytic properties of orbits of points under iteration of self-maps of $\mathbb R$ or $\mathbb C$ (or more generally self-maps of a real or complex manifold). In arithmetic dynamics, a more recent subject, one likewise studies properties of orbits of self-maps, but with a number theoretic flavor. Many of the motivating problems in arithmetic dynamics come via analogy with classical problems in arithmetic geometry: rational and integral points on varieties correspond to rational and integral points in orbits; torsion points on abelian varieties correspond to periodic and preperiodic points of rational maps; and abelian varieties with complex multiplication correspond to post-critically finite rational maps.

This analogy focuses on forward iteration, but sometimes surprising and interesting results can be found by thinking instead about pre-images of rational points under iteration. In this talk, we will give some background and motivation for the field of arithmetic dynamics in order to describe some of these "backwards iteration" results, including uniform boundedness for rational pre-images and open image results for Galois representations associated to dynamical systems.

We develop an efficient and accurate level set method to study numerically a crawling eukaryotic cell using a minimal model. This model describes the cell polarity and movement using a reaction-diffusion system coupled with a sharp-interface model. 

We employ an efficient finite difference method for the reaction-diffusion equations with no-flux boundary conditions. This results in a symmetric positive definite system, which can be solved by the conjugate gradient method accelerated by preconditioners. To track the long-time dynamics, we employ techniques of the moving computational window to keep the efficiency. Our level-set simulations capture well the cell crawling, the straight line trajectory, the circular trajectory, and other features. 

Our efficient and accurate computational techniques can be extended to a broad class of biochemical descriptions of cell motility, for which problems are posed on moving domains with complex geometry and fast simulations are very important. This is a joint work with Li-Tien Cheng and Bo Li.

We give a method for sampling points from an affine algebraic variety over a local field with a prescribed probability distribution. In the spirit of the previous work by Breiding and Marigliano on real algebraic manifolds, our method is based on slicing the given variety with random linear spaces of complementary dimension. We also provide an implementation of our sampling method and discuss a few applications, in particular we sample from algebraic p-adic matrix groups and modular curves.