Week of November 28, 2021

Mon Nov 29, 2021
12:00pm - Zoom ID 989 6529 0738. Passcode: the last four digits of the zoom ID in the reverse order - Probability and Analysis Webinar
Joe Neeman - (UT Austin)
Large deviations for triangle densities

Take a uniformly random graph with a fixed edge density e. Its triangle density will typically be about e^3, and we are interested in the large deviations behavior: what's the probability that the triangle density is about e^3 - delta? The general theory for this sort of problem was studied by Chatterjee-Varadhan and Dembo-Lubetzky, who showed that the solution can be written in terms of an optimization over certain integral kernels. This optimization is difficult to solve explicitly, but Kenyon, Radin, Ren and Sadun used numerics to come up with a fascinating and intricate set of conjectures regarding both the probabilities and the structures of the conditioned random graphs. We prove these conjectures in a small region of the parameter space.


Joint work with Charles Radin and Lorenzo Sadun

2:00pm to 3:00pm - 510R - Combinatorics and Probability
Olya Mandelshtam - (University of Waterloo)
The multispecies zero range process and modified Macdonald polynomials

Over the last couple of decades, the theory of interacting particle systems has found some unexpected connections to orthogonal polynomials, symmetric functions, and various combinatorial structures. The asymmetric simple exclusion process (ASEP) has played a central role in this connection. Recently, Cantini, de Gier, and Wheeler found that the partition function of the multispecies ASEP on a circle is a specialization of a Macdonald polynomial $P_{\lambda}(X;q,t)$. Macdonald polynomials are a family of symmetric functions that are ubiquitous in algebraic combinatorics and specialize to or generalize many other important special functions. Around the same time, Martin gave a recursive formulation expressing the stationary probabilities of the ASEP on a circle as sums over combinatorial objects known as multiline queues, which are a type of queueing system. Shortly after, with Corteel and Williams we generalized Martin's result to give a new formula for $P_{\lambda}$ via multiline queues.


The modified Macdonald polynomials $\widetilde{H}_{\lambda}(X;q,t)$ are a version of $P_{\lambda}$ with positive integer coefficients. A natural question was whether there exists a related statistical mechanics model for which some specialization of $\widetilde{H}_{\lambda}$ is equal to its partition function. With Ayyer and Martin, we answer this question in the affirmative with the multispecies totally asymmetric zero-range process (TAZRP), which is a specialization of a more general class of zero range particle processes. We introduce a new combinatorial object in the flavor of the multiline queues, which on one hand, expresses stationary probabilities of the mTAZRP, and on the other hand, gives a new formula for $\widetilde{H}_{\lambda}$. We define an enhanced Markov chain on these objects that lumps to the multispecies TAZRP, and then use this to prove several results about particle densities and correlations in the TAZRP.

4:00pm to 5:00pm - RH 306 - Applied and Computational Mathematics
Eun-Jae Park - (Yonsei University)
Staggered discontinuous Galerkin methods on polygonal meshes

Recently, polygonal finite element methods have received considerable attention. This is because general meshes offer a very convenient framework for mesh generation, mesh deformations, fracture problems, composite materials, topology optimizations, mesh refinements and coarsening; for instance, to handle hanging nodes, different cell shapes within the same mesh and  non-matching interfaces. 

In this talk,  a new computational paradigm for discretizing PDEs  is presented via the staggered Galerkin method on general meshes.
First, a class of locally conservative, lowest order staggered discontinuous Galerkin method on general quadrilateral/polygonal
meshes for elliptic problems are proposed.  The method can be flexibly applied to rough grids such as highly distorted  meshes.
Next, adaptive mesh refinement is an attractive tool for general meshes due to their flexibility and simplicity in handling hanging
nodes. We derive a simple residual-type error estimator. Numerical results indicate that optimal convergence can be achieved for both
the potential  and vector variables, and the singularity can be well-captured by the proposed error estimator. Then, some
applications to diffusion equations, Stokes equations, and linear elasticity equations are considered. Finally, we extend this
approach to high-order polynomial approximations on general meshes. 

Tue Nov 30, 2021
4:00pm - NS2 1201 - Differential Geometry
Hongyi Sheng - (UC Irvine)
Deformations of the scalar curvature and the mean curvature
In Riemannian manifold $(M^n, g)$, it is well-known that its minimizing hypersurface is smooth when $n\leq 7$, and singular when $n\geq 8$. This is one of the major difficulties in generalizing many interesting results to higher dimensions, including the Riemannian Penrose inequality. In particular, in dimension 8, the minimizing hypersurface has isolated singularities, and Nathan Smale constructed a local perturbation process to smooth out the singularities. However, Smale’s perturbation will also produce a small region with possibly negative scalar curvature. In order to apply this perturbation in general relativity, we constructed a local deformation prescribing the scalar curvature and the mean curvature simultaneously. In this talk, we will discuss how the weighted function spaces help us localize the deformation in complete manifolds with boundary, assuming certain generic conditions. We will also discuss some applications of this result in general relativity.
Wed Dec 1, 2021
2:00pm to 3:00pm - Rowland Hall 510R - Combinatorics and Probability
Roman Vershynin - (UCI)
Mathematics of synthetic data and privacy

An emerging way to protect privacy is to replace true data by synthetic data. Medical records of artificial patients, for example, could retain meaningful statistical information while preserving privacy of the true patients. But what is synthetic data, and what is privacy? How do we define these concepts mathematically? Is it possible to make synthetic data that is both useful and private? I will tie these questions to a simple-looking problem in probability theory: how much information about a random vector X is lost when we take conditional expectation of X with respect to some sigma-algebra? This talk is based on a series of papers joint with March Boedihardjo and Thomas Strohmer, mainly this one: https://arxiv.org/abs/2107.05824 

Thu Dec 2, 2021
9:00am to 10:00am - Zoom - Inverse Problems
Yi-Hsuan Lin - (National Yang Ming Chiao Tung University)


10:00am to 10:50am - Zoom - Number Theory
Ofir Gorodetsky - (Oxford University)
11:00am to 11:50am - Zoom ID: 949 5980 546, Password: the last four digits of ID in the reverse order - Harmonic Analysis
Gabriel Rivière - (Université de Nantes)
Meromorphic continuation of Poincaré series

Poincaré series are natural functions that arise in Riemannian geometry 
when one wants to count the number of geodesic arcs of length less than 
T between two given points on a compact manifold. I will begin with an 
introduction on this topic. Then I will discuss some recent results with 
N.V. Dang (Univ. Paris Sorbonne) showing that, in the case of negatively 
curved manifolds, these series have a meromorphic continuation to the 
whole complex plane. This can be shown by relating Poincaré series with 
the resolvent of the geodesic vector field and by exploiting recent 
results on this resolvent obtained through microlocal methods. If time 
permits, I will also explain how the genus of a surface can be recovered 
from the analysis of these series.

4:00pm - RH 306 - Colloquium
Jinchao Xu - (Penn State University)