### Random matrix theory and supersymmetry techniques

- Series
- Job Candidate Talk
- Time
- Monday, January 6, 2020 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Tatyana Shcherbyna – Princeton University – tshcherbyna@princeton.edu

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- Series
- Job Candidate Talk
- Time
- Monday, January 6, 2020 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Tatyana Shcherbyna – Princeton University – tshcherbyna@princeton.edu

Starting from the works of Erdos, Yau, Schlein with coauthors, significant progress in understanding universal behavior of many random graph and random matrix models were achieved. However for random matrices with a spatial structure, our understanding is still very limited. In this talk I am going to overview applications of another approach to the study of the local eigenvalue statistics in random matrix theory based on so-called supersymmetry techniques (SUSY). The SUSY approach is based on the representation of the determinant as an integral over the Grassmann (anticommuting) variables. Combining this representation with the representation of an inverse determinant as an integral over the Gaussian complex field, SUSY allows to obtain an integral representation for the main spectral characteristics of random matrices such as limiting density, correlation functions, the resolvent's elements, etc. This method is widely (and successfully) used in the physics literature and is potentially very powerful but the rigorous control of the integral representations, which can be obtained by this method, is quite difficult, and it requires powerful analytic and statistical mechanics tools. In this talk we will discuss some recent progress in application of SUSY to the analysis of local spectral characteristics of the prominent ensemble of random band matrices, i.e. random matrices whose entries become negligible if their distance from the main diagonal exceeds a certain parameter called the band width.

- Series
- Geometry Topology Seminar
- Time
- Monday, January 6, 2020 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skile 006
- Speaker
- Melissa Zhang – UGA

When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will explore how link symmetries are reflected in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. In joint work with Matthew Stoffregen, we use Lawson-Lipshitz-Sarkar's construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links.

- Series
- Job Candidate Talk
- Time
- Tuesday, January 7, 2020 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Xiaochuan Tian – University of Texas at Austin – xtian@math.utexas.edu

Nonlocal models are experiencing a firm upswing recently as more realistic alternatives to the conventional local models for studying various phenomena from physics and biology to materials and social sciences. In this talk, I will describe our recent effort in taming the computational challenges for nonlocal models. I will first highlight a family of numerical schemes -- the asymptotically compatible schemes -- for nonlocal models that are robust with the modeling parameter approaching an asymptotic limit. Second, I will discuss nonlocal-to-local coupling techniques so as to improve the computational efficiency of using nonlocal models. This also motivates the development of new mathematical results -- for instance, a new trace theorem that extends the classical results.

Although new nonlocal models have been gaining popularity in various applications, they often appear as phenomenological models, such as the peridynamics model in fracture mechanics. Here I will illustrate how to characterize the origin of nonlocality through homogenization of wave propagation in periodic media.

- Series
- School of Mathematics Colloquium
- Time
- Thursday, January 9, 2020 - 11:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- June Huh – Princeton University – junehuh@ias.edu

Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. Although no specific background beyond linear algebra and multivariable calculus will be needed to enjoy the presentation, I advertise the talk to people with interests in at least one of the following topics: graphs, convex bodies, stable polynomials, projective varieties, Potts model partition functions, tropicalizations, Schur polynomials, highest weight representations. Based on joint works with Petter Brändén, Christopher Eur, Jacob Matherne, Karola Mészáros, and Avery St. Dizier.

- Series
- Job Candidate Talk
- Time
- Thursday, January 9, 2020 - 14:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Antonio De Rosa – NYU – derosa@cims.nyu.edu

Elliptic integrands are used to model anisotropic energies in variational problems. These energies are employed in a variety of applications, such as crystal structures, capillarity problems and gravitational fields, to account for preferred inhomogeneous and directionally dependent configurations. After a brief introduction to variational problems involving elliptic integrands, I will present an overview of the techniques I have developed to prove existence, regularity and uniqueness properties of the critical points of anisotropic energies. In particular, I will present the anisotropic extension of Allard's rectifiability theorem and its applications to the Plateau problem. Furthermore, I will describe the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points. Finally, I will mention some of my ongoing and future research projects.

- Series
- Stochastics Seminar
- Time
- Thursday, January 9, 2020 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- Cheng Mao – Georgia Institute of Technology – cmao35@gatech.edu

In various applications involving ranking data, statistical models for mixtures of permutations are frequently employed when the population exhibits heterogeneity. In this talk, I will discuss the widely used Mallows mixture model. I will introduce a generic polynomial-time algorithm that learns a mixture of permutations from groups of pairwise comparisons. This generic algorithm, equipped with a specialized subroutine, demixes the Mallows mixture with a sample complexity that improves upon the previous state of the art.

- Series
- PDE Seminar
- Time
- Thursday, January 9, 2020 - 15:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Mahir Hadzic – University College London – m.hadzic@ucl.ac.uk

The basic model of an isolated self-gravitating gaseous star is given by the gravitational Euler-Poisson system. For any value of the adiabatic index strictly between 1 and 4/3 we construct an infinite-dimensional family of collapsing solutions to the Euler-Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface in the Lagrangian coordinates. The leading order singular behaviour is driven by collapsing dust solutions. This is a joint work with Yan Guo (Brown) and Juhi Jang (USC).

- Series
- ACO Student Seminar
- Time
- Friday, January 10, 2020 - 13:05 for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Jinyan Liu – University of Hong Kong – jyliu678@gmail.com

We consider the problem of learning optimal reserve price in repeated auctions against non- myopic bidders, who may bid strategically in order to gain in future rounds even if the single- round auctions are truthful. Previous algorithms, e.g., empirical pricing, do not provide non- trivial regret rounds in this setting in general. We introduce algorithms that obtain a small regret against non-myopic bidders either when the market is large, i.e., no single bidder appears in more than a small constant fraction of the rounds, or when the bidders are impatient, i.e., they discount future utility by some factor mildly bounded away from one. Our approach carefully controls what information is revealed to each bidder, and builds on techniques from differentially private online learning as well as the recent line of works on jointly differentially private algorithms.

- Series
- Combinatorics Seminar
- Time
- Friday, January 10, 2020 - 15:00 for 1 hour (actually 50 minutes)
- Location
- Skiles 202
- Speaker
- Lutz Warnke

We shall discuss the recent breakthrough of Annika Heckel on the chromatic number of the binomial random graph G(n,1/2), showing that it is not concentrated on any sequence of intervals of length n^{1/4-o(1)}.

To put this into context, in 1992 Erdos (and also Bollobás in 2004) asked for any non-trivial results asserting a lack of concentration, pointing out that even the weakest such results would be of interest.

Until recently this seemed completely out of reach, in part because there seemed to be no obvious approach/strategy how to get one's foot in the door.

Annika Heckel has now found such an approach, based on a clever coupling idea that compares the chromatic number of G(n,1/2) for different n.

In this informal talk we shall try to say a few words about her insightful proof approach from https://arxiv.org/abs/1906.11808

Please note the unusual room (Skiles 202)

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