2025 FALL ABSTRACTS

Sep 02 @ Temple: Felix Höfer (Princeton)

Markov Perfect Equilibria in discrete finite-player and mean-field games

We study dynamic finite-player and mean-field stochastic games within the framework of Markov perfect equilibria (MPE). Unlike their continuous-time analogues, discrete-time finite-player games generally do not admit unique MPE. However, we show that uniqueness is remarkably recovered when the time steps are sufficiently small, and we provide examples demonstrating the necessity of this assumption. This result, established without relying on any monotonicity conditions, underscores the importance of inertia in dynamic games. Furthermore, we discuss different learning algorithms and prove their convergence to the unique MPE.

Sep 09 @ Penn:  Andres A. Contreras Hip (University of Chicago)

Gaussian curvature for Liouville Quantum Gravity and random planar maps

Liouville quantum gravity is a canonical model for random surfaces conjectured to be the scaling limit of various planar maps. Since curvature is a central concept in Riemannian geometry, it is natural to ask whether this can be extended to LQG surfaces. In this talk, we introduce a notion of Gaussian curvature for LQG surfaces, despite their low regularity, and study the relations with its discrete counterparts. We conjecture that this definition of Gaussian curvature is the scaling limit of the discrete curvature. In support of this conjecture, we prove that the discrete curvature on the $\epsilon$-CRT map with a Poisson vertex set integrated with a smooth test function is of order $\epsilon^{o(1)},$ and show the convergence of the total discrete curvature on a CRT map cell when scaled by $\epsilon^{1/4}.$ Joint work with E. Gwynne.

Sep 16 @ Temple:  Eric Thoma (Stanford)

A mean value inequality for the Coulomb gas

The Coulomb gas is a statistical physics model consisting of N particles interacting with electrostatic repulsion and with a confining potential. I will first review results on the microscopic structure on the gas. Then, I will show how a certain subharmonic structure associated with the k-point correlation function arises. This structure implies new bounds on quantities such as the furthest particle from the origin while generalizing bounds known for the Ginibre ensemble, and it also explains how Poisson point process statistics take over in the high-temperature regime.

Sep 23 @ Penn: Ani Sridhar (NJIT)

Detecting Super-spreaders in Network Cascades

Suppose that a continuous-time, stochastic diffusion (i.e., the Susceptible-Infected process) spreads on an unknown graph. We only observe the time at which the diffusion reaches each vertex, i.e., the set of infection times. What can be learned about the unknown graph from the infection times? While there is far too little information to learn individual edges in the graph, we show that certain high-level properties -- such as the number of vertices of sufficiently high degree, or super-spreaders -- can surprisingly be determined with certainty. To achieve this goal, we develop a suite of algorithms that can efficiently detect vertices of degree asymptotically greater than sqrt(n) from infection times, for a natural and general class of graphs with n vertices. To complement these results, we show that our algorithms are information-theoretically optimal: there exist graphs for which it is impossible to tell whether vertices of degree larger than n^{1/2 - \epsilon} exist from vertices' infection times, for any \epsilon > 0. Finally, we discuss the broader implications of our ideas for change-point detection in non-stationary point processes. This talk is based on joint work with Anna Brandenberger (MIT) and Elchanan Mossel (MIT).

Sep 24 @Penn:  Jiaoyang Huang (UPenn)

[Note that this is a Penn Mathematics Colloquium that will take place in DRL A4 ]

Ramanujan Property and Edge Universality of Random Regular Graphs

Extremal eigenvalues of graphs are of particular interest in theoretical computer science and combinatorics. Specifically, the spectral gap—the difference between the largest and second-largest eigenvalues—measures the expansion properties of a graph. In this talk, I will focus on random d-regular graphs.

I will begin by providing background on the eigenvalues of random d-regular graphs and their connections to random matrix theory. In the second part of the talk, I will discuss our recent results on eigenvalue rigidity and edge universality for these graphs. Eigenvalue rigidity asserts that, with high probability, each eigenvalue concentrates around its classical location as predicted by the Kesten-McKay distribution. Edge universality states that the second-largest eigenvalue and the smallest eigenvalue of random d-regular graphs converge to the Tracy-Widom distribution from the Gaussian Orthogonal Ensemble. Consequently, approximately 69% of d-regular graphs are Ramanujan graphs. This work is based on joint work with Theo McKenzie and Horng-Tzer Yau.

Sep 30 @ Temple: Elliot Paquette (McGill) 

From magic squares, through random matrices, and to the multiplicative chaos 

In 2004, motivated by connections of random matrix theory to number theory, Diaconis and Gamburd showed a fascinating connection between the enumeration problem of magic squares (squares filled with integers with row and column sum constraints) and the moments of the ‘secular coefficients’ of random matrices, when the size of the matrix tends to infinity. These are the coefficients in the monomial expansion of a characteristic polynomial, or equivalently, the elementary symmetric polynomials of the eigenvalues of this random matrix. It turns out that this characteristic polynomial has a limit, when the matrix size tends to infinity. It converges to a random fractal, the holomorphic multiplicative chaos. We describe this process on the unit circle, and show how it can be connected even more strongly to random matrices, and how magic square combinatorics are a type of ‘signature’ of this holomorphic multiplicative chaos. We’ll review some open questions about these objects, and discuss some links between this and other stochastic processes such as the Gaussian multiplicative chaos, the circular beta-ensemble and random multiplicative function.

Oct 07 @ Penn:  Christophe Garban (Université Lyon 1 / NYU)

One-arm exponents of the high-dimensional Ising model

In a joint work with Diederik van Engelenburg, Romain Panis and Franco Severo, we study the probability that the origin is connected to the boundary of the box of size $n$ (the one-arm probability) in several percolation models related to the Ising model. We prove that different universality classes emerge at criticality and that the FK-Ising model has upper-critical dimension equal to 6, in contrast to the Ising model, where it is known to be (less or) equal to 4. I will start the talk with a short introduction on the Ising model on Z^d. 

Oct 14 @ Penn:  Cosmin Pohoata (Emory)

The Erdős-Szekeres problem  

For every natural number n, if we start with sufficiently many points in R^d in general position there will always exist n points in convex position. The problem of determining quantitative bounds for this statement is known as the Erdős-Szekeres problem, and is one of the oldest problems in extremal combinatorics (sometimes also called the “Happy Ending Problem”). We will discuss some of its history, some recent developments in the plane and in higher dimensions, as well as some connections with a few other topics in combinatorics and beyond. 

Oct 21 @ Temple:  Xiaoqin Guo (University of Cincinnati)

On the rate of the homogenization of a non-divergence form difference operator in random environments

We consider the convergence rate of a random non-divergence form difference equation on ℤd to its "effective" differential equation on ℝd. We will discuss the optimal convergence rate when the coefficient field has a finite range of dependence. Moreover, when the coefficient field is i.i.d. and exhibits isotropic symmetry, we prove strictly faster convergence, improving the generic finite-range rate. Joint work with Hung V. Tran (Wisconsin) and Timo Sprekler (Texas A&M).

Oct 28 @ Temple: M. Alper Gunes (Princeton)

TBA

Nov 04 @ Penn: Li-Cheng Tsai (Utah)

TBA

Nov 11 @ Temple: Kevin Hu (Columbia)

TBA

Nov 18 @ Penn:  Daniel Lacker (Columbia)

TBA

Nov 25 @ No seminar

TBA

Dec 02 @ Temple:

TBA