Date/Time : 15 Oct 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Milind Hegde , Nanyang Technological University

A recurring theme in probability theory is that of universality: when extremely different looking systems have the same large scale statistical behavior. In the last few decades, an important new universality class has been discovered, called the Kardar-Parisi-Zhang (KPZ) universality class. However, the universality is only putative as only a handful of “metric” type models have been shown to lie in it in the strongest sense.

In this talk we will discuss a recent proof of membership in the KPZ class of the first non-metric type of model, namely the colored stochastic six-vertex model. The model arises naturally in probability theory and has connections to many areas of statistical physics and quantum integrable systems; e.g., the first form of the six-vertex model was introduced by Pauling in 1935 to model the crystal structure of ice. The Yang-Baxter equation and line ensembles (collections of random non-intersecting curves) will play fundamental roles in our discussion, but no prior background will be assumed. This is based on joint work with Amol Aggarwal and Ivan Corwin.

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Date/Time : 15 Oct 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Luca San Mauro, University of Bari

Borel reducibility has been a decisive tool in assessing whether familiar classes of countable structures can be neatly classified. However, when the corresponding isomorphism problems admit only countably many isomorphism types, they all share the same Borel complexity. To study such cases, we propose a finer framework, inspired by algorithmic learning theory. In this setting, a countable family of countable structures is learnable exactly when its isomorphism problem continuously reduces to E0 (the relation of eventual agreement on reals). Replacing E0 with other equivalence relations then yields a natural hierarchy for ranking isomorphism problems that escape the Borel analysis. Within this framework, we investigate both classical benchmark relations and those arising from established learning criteria. We also revisit the Friedman-Stanley tower from a learning-theoretic viewpoint. This talk is based on a series of joint works with N. Bazhenov, V. Cipriani, E. Fokina, S. Jain, A. Marcone, and F. Stephan.

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Date/Time : 15 Oct 2025 14:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Kai Dong, National University of Singapore

In this talk, we give a new proof of the past nonlinear stability of Kasner solutions for $3+1$-dimensional Einstein vacuum equations with polarized $U(1)$-symmetry. Our method, following Beyer, Oliynyk, and Arturo Olvera-Santamaria \cite{beyer2021fuchsian}, relies on energy estimates for a class of first order hyperbolic PDEs, namely Fuchsian systems.

We show that the perturbed solutions are asymptotically Kasner, geodesically incomplete and crushing at the big bang singularity. This is achieved by reducing the $3+1$ Einstein vacuum equations to a Fuchsian system coupled with several constraint equations, with the symmetry assumption playing an important role in the reduction. Using Fuchsian theory together with finite speed of propagation, we obtain global existence and precise asymptotic behavior of the solutions up to the singularities.

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Date/Time : 15 Oct 2025 15:00
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Liang Zhao, National University of Singapore

In this talk, we will consider the simplification from the multi-dimensional (M-D) bipolar Euler–Poisson (2EP) equations to the unipolar Euler–Poisson (1EP) equations of electrons, which are both important models in plasma physics and semiconductors. Based on some rigorous convergence analysis of certain physical parameters, we are able to justify these simplifications for two classes of global-in-time solutions of the M-D 2EP system. The first is the smooth solution of the damped M-D 2EP system near the constant equilibrium state, and the second is the regular solution to the Cauchy problem for the undamped M-D 2EP system with far-field vacuum. The global-in-time error estimates between solutions to 2EP and 1EP are also established.

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Date/Time : 14 Oct 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Lo Chi-Heng, National University of Singapore

In this talk, I will first introduce the closure ordering conjecture for local Arthur packets. Then I will present a proof using the combinatorial structure of Ψ(π) that we studied last week.  I will also discuss two important consequences:

(1) a framework for proving the enhanced Shahidi conjecture for general groups, and
(2) a proof of Vogan’s conjecture on the coincidence of A-packets and ABV-packets of Arthur type for classical groups, under certain assumptions.

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Date/Time : 13 Oct 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : :Si Ying Lee, National University of Singapore

I will talk about constructing integral models for Hecke correspondences in both the global and local settings, using the theory of F-gauges. Using these integral models, I will explain how one can construct integral Hecke actions on coherent cohomology for compact abelian type Shimura varieties, proving a conjecture of Fakhruddin-Pilloni. This is joint work in progress with Keerthi Madapusi.

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Date/Time : 10 Oct 2025 16:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Liu Liu, The Chinese University of Hong Kong

In this talk, we introduce a bi-fidelity algorithm for velocity discretization of Boltzmann-type kinetic equations under multiple scalings. The proposed method employs a simpler and computationally cheaper low-fidelity model to capture a small set of significant velocity points through the greedy approach, then evaluates the high-fidelity model only at these few velocity points and to reconstruct a bi-fidelity surrogate. This method, inspired by its previous application in solving uncertainty quantification problem for kinetic models, integrates a simpler collision term of relaxation type in the low-fidelity model and an asymptotic-preserving scheme in the high-fidelity update step. Both linear Boltzmann under diffusive scaling and the nonlinear full Boltzmann in hyperbolic scaling are discussed. The weak asymptotic-preserving property and empirical error bound estimates are shown. Extensive numerical experiments on Boltzmann problems with smooth or discontinuous initial conditions and under various regimes have been carefully studied, which demonstrates the effectiveness and efficiency of our designed numerical schemes. This is a joint work with Nicolas Crouseilles and Zhen Hao.

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Date/Time : 08 Oct 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Peter Markowich, King Abdullah University of Science and Technology

We propose a partial differential-integral equation (PDE) framework for deep neural networks (DNNs) and their associated learning problems by taking the continuum limits of both network width and depth. The proposed model captures the complex interactions among hidden nodes, overcoming limitations of traditional discrete and ordinary differential equation (ODE)-based models. We explore the well-posedness of the forward propagation problem, analyze the existence and properties of minimizers for the learning task, and provide a detailed examination of necessary and sufficient conditions for the existence of critical points. Controllability and optimality conditions for the learning task with its associated PDE forward problem are established using variational calculus, the Pontryagin Maximum Principle, and the Hamilton-Jacobi-Bellman equation, framing the deep learning process as a PDE-constrained optimization problem. In this context, we prove the existence of viscosity solutions for the latter and we establish optimal feedback controls based on the value functional. This approach facilitates the development of new network architectures and numerical methods that improve
upon traditional layer-by-layer gradient descent techniques. The paper provides a mathematical foundation for connecting neural networks, PDE theory, variational analysis, and optimal control. By integrating these fields, we offer a robust framework that enhances deep learning models’ stability, efficiency, and interpretability.

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Date/Time : 07 Oct 2025 13:00
Venue : S16 #05-18
Speaker : Jiakun Liu, The University of Sydney

In this talk, we will introduce the optimal transportation with some interesting applications in various fields including a reconstruction problem in cosmology; a brief proof of isoperimetric and Brunn-Minkowski inequalities in geometry; and an application in image recognition relating to a transport between hypercubes. We will also discuss our recent research on the regularity of free boundaries in optimal transportation.

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