Date/Time : 10 Sep 2025 16:30
Venue : S17 #06-11 (Seminar Room 6)
Speaker : Wang Zhongjian, Nanyang Technological University

In this talk I will discuss the role of regularity estimate for viscous Hamilton Jacobi equation in showing the optimal iteration complexity (square root of dimension) of the Diffusion model. In our approach to the optimal bound, we only assume the Gaussian tail assumption, which does not require log-concaveness of the distribution. Our Gaussian tail assumption admits distribution of early stopping setting and scalable to infinite dimensions. More recently, we also provide analysis for Follmer process and show the deterministic heat flow is a Lipschitz change of variable between target and Gaussian under the Gaussian tail assumption.

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Date/Time : 08 Sep 2025 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Dai Wenhan, National University of Singapore

Under categorical local Langlands, certain natural coherent sheaves on Vogan stacks give rise to matching automorphic sheaves on Bun_G, which are generally hard to understand. Understanding the stalks of these automorphic sheaves at points of Bun_G would crucially provide a mechanism for the explicit matching of objects along the categorical local Langlands functor. Despite its importance, the general behavior of these stalks remains largely mysterious. In this talk we will study these sheaves for G=GLn at the Vogan stack of the Steinberg parameter. Some recent conjectures predict that at basic points, these stalks are given by generalized Steinberg representations of inner forms of G, up to degree shifts. After formulating the main conjecture, I will report some progress towards its proof in a special case on the degree-zero component of Bun_G, achieved through combinatorial computations inspired by Hansen. If time permits, I will also present an application of the result to cohomology of local Shimura varieties.

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Date/Time : 04 Sep 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Khoat Than, Hanoi University of Science and Technology

Deep neural networks (DNNs) have driven many of the most significant breakthroughs in artificial intelligence in recent years, fundamentally transforming the way we live and work. Tasks that seemed intractable just a decade ago can now be solved with remarkable accuracy by large-scale DNNs. A central reason for their success lies in their ability to generalize: after learning from a finite training set, DNNs often perform surprisingly well on unseen data. Yet, despite this empirical success, the theoretical foundations of generalization in deep learning remain poorly understood, posing a long-standing challenge.

In this talk, I will provide an overview of recent theoretical advances, many of which, despite their elegance, turn out to yield vacuous or impractical guarantees for modern networks. I will then introduce our new approaches, which establish explicit connections between the local behaviors of a DNN across different regions of the data space, the complexity of the data distribution, and the model’s generalization performance. These results reveal novel structural insights and suggest that such perspectives could bring us closer to resolving one of the most fundamental open problems in deep learning.

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Date/Time : 03 Sep 2025 16:30
Venue : S17 #04-04 (SR3)
Speaker : Tuan Dam, Hanoi University of Science and Technology

This work presents novel Monte Carlo Tree Search (MCTS) algorithms for stochastic Markov Decision Processes (MDPs) that address fundamental limitations in existing methods like UCT and Fixed-Depth-MCTS. The proposed approaches incorporate power mean estimators as value backup operators to achieve more accurate value estimation between the biased extremes of weighted sum (underestimation) and maximum valuation (overestimation). Combined with polynomial exploration bonuses that properly balance exploration-exploitation in non-stationary settings, these algorithms achieve polynomial convergence rates of O(n^(-1/2)) for discrete spaces and O(n^(-ζ)), ζ ∈ (0,1/2) for continuous state-action spaces, where n represents visited trajectories. The methods extend existing theoretical guarantees from deterministic to stochastic environments, with empirical validation demonstrating effectiveness across various stochastic MDP tasks in both discrete and continuous domains.

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Date/Time : 03 Sep 2025 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Laura DeMarco, Harvard University

In holomorphic dynamical systems, one studies maps on the Riemann sphere (or other complex manifolds) with focus on their Julia sets and invariant measures. From this point of view, the Lattès maps — those that are quotients of maps on elliptic curves — are rather uninteresting; their dynamical features are well understood. But viewed algebraically, there are still many unanswered questions. I’ll begin the talk with some history of these maps. Then I’ll describe a recent question about the geometry of torsion points on elliptic curves and how it has led to interesting complex-dynamical questions about other families of maps and, in turn, new perspectives on the arithmetic side.

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Date/Time : 02 Sep 2025 14:00
Venue : S16 #05-17
Speaker : Wang Yamin, National University of Singapore

The classical Alt-Phillips problem, appearing in the study of permeable catalysts and population dynamics, was first studied by Phillips [1983] and Alt-Phillips [1986]. It embeds two of the most well studied elliptic free boundary problems as special cases, namely, the Alt-Caffarelli problem when the parameter $gamma=0$ and the obstacle problem when $gamma=1$.

In this talk, we study the boundary behavior of a solution to the fully nonlinear Alt-Phillips problem with $\gamma\in (1,2)$. We show that the free boundary intersects the fixed boundary tangentially where the Dirichlet data vanish. For this range of $\gamma$, our result is new even when the operator is the Laplacian. This is joint work with Yu Hui.

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Date/Time : 02 Sep 2025 14:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Yang Ruotao, Academy of Mathematics and Systems Science, Chinese Academy of Sciences

The Gaiotto conjecture says that the category of representations of the quantum supergroup can be realized as a category of certain sheaves on the affine Grassmannian. It is the quantum version of a particular case of the relative Langlands conjecture by Ben-Zvi-Sakellaridis-Venkatesh. In this talk, we will mainly focus on the tamely ramified version of the Gaiotto conjecture, and see the application of the factorization method in its study. The talk is organized as follows, first we will review the background and explain the statement of the conjecture; then we will roughly sketch the strategy of the proof using the factorization method; if time permits, we will explain the application of the factorization method in the general relative Langlands program. The talk is based on ongoing works with Michael Finkelberg and Roman Travkin.

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Date/Time : 02 Sep 2025 16:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Yilin Wang, ETH Zürich

The goal of this talk is to showcase how we can use stochastic processes to study the geometry of surfaces. After recalling basic facts about surfaces with constant curvature, their length spectrum, and Brownian motion on them, we use the Brownian loop measure to express the lengths of closed geodesics on a hyperbolic surface and zeta-regularized determinant of the Laplace-Beltrami operator. This gives a tool to study the length spectra of a hyperbolic surface and we obtain a new identity between the length spectrum of a compact surface and that of the same surface with an arbitrary number of additional cusps. This is mainly based on a joint work with Yuhao Xue (IHES).

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Date/Time : 01 Sep 2025 15:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Wenqiang Xu, New York University

Are there infinitely many quadratic characters (for instance, the Legendre symbol mod p) for which the partial sums are always non-negative? This is a classical question that goes back to Fekete and Polya in 1911. Although only 0% of characters can have this property, numerical work (most recently by Kalmynin) suggests that such characters are nevertheless plentiful. For instance, computations show that there are many more examples than may be expected by modeling such sums by a simple random walk. I will discuss joint work with Angelo and Soundararajan which obtains new upper bounds for the number of such characters, by studying a related problem on maxima of L-values (closely connected to the Fyodorov–Hiary-Keating conjectures). I will give a heuristic explanation for why such characters are plentiful, which suggests that our upper bounds may not be too far from the truth.

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