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.

Add to calendar :

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).

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

Date/Time : 28 Aug 2025 16:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Zhang Zhen, Southern University of Science and Technology

We propose a novel phase-field model for N-phase problems by using a dichotomic representation. Derived from Onsager’s variational principle, the proposed model enjoys mechanic, energetic, algebraic and dynamic consistencies and is shown to asymptotically converge to its sharp interface limit. An operator-splitting based numerical method is developed to simulate the model with both second-order accuracy and energy stability. Various numerical results are presented to validate theoretical derivation and show interesting phenomena.

Add to calendar :

Date/Time : 28 Aug 2025 10:30
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Huy Tuan Pham, California Institute of Technology

Sumset is a fundamental object in additive combinatorics. Since Freiman’s celebrated theorem which characterizes the structure of integer sets with small sumsets, the structure theory of sets with small sumsets has gathered significant development, breakthroughs and applications.

In this talk, I will discuss new combinatorial and probabilistic perspectives on sumsets. Motivated from the study of random Cayley graphs, these perspectives provide new ways to quantify the structure and complexity of sets with small sumsets that are robust, flexible and quantitative. Emerging from this picture is the rich interaction between additive combinatorics and probabilistic combinatorics, which leads to new understanding of Cayley graphs and random Cayley graphs in general groups, as well as the resolution of longstanding questions in additive combinatorics

Add to calendar :

Date/Time : 27 Aug 2025 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Sun Hongpeng, Renmin University of China

This paper introduces a preconditioned convex splitting algorithm enhanced with line search techniques for nonconvex optimization problems. The algorithm utilizes second-order backward differentiation formulas (BDF) for the implicit and linear components and the Adams-Bashforth scheme for the nonlinear and explicit parts of the gradient flow in variational functions. The proposed algorithm, resembling a generalized difference-of-convex function approach, involves a changing set of convex functions in each iteration. It integrates the Armijo line search strategy to improve performance. The study also discusses classical preconditioners such as symmetric Gauss-Seidel, Jacobi, and Richardson within this context. The global convergence of the algorithm is established through the Kurdyka-Łojasiewicz properties, ensuring convergence within a finite number of preconditioned iterations. Numerical experiments demonstrate its high efficiency. This is a joint work with Xinhua Shen and Zaijiu Shang.

Add to calendar :

Date/Time : 27 Aug 2025 17:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Yang Yue, National University of Singapore

I will talk about Halpern-Läuchli Theorem (HL) and its proof(s). This is an ongoing project with Chong Chitat and Li Wei, aiming to find the proof-theoretic strength of HL. I will sketch a proof that under Σ3-induction, there is a computable solution for HL.

Add to calendar :