Date/Time : 12 Jan 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Wang Zhihan, Cornell University

Singularities arise naturally in geometric variational problems and evolution equations. A fundamental question is to understand the local structure of these singular sets, both in general settings and under generic assumptions. In this talk, I will discuss recent progress on the analysis of singularities for the area functional, focusing on structural results and generic regularity phenomena in minimal surfaces and mean curvature flow.

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Date/Time : 08 Jan 2026 15:00
Venue : S17 #04-06 (Seminar Room 1)
Speaker : Yang Lei, National University of Singapore

A major challenge in homogeneous dynamics is to quantify fundamental theorems in this area, including Margulis’s non-divergence theorem and Ratner’s equidistribution theorem. Thanks to the deep connection between homogeneous dynamics and number theory, results in this direction have important consequences in number theory. In recent years, new ideas and tools have been introduced to this area, which leads to exciting developments in the study of quantitative behavior of unipotent orbits and their variants. I will give a brief summary of recent developments and explain my works in this direction.

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Date/Time : 02 Jan 2026 10:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Wei Zhang, MIT

In their 1986 paper, Gross and Zagier proved a formula relating the height of Heegner point on an elliptic curve to the first derivatives of the L-function of that elliptic curve. Since then, the problem of generalizing this fundamental result to higher dimensional algebraic varieties has been of great interest. In this talk we will present some of the generalizations with relatively recent results, with an emphasis on Kudla’s program and the arithmetic Gan-Gross-Prasad conjecture.

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Date/Time : 24 Dec 2025 10:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Yi Zhou, Fudan University

We propose a new physical space approach to bilinear estimates for dispersive equations. We can use this method to give alternative proofs other than the Bourgain space method of low regularity well posedness for dispersive equations. We can also use this method to give alternative proofs of global existence with small initial data for nonlinear wave equations. Moreover, we get new results including the global existence for one dimensional periodic Schrodinger map flow and global well psodness in the critical Besov space of Skew mean curvature flow in high dimensions.

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Date/Time : 24 Dec 2025 11:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Hong Kiat Tan, University of California, Los Angeles

2 x 2 systems of hyperbolic conservation laws arise throughout the physical sciences, such as in isentropic gas dynamics (the p-system), vehicular traffic flow models, shallow water equations, and particle-laden flows equations. A fundamental question for such systems concerns the structural stability of Riemann solutions, that is, whether the qualitative structure of shock and rarefaction waves persists under perturbations of the flux function and initial data.

In this talk, I will present a proof of generic structural stability for Riemann solutions to 2 × 2 systems of hyperbolic conservation laws in one spatial dimension. Specifically, we show that for almost every left and right states, shocks and rarefaction solutions of the same wave type are preserved under perturbations of the flux functions and the initial data. The proof requires the standard assumptions of strict hyperbolicity and genuine nonlinearity, together with geometric conditions motivated by differential topology, such as the regular manifold assumption.

The proof proceeds in two stages. First, we establish that transversality of the wave curves (the Hugoniot loci and rarefaction curves) implies structural stability of the Riemann solution. Next, we invoke a variant of Thom’s parametric transversality theorem to show that this transversality condition holds generically in the left and right states. We conclude by discussing applications of the theorem to numerical analysis and approximation schemes for 2 × 2 systems of conservation laws.

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Date/Time : 22 Dec 2025 15:00
Venue : S17 #05-11 (Seminar Room 5)
Speaker : Xinwen Zhu, Stanford University

Let Sh_G be a Hodge type Shimura variety and let p be a prime such that G_{Q_p} is umramified. I will discuss the torsion vanishing type results for the Whitaker coefficients of depth zero part of cohomology of Sh_G as G(Q_p) representations. This is a joint work in progress with Ian Gleason and Xiangqian Yang. Title: Torsion vanishing of local Whittaker coefficients of cohomology of Shimura varieties. Abstract: Let Sh_G be a Hodge type Shimura variety and let p be a prime such that G_{Q_p} is umramified. I will discuss the torsion vanishing type results for the Whitaker coefficients of depth zero part of cohomology of Sh_G as G(Q_p) representations. This is a joint work in progress with Ian Gleason and Xiangqian Yang.

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