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 : 09 Jan 2026 16:30
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Kuang Yang, Guangdong University of Technology

Complex Langevin (CL) is a computational method to circumvent the numerical sign problem with applications in finite-density quantum chromodynamics and the real-time dynamics of quantum field theories. It has long been known that, depending on the simulated system, CL does not always converge correctly. We study the regularization of the complex Langevin method via augmenting the action with a stabilization term. We also show that, in certain situations, this regularization can introduce biases, and we propose several extensions to mitigate these effects and further enhance accuracy. Several numerical experiments in the lattice field theory are carried out to show the effectiveness of our approaches.

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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 : 05 Jan 2026 15:00
Venue : S17 #04-05 (Seminar Room 2)
Speaker : Qinglin Tang, Sichuan University

The discrete normalized gradient flow with semi-implicit time discretization (GFSI), introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674-1697], has become one of the most widely used numerical method for computing the ground states (GS) of Bose-Einstein condensates (BECs). In spite of strong numerical evidence of its energy dissipation and global convergence, a rigorous proof of these two properties for the GFSI remains an open challenge. In this talk, by reformulating the GFSI into an equivalent form in terms of the normalized numerical solution, we will rigorously establish the energy dissipation property of the algorithm. Based on this, we then establish its global convergence to a stationary state. Furthermore, revisiting the equivalent form in its continuous version, we will propose a more efficient semi-implicit scheme to compute the GS. Mass-preserving and energy-diminishing properties of the scheme will also be proved.

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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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Date/Time : 16 Dec 2025 14:00
Venue : S17 #05-12 (Seminar Room 4)
Speaker : Yongchun Li, The Chinese University of Hong Kong

The low-rank constrained optimization arises in various machine learning and operations research problems. It minimizes a linear objective function subject to multiple two-sided linear inequalities and a low-rank domain set. Although the low-rank constrained optimization is generally NP-hard, a viable approach is to convexify the domain set (i.e., replace the domain set with its convex hull), known as “partial convexification.” Partial convexification often leads to a tractable convex relaxation; however, its solution quality lacks theoretical guarantees. To fill this gap, (i) we establish the necessary and sufficient condition under which the partial convexification matches the original low-rank constrained optimization; and (ii) we derive an upper bound on the minimum rank among all the optimal solutions of the partial convexification and prove its tightness. To efficiently solve the partial convexification, we develop a column generation algorithm combined with a rank-reduction algorithm. This combination ensures that the output solution satisfies the theoretical guarantees. Finally, our numerical experiments validate the strength of the partial convexification and the effectiveness of the proposed algorithms.

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