Date/Time : 26 Mar 2025 16:30
Venue : S17 #06-11 (SR6)
Speaker : Wang Chushan, National University of Singapore

The nonlinear Schrödinger equation (NLSE) arises from various applications in quantum physics and chemistry, nonlinear optics, plasma physics, Bose-Einstein Condensates, etc. In these applications, it is necessary to incorporate low-regularity or singular potential and nonlinearity into the NLSE. Typical examples of such potential and nonlinearity include the discontinuous square-well potential, the singular Coulomb potential, the non-integer power nonlinearity, and the logarithmic nonlinearity. Such low regularity and singularity pose significant challenges in the analysis of standard numerical methods and the development of accurate, efficient, and structure-preserving schemes.

In this talk, I will introduce several new analysis techniques to establish optimal error bounds for some widely used numerical methods under optimally weak regularity assumptions. Based on the analysis, we also propose novel temporal and spatial discretizations to handle the low regularity and singularity more effectively.

Add to calendar :

Date/Time : 25 Mar 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Jiuzu Hong, University of North Carolina at Chapel Hill

Global Schubert varieties of parahoric Bruhat-Tits group schemes were introduced by Xinwen Zhu in his proof of the coherence conjecture of Pappas-Rapoport. These global Schubert varieties provide flat families relating spherical Schubert varieties in affine Grassmannians and other Schubert varieties or their unions in partial affine flag varieties. In a series of talks, I will give an introduction to this topic and discuss several applications based on my own research.

In the first talk, I will cover the basics of affine Grassmannian and Schubert varieties.
In the second talk, I will discuss parahoric group schemes and flag varieties of loop groups.
In the last talk, I will introduce the global Schubert varieties of parahoric Bruhat-Titis group schemes and discuss some of their applications.

Add to calendar :

Date/Time : 24 Mar 2025 14:00
Venue : S17 #05-11 (SR5)
Speaker : Wei-Hsuan Yu, National Central University

We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main results, we determine the exact size of maximal binary codes with two distances for all lengths n≥6 as well as the exact size of maximal binary constant weight codes with 2,3, and 4 distances for several values of the weight and for all but small lengths.

Add to calendar :

Date/Time : 24 Mar 2025 15:00
Venue : S17 #05-11 (SR5)
Speaker : Chia-En Chiang, National Taiwan Normal University

An almost k-cover of the hypercube Q^n = {0,1}^n is a collection of hyperplanes that avoids the origin and covers every other vertex at least k times. When k is large with respect to the dimension n, Clifton and Huang asymptotically determined the minimum possible size of an almost k-cover. Central to their proof was an extension of the LYM inequality, concerning a weighted count of hyperplanes.

In this paper we completely characterise the hyperplanes of maximum weight, showing that there are $\binom{2n-1}{n}$ such hyperplanes. We further provide stability, bounding the weight of all hyperplanes that are not of maximum weight. These results allow us to effectively shrink the search space when using integer linear programming to construct small covers, and as a result we are able to determine the exact minimum size of an almost k-cover of Q^6 for most values of k. We further use the stability result to improve the Clifton–Huang lower bound for infinitely many choices of k in every sufficiently large dimension n.

Add to calendar :

Date/Time : 19 Mar 2025 14:00
Venue : S17 #04-04 (SR3)
Speaker : Viet-Anh Nguyen, University de Lille

In 1992 Demailly raised the problem of developing a theory of intersection for positive closed currents. During the period 2009-2012, Dinh and Sibony gave two remarkable answers to this problem via their two new theories, namely, the super-potential theory and the theory of tangent currents. We focus more on the latter theory. More concretely, given m (≥2) positive closed currents T_i of bidegree (p_i,p_i) for 1≤i≤m on a compact Kahler manifold X, under theoretically reasonable assumptions, Dinh-Sibony can define the intersection T1 ⋏ . . . ⋏ Tm in the sense of the theory of tangent currents. In continuation of Dinh-Sibony’s pioneering works, we address/study, in this talk, the following basic questions:

Problem 1 (Existence of the intersection). Find an effective sufficient condition so that m(≥ 2) positive closed currents T_i of bidegree (p_i,p_i) for 1≤ i≤ m on a compact Kahler manifold X are wedgeable in the sense of the theory of tangent currents, that is, the current T1 ⋏ . . . ⋏ Tm exists.

Problem 2 (Continuity of the intersection). For m≥ 2 and for 1≤i≤m let (T_{i,n})_{n\in\N} be a sequence of positive closed currents of bidegree (p_i,p_i) on a compact Kahler manifold X so that \lim_{n\to\infty} T_{i,n} exists in the sense of currents. Find an effective sufficient condition imposed on these $m$ sequences so that the following equality holds:

\lim_{n\to\infty} ( T_{1,n}⋏ …⋏ T_{m,n}) =(\lim_{n\to\infty} T_{1,n})⋏…… ⋏\lim_{n\to\infty} T_{m,n})

Our approach relies on a new tool: the generalized Lelong numbers.

Key Words: positive closed/pluriharmonic/plurisubharmonic current, horizontal dimension, tangent currents, Lelong-Jensen formula, generalized Lelong numbers.

Add to calendar :

Date/Time : 19 Mar 2025 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jiajie Chen, New York University

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is a long-standing open problem in mathematical fluid mechanics. In this talk, I will begin by providing an overview of this problem and then introduce a framework for stable, nearly self-similar blowup, developed in joint work with Tom Hou (Caltech). Using this framework, we establish singularity formation in the incompressible Euler equations with smooth initial data and boundary. Additionally, I will briefly present recent results on constructing singularities in fluids and related equations using the self-similar blowup method.

Add to calendar :

Date/Time : 19 Mar 2025 10:00
Venue : S17 #05-11 (SR5)
Speaker : Edmund Karasiewicz, National University of Singapore

In this mini course we will study two topics. The first is the PGL2xF4 exceptional theta correspondence over a p-adic field. The second is the dual group of a spherical variety. Over the course of four lectures we will:

1. Introduce exceptional groups, focusing on constructions and tools relevant to the exceptional theta correspondence.
2. Describe the PGL2xF4 exceptional theta correspondence over a p-adic field.
3. Present the combinatorial construction of the dual group of a spherical variety due to Knop-Schalke.
4. Connect the exceptional theta correspondence to the F4/Spin9 spherical variety.

Join Zoom Meeting
https://nus-sg.zoom.us/j/84884850855?pwd=bvokLMShAf5wFeAHh2zemmel1dvb9f.1

Meeting ID: 848 8485 0855
Passcode: 226224

Add to calendar :

Date/Time : 19 Mar 2025 15:00
Venue : S17 #04-06 (SR1)
Speaker : Robert Lazarsfeld, Stony Brook University

Recall that two algebraic varieties of the same dimension n are birationally equivalent if they contain isomorphic Zariski-open subsets. In particular, a variety is rational if it is birational to projective space. In the past few decades there has been a great deal of interest and progress in the study of rationality, but most varieties aren’t rational. I will survey a body of recent work going in a complementary direction, namely: how can one measure and control “how irrational” a given variety might be? More generally, how can one quantify the “birational distance” between two varieties?

Add to calendar :

Date/Time : 17 Mar 2025 16:00
Venue : S16 #04-35
Speaker : Jiajie Chen , New York University

While it is known that a pre-shock or implosion singularity can form in finite time for the compressible Euler equations from smooth initial data, whether the vorticity blows up in finite time has remained an open problem. In this talk, we will explore the Euler equations with axisymmetry. Using (nearly) axisymmetric flow, we construct vorticity blowup in the compressible Euler equations in R^d, d >=2 with smooth, localized, and non-vacuous initial data. The vorticity blowup occurs at the time of the first singularity and is accompanied by a (nearly) axisymmetric implosion, in which the swirl velocity exhibits full stability as opposed to finite co-dimension stability.

Add to calendar :