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.

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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.

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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.

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

Meeting ID: 848 8485 0855
Passcode: 226224

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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?

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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.

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Date/Time : 14 Mar 2025 11:00
Venue : S17 #04-04 (SR3)
Speaker : Wang Yutong, National University of Singapore

In the previous sections, we have introduced some recent LLM tools in Lean 4, especially the ones for auto-formalization and auto-proving. In this lecture, I will show you the basic concepts and techniques behind these tools, and how you can use them to improve your efficiency in both formal and informal research. I will also try to introduce to you some recent progress in the field of LLM reasoning in mathematics.

Zoom link: https://nus-sg.zoom.us/j/84932999912?pwd=tSka7Th1z2McUW3zhjquuUQ7hcLfab.1
Zoom ID: 849 3299 9912
Pwd: 475364

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Date/Time : 13 Mar 2025 14:00
Venue : S17 #06-11 (SR6)
Speaker : Didac Martinez Granado, University of Luxembourg

In geometric group theory, one typically studies whether a given group G admits particular geometric structures—well-behaved actions on real trees, measured wall structures, or admissible random walks—all of which give rise to (invariant) metrics on G. This mini course adopts a broader perspective: rather than focusing on each structure in isolation, we consider *all* possible metrics on G at once and investigate the global geometry of this “space of metrics.” Introduced by Furman in the early 2000s, it has recently undergone an intense study by Reyes, Cantrell-Tanaka, Cantrell-Reyes, and myself with collaborators, among others.

When G is a surface group, a motivating analogy comes from the Teichmüller space of a surface, which parametrizes all marked hyperbolic metrics and is a finite-dimensional ball. However, the space of metrics of a surface group is infinitely dimensional, incorporating diverse constructions such as hyperbolic metrics, word-metrics, geodesic currents and Green metrics. We will describe some of these metrics, focusing on geodesic currents as a motivating example. Another avenue to obtain metrics on G from measures is from certain “admissible” random walks with respect to probability measures on G. These induce “Green metrics” which naturally sit in this space. We will introduce and analyze a natural distance on the space of metrics, discussing in particular how both word-metrics and those arising from admissible random walks are dense (Cantrell-MG-Reyes).

We believe this course could be of interest to people interested in hyperbolic geometry, geometric group theory, dynamics, or probability.

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Date/Time : 12 Mar 2025 15:00
Venue : S17 #05-12 (SR4)
Speaker : Svetlana Katok, Pennsylvania State Unviersity

Given a closed, orientable surface of constant negative curvature and genus g ≥ 2, we study a family of generalized Bowen–Series boundary maps and their two dynamical invariants: the topological entropy and the measure-theoretic entropy with respect to their smooth invariant measure. Each such map is defined for a particular fundamental polygon and a particular multi-parameter. We prove two strikingly different results: rigidity of topological entropy and flexibility of measure-theoretic entropy. The topological entropy is constant in this family and depends only on the genus of the surface. We give an explicit formula for this entropy and show that it stays constant both within our parameter space and within the Teichmüller space of the surface. We obtain an explicit formula for the measure-theoretic entropy for the smooth invariant probability measure on the circle that only depends on the genus of the surface and the perimeter of the (8g−4)-sided fundamental polygon, and prove that it varies in the Teichmüller space and takes all values between 0 and maximum that is achieved on the surface which admits a regular fundamental (8g-4)-gon, and stays constant on a subset of the parameter space. The rigidity proof uses conjugation to maps of constant slope, while the flexibility proof – valid only for certain multi-parameters – uses the realization of the geodesic flow on the surface as a special flow over the natural extension of the boundary map, and the Reduction Theory for maps associated to Fuchsian groups.

In particular, we show that the smooth invariant probability measure is not the measure of maximal entropy (the Parry measure) for the considered boundary maps.

This is joint work with Adam Abrams and Ilie Ugarcovici.

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Date/Time : 12 Mar 2025 16:30
Venue : S17 #05-12 (SR4)
Speaker : Zhang Junyu, National University of Singapore

Mirror descent (MD) and its stochastic variants are well-understood first order methods in convex optimization under appropriate smoothness conditions and have been extended to general relative smooth and nonconvex problems in recent years. Despite being widely studied, there are still many issues in the existing research.

In this talk, we will first reveal the inappropriateness of the currently widely adopted optimality measures, which casts doubts about the validity of current complexity results, for both deterministic and stochastic MD. Second, due to the absence of Lipschitz continuity, most existing results on nonconvex stochastic mirror descent (SMD) do not allow popular acceleration techniques such as random shuffling and variance reduction, etc. To fix these issues, we propose a novel regularity condition that is widely satisfied by most of the popular kernels. Combined with the dual space perspective of mirror descent, a new optimality measure has been proposed and a Lipschitz-like gradient bound has been established. We elaborate how these concepts can help the analysis SMD by establishing new and improved complexity results for SMD with variance reduction.

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