Date/Time : 20 Aug 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sung-Soo Byun and Seong-Mi Seo , Seoul National University and Chungnam University

This is a lecture series on Coulomb gases on Riemann surfaces. We will in particular focus on partition functions of two-dimensional Coulomb gases on Riemann surfaces and correlations of determinantal point processes on complex manifolds. We will discuss the macroscopic properties of Coulomb gases, which can be effectively described using logarithmic potential theory. A key topic in Coulomb gas theory is free energy expansion as the system size tends to infinity. This asymptotic expansion plays a crucial role in establishing the law of large numbers and central limit theorems, and it also finds applications in several large deviation probabilities, such as hole probabilities. Additionally, the coefficients in this expansion encapsulate potential theoretic, topological, and conformal geometric aspects of the model. Throughout these lectures, we will explore predictions made by the physics community, as well as recent advancements within the mathematics community, with a focus on the planar case. On the other hand, in random matrix theory, the eigenvalues of a normal matrix form a determinantal point process that can be described as a system of Coulomb gases on the complex plane. This lecture series will introduce a generalization of this random normal matrix ensemble to complex manifolds, while preserving its determinantal structure. We will explore the global and local behaviour of the particle system in the limit of many particles, with a focus on the scaling limits of correlation functions and their universality.

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Date/Time : 19 Aug 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Sung-Soo Byun and Seong-Mi Seo , Seoul National University and Chungnam University

This is a lecture series on Coulomb gases on Riemann surfaces. We will in particular focus on partition functions of two-dimensional Coulomb gases on Riemann surfaces and correlations of determinantal point processes on complex manifolds. We will discuss the macroscopic properties of Coulomb gases, which can be effectively described using logarithmic potential theory. A key topic in Coulomb gas theory is free energy expansion as the system size tends to infinity. This asymptotic expansion plays a crucial role in establishing the law of large numbers and central limit theorems, and it also finds applications in several large deviation probabilities, such as hole probabilities. Additionally, the coefficients in this expansion encapsulate potential theoretic, topological, and conformal geometric aspects of the model. Throughout these lectures, we will explore predictions made by the physics community, as well as recent advancements within the mathematics community, with a focus on the planar case. On the other hand, in random matrix theory, the eigenvalues of a normal matrix form a determinantal point process that can be described as a system of Coulomb gases on the complex plane. This lecture series will introduce a generalization of this random normal matrix ensemble to complex manifolds, while preserving its determinantal structure. We will explore the global and local behaviour of the particle system in the limit of many particles, with a focus on the scaling limits of correlation functions and their universality.

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Date/Time : 15 Aug 2024 17:00
Venue : S17 #05-12 (SR4)
Speaker : Qingchao Yu, The University of Hong Kang

Steinberg’s cross-section is a transversal slice of the regular unipotent orbit of connected reductive group. It has been generalized to unipotent orbits by the work of He-Lusztig and Duan, using Lusztig’s map and the notion of good position elements. In this talk, we first recall the theory of Steinberg’s cross-section and then
generalize it using the notion of convex elements of the Weyl group. We show that the U-conjugation property, which plays an essential role in the construction of He-Lusztig, holds for convex elements. Moreover, all good position elements are convex.

This is based on my recent joint work with Sian Nie and Panjun Tan.

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Date/Time : 15 Aug 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Felix Schremmer, The University of Hong Kong

If a Shimura variety is fully Hodge-Newton decomposable, its mod p reduction enjoys particularly nice properties. These cases are well-understood and characterized by a certain inequality known as the minute condition. In a joint project in progress with Xuhua He and Eva Viehmann, we study the next larger cases, where the minute condition is barely not satisfied any more. I will give a classification of these cases and explain the geometric properties of the corresponding Ekedahl-Oort and Kottwitz-Rapoport strata. If time permits, I will also talk about the local model in the case of Iwahori level structure, and where to find it in the affine flag variety.

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Date/Time : 14 Aug 2024 14:00
Venue : S17 #04-06 (SR1)
Speaker : Zhang Tengren, National University of Singapore

Anosov subgroups are a special class of discrete subgroups of a non-compact semisimple Lie group. This notion was first discovered by Labourie, who used it to prove the discreteness of Hitchin representations. In this talk, I will discuss my recent contributions to our understanding of Anosov subgroups and their various generalizations, as well as some applications to studying character varieties.

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Date/Time : 14 Aug 2024 14:00
Venue : S17 #04-04 (SR3)
Speaker : Jie Han, Beijing Institute of Technology

Random graphs are important objects in discrete probability theory. There has been a growing interest in extending graph theory theorems into the random graph setting. We will survey this line of problems and mention some recent developments.

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Date/Time : 14 Aug 2024 16:15
Venue : S17 #04-05 (SR2)
Speaker : Trung Tuyen Truong, University of Oslo

This talk presents a new equivalence and approach to the Riemann hypothesis, via a relatively new root finding method called Backtracking New Q-Newton’s method (BNQN), which is a variant of Newton’s method. Besides having strong global guarantee for finding roots of meromorphic functions, this new method also has connections to Voronoi’s diagrams, which is an important input for the new approach. This is joint work with Thuan Quang Tran (University of Oslo).  

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Date/Time : 14 Aug 2024 10:00
Venue : S17 #06-11 (SR6)
Speaker : Jinxin Xue, Tsinghua University

The mean curvature flow is to evolve a hypersurface in Euclidean space using the mean curvatures at each point as the velocity field. The flow has good smoothing property, but also develops singularities. The singularities are modeled on an object called shrinkers, which give homothetic solutions to the flows. As there are infinitely many shrinkers that seem impossible to classify, it is natural to explore the idea of generic mean curvature flows that is to introduce a generic perturbation of the initial conditions. In this talk, we shall explain our work on this topic, including perturbing away nonspherical and noncylindrical shrinkers, and generic isolatedness of cylindrical singularities. The talk is based on a series of works jointly with Ao Sun.

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Date/Time : 14 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Koh Heer Tern, Nanyang Technological University

We explore a recent program in comparing effective topological spaces up to (not necessarily computable) homeomorphism. We provide a survey of some earlier results separating some of the common notions of presentations for Polish spaces. By pushing existing definability techniques and effective versions of Stone duality, we also provide some new counterexamples. Most interestingly, that there exists a Polish space both left-c.e. and right-c.e. presentable but not computably presentable.

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