Date/Time : 21 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 : 21 Aug 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Yang Tianchi, National University of Singapore

The Turan-type problem is fundamental in extremal graph theory, guiding much of the field’s progress. In this talk, we extend these classic results to hypergraphs, focusing on bipartite structures like 4-cycles and complete bipartite graphs $K_{s,t}$. Our result highlights a rare instance in hypergraph Turan problems where the solution depends on the parity of the uniformity.

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Date/Time : 21 Aug 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Van Tien Nguyen, National Taiwan University

Advanced mathematical techniques have been developed in the last two decades for precise construction of blowup solutions within the class of finite energy. However, the question of a complete classification of blowup dynamics remains open. In this talk, we consider the energy-critical nonlinear heat equation in $\mathbb{R}^3$  as a model example to illustrate our technique to archive such a blowup classification. It consists of two parts: 1) a review of different techniques for constructing finite-energy blowup solutions, and 2) a rigidity result of the finite-energy blowup.  

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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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