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 : 21 Aug 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Shahar Mendelson, Australian National University

Structure recovery is at the heart of modern Data Science. Roughly put, the goal in structure recovery problems is to identify (or at least approximate) an unknown object using limited, random information – e.g., a random sample of the unknown object.

As it happens, key questions on recovery are fundamental (open) problems in Asymptotic Geometric Analysis and High Dimensional Probability. In this talk I will give one example (out of many) that exhibits the rather surprising ties between those seemingly unrelated areas.

I will explain why noise-free recovery is dictated by the geometry of the natural function class extension of a random projection of a subset of 𝑅^𝑑 : for a class of functions 𝐹 and n i.i.d random variables 𝜎 = (𝑋_1,…,𝑋_n), the corresponding coordinate projection of 𝐹 is the set 𝑃_𝜎 (𝐹) = { (𝑓(𝑋_1),….,𝑓(𝑋_n)) : 𝑓 ∈ 𝐹 } ⊂ 𝑅^n.

I will outline a (sharp) estimate on the structure of a typical 𝑃_𝜎 (𝐹) that leads to the solution of the noise-free recovery problem under minimal assumptions. I will explain why the same estimate resolves various questions in high dimensional probability (e.g., the smallest singular values of certain random matrices) and high dimensional geometry (e.g., the Gelfand width of a convex body).

The optimality of the solution is implied by a exposing a “hidden extremal structure” contained in 𝑃_𝜎 (𝐹), which in turn is based on a complete answer to Talagrand’s celebrated entropy problem.

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Date/Time : 21 Aug 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Chung Pang Mok, Soochow University

Using the arithmetic of elliptic curves over finite fields, we present an algorithm for the efficient generation of uniform pseudorandom vectors in high dimensions. We will also describe some numerical applications of the pseudorandom vector construction, including numerical integration in high dimensions and the numerical evaluation of Feynman-Kac formulas.

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Date/Time : 21 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Vo Ngoc Thieu, National University of Singapore

The main aim of this talk is to introduce our recent results on computational problems related to DAEs, including the effective differential Nullstellensatz, effective differential elimination, and finding general solutions of low-order algebraic ODEs. The effective differential Nullstellensatz involves finding a positive integer N for a given DAE system, such that one can check the consistency of the system by performing N differentiations and polynomial eliminations. Differential elimination involves removing independent variables from a DAE system. Differential Nullstellensatz and elimination are two fundamental problems in differential algebra and differential algebraic geometry. Since the number N represents the computational complexity of the effective differential Nullstellensatz and elimination, finding an upper bound for N is crucial. We will present our recent investigations into the problem of determining an upper bound for N. In addition, our results on the problem of determining algebraic/rational general solutions of first-order algebraic ODEs, as well as their connection with the Poincaré problem, will also be presented.

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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 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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