Date/Time : 23 Aug 2024 10:00
Venue : S17 #06-11 (SR6)
Speaker : Yixuan Wang, Caltech

We drew inspiration from the Kolmogorov–Arnold representation theory, using a composition of sum of 1D functions to represent higher dimensional functions, and proposed KANs, which has learnable activation functions parameterized by splines. Compared with MLPs, it will have much better scaling laws and interpretability since we use a much smaller network with 1D functions, provided that the underlying function has such a smooth representation, and in practice we can make KANs arbitrarily wide and deep. We demonstrate the wide applicability of this paradigm by examples of function pitting, PDE solving (PINNs), and applications to scientific problems. I will also discuss the most up-to-date developments of KANs and present some theoretical understandings.

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Date/Time : 23 Aug 2024 16:00
Venue : S17 #04-05 (SR2)
Speaker : Yixuan Wang, Caltech

We provide a general framework that could be potentially amenable to the stability of a large class of problems with type I singularities in a weighted Sobolev space and use nonlinear heat and complex Ginzburg-Landau equations as examples, which are joint works with Tom Hou, Van Tien Nguyen, and Jiajie Chen. A generalized dynamic rescaling formulation is introduced, with modulation parameters capturing the spatial translation and rotation symmetries of the equation and novel additional modulation parameters perturbing the scaling symmetry. This new formulation provides enough degrees of freedom to impose vanishing conditions on the rescaled solution, completely eliminating the unstable and neutrally stable modes of the linearized operator around the blowup profile. We then establish the full stability of the blowup using weighted energy estimates, without relying on a topological argument or a spectrum analysis. The log correction for the blowup rate is automatically inferred by refined estimates of the modulation parameters.

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Date/Time : 23 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 : 22 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 : 22 Aug 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Taiwang Deng, Beijing Institute of Mathematical Sciences and Applications

In this talk, I will explain the construction of a functor from the category of perverse sheaves on quiver varieties of type A to the category of perverse sheaves on flag varieties.We expect this functor to be dual to a functor introduced by K.Y. Chan and K. D. Wang recently which goes from the category of Harish-Chandra modules of GL(n,C) to the category of finite-dimensional modules of a graded Hecke algebra of type A. Moreover, dual (partial) Bernstein-Zelevinsky derivatives can be realized on quiver varieties via Lusztig induction and the latter can be shown to relate to the (dual) translation functor through our functor. This talk is based on my previous work as well as ongoing work with Chang Huang, Qixian Zhao, and Bin Xu.

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Date/Time : 22 Aug 2024 17:00
Venue : S17 #05-12 (SR4)
Speaker : Chung Pang Mok, Soochow University

We justify (at least in a special case) the pseudorandomness of the vectors as constructed in the previous talk, by giving explicit discrepancy estimates. The main inputs are Erdos-Turan-Koksma type inequality for discrepancy estimation, and Bombieri-Weil bounds on certain kinds of exponential sums on elliptic curves over finite fields.

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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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