Date/Time : 28 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Linus Richter, National University of Singapore

I will outline a few connections between various notions of definability (which vary in degree of logical formality), give examples, and describe some open questions at the intersection of logic and classical mathematics.

Add to calendar :

Date/Time : 28 Aug 2024 14:00
Venue : S17 #05-12 (SR4)
Speaker : Lim Jeck, California Institute of Technology

We show that if l_1, … , l_k are algebraic numbers, then, for any finite subset A of ℂ,

|A + l_1·A + … + l_k·A| >= H(l_1,…,l_k)|A|-o(|A|),

where H(l_1,…,l_k) is an explicit constant that is best possible. The case k=1 already confirms a conjecture of Krachun and Petrov. Joint work with David Conlon.

Add to calendar :

Date/Time : 28 Aug 2024 16:30
Venue : S17 #04-05 (SR2)
Speaker : Zhang Xiaolong, Nanyang Technological University & Hunan Normal University

The Logarithmic Schrödinger Equation (LogSE) exhibits rich dynamics and possesses unique properties compared to the standard Schrödinger equation with cubic nonlinearity. However, the logarithmic nonlinear term $f(u)=uln(∣u∣^2)$ presents significant challenges for both numerical solutions and error analysis, primarily due to low regularity and singularity. In this talk, we will characterize this low regularity within suitable fractional Sobolev spaces and derive the first set of error estimates for the time-splitting Fourier spectral methods with initial data of fractional-order regularity. These results are supported by recent regularity estimates for the LogSE from the PDE community and ample numerical evidence.(Joint work with Prof. Li-Lian Wang, NTU)

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :

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.

Add to calendar :