Date/Time : 29 Aug 2024 17:10
Venue : S17 #05-12 (SR4)
Speaker : Ma Jia Jun, Xiamen University and Xiamen University Malaysia

The speaker will share his personal experiences with Lean through this presentation. As undergraduate final year projects at Xiamen University Malaysia Campus, we have successfully formalized key concepts in game theory, including the Second-Price Auction and von Neumann’s Minimax Theorem. In collaboration with students from Xiamen University and the National University of Singapore, we have also made significant progress in formalizing results related to Lie algebras, Coxeter groups, and Kazhdan-Lusztig polynomials. The speaker will reflect on how formalization can deepen our understanding of mathematical concepts and its potential in teaching and research.

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Date/Time : 29 Aug 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Minsung Kim, Centro di Ricerca Matematica Ennio De Giorgi

The deviation of the Birkhoff integral for area-preserving flow on compact surfaces was first studied by Forni. He proved that the deviation spectrum was determined by the Lyapunov exponents of a renormalization cocycle so-called Kontsevitch-Zorich cocycle. This deviation result was later proved again by Bufetov and Frączek-Ulcigrai for translation flows and locally Hamiltonian flows for non-degenerate types. In this talk, we study the spectrum for deviations of Birkhoff integrals for locally Hamiltonian flows beyond the case of Forni where the observable vanishes at the singularities. Our new developments include a better understanding of the asymptotics at singularities (degenerate type) and the appearance of a new deviation spectrum. This is a joint work with Krzysztof Frączek.

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Date/Time : 29 Aug 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Zeng Haozhi , Huazhong University of Science and Technology

The Peterson variety is introduced by Dale Peterson to describe the quantum cohomology rings of all the partial flag varieties. In this talk, we discuss an explicit morphism from the Peterson variety to the Blume toric orbifold. As an application, we calculate the totally nonnegative part of the Peterson variety in Lie type A and give a proof of Rietsch’s conjecture. This is joint work with Hiraku Abe.

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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.

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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.

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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)

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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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