Date/Time : 20 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Jin Zeyu, Peking University

A promising approach to investigating high-dimensional problems is to identify their intrinsically low-dimensional features, which can be achieved through recently developed techniques for effective low-dimensional representation of functions such as machine learning. Based on available finite-dimensional approximate solution manifolds, we propose a novel model reduction framework for kinetic equations. The method employs projections onto tangent bundles of approximate manifolds, naturally resulting in first-order hyperbolic systems. Under certain conditions on the approximate manifolds, the reduced models preserve several crucial properties, including hyperbolicity, conservation laws, entropy dissipation, finite propagation speed, and linear stability. For the first time, we rigorously discuss the relation between the H-theorem of kinetic equations and the linear stability conditions of reduced systems, determining the choice of Riemannian metrics involved in the model reduction. The framework is widely applicable for the model reduction of many models in kinetic theory. Joint work with Ruo Li.

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Date/Time : 20 Mar 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Sun Mengzhou,

A general question in the model theory of arithmetic is: For each theories S, T and natural number n, is it true that every countable sufficiently saturated model of S has a proper n-elementary end extension to a model of a T? Efforts over the past four decades have revealed answers to this question for S and T in the induction-collection hierarchy IΣ_n, BΣ_n, except the following instance by Clote and Kaufmann: Is it true that, given any integer n, every countable model of BΣ_n+2 has a proper n-elementary end extension to a model of BΣ_n+1? We present a positive answer to the Kaufmann-Clote question. The proof consists of a second-order ultrapower construction based on a low basis theorem. We also include a survey on the results related to the general question above.

This is a joint work with Tin Lok Wong and Yue Yang.

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Date/Time : 20 Mar 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Fabrizio Bianchi, Università di Pisa

The theory of parabolic implosion – i.e., the study of perturbations of parabolic fixed points – has been at the centre of the research in one-dimensional holomorphic dynamics in the last several decades. I will discuss recent progress on the generalization of this theory in two complex variables, as well as some applications.

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Date/Time : 20 Mar 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Wang Shu, Beijing University of Technology

In this talk we discuss quasi-neutral limit problem of Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics and review the known main progress on rigorously establishing quasi-neutrality theory from the view of mathematics. Quasi-neutrality is one basic assumption in physics such as semiconductors and plasma, which is firstly proposed by W.Van Roosbroech in Bell System Tech. J., 1950. Firstly, we introduce some multi-scaling structure stability theory on quasi-neutral limit and boundary layer, initial layer and mixed layer problem for drift-diffusion models in semiconductors. Then, we give some recent progresses on well-posedness theory and quasi-neutral limit problem on Planck-Nernst-Poisson-Navier-Stokes equations for electro-hydrodynamics.

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Date/Time : 18 Mar 2024 16:00
Venue : S16 #03-04
Speaker : Stefan Junk, Gakushuin University

We consider the directed polymer model, which describes paths affected by a random space-time environment in spatial dimension d>2. The model undergoes a phase transition between a high-temperature, weak disorder phase and a low-temperature, strong disorder phase, which is characterized by whether the associated (normalized) partition function converges to zero. From the physical point of view, it is more natural to consider a very-strong-disorder regime, characterized by whether the partition function converges to zero exponentially fast. It has been a long-standing conjecture that these two notions are equivalent and we now give a proof of this. Moreover, our proof reveals that weak disorder holds at the critical value. Joint work with Hubert Lacoin.

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Date/Time : 14 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Le Nhat Hoang, National University of Singapore

The Gan-Gross-Prasad (GGP) conjecture studies a family of restriction problems for classical groups and proposes precise answers to these problems using the local and global Langlands correspondences. It also has a twisted variant in the Fourier-Jacobi case, which is called the twisted Gan-Gross-Prasad conjecture. In this talk, I will report my progress on the local twisted GGP conjecture for tempered representations. The strategy is to adapt Waldspurger and Beuzart-Plessis’s method to develop a local relative trace formula as well as a twisted trace formula and compare their elliptic parts. Although the geometric sides of both trace formulae have not been developed, one can use a partial comparison and an instance of an elliptic L-packet to prove the statement. Moreover, I will propose a geometric multiplicity formula for the local twisted GGP and necessary ingredients to prove it.

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Date/Time : 13 Mar 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Willie Wong, Michigan State University

Most of us will agree that solving ordinary differential equations is easier than solving partial differential equations. As a consequence, many predictions of physical theories are obtained through considering regimes where the equations of motion reduce to a finite dimensional dynamical system. A specific example is in our understanding of the early universe, where the prevalent paradigm is based in part on cosmological big bang solutions in general relativity. This cosmological ansatz where the universe is assumed to be spatially homogeneous is often justified physically by the “working assumption” that when viewed from the largest scales, the inhomogeneities (stars, planets, galaxies) average out, and the matter content of the universe can be approximated by a compressible fluid. In a joint work with Shih-Fang Yeh, we probed whether this “working assumption” is justified, from a mathematical point of view. Our results show that on the cosmological timescale, certain cosmological big bang solutions are susceptible to instabilities generated through nonlinear self-interactions of the constituent inhomogeneous matter. The goal of this talk is to present the mathematical context of this result and briefly describe the instability mechanism involved, focusing on the relevance of the conformal (or causal) geometry of the big bang solution.

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Date/Time : 13 Mar 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Taoran He, National University of Singapore

In this talk, we will report and propose a new angular momentum memory effect of gravitational waves along null infinity based on recent progress in proving stability of Minkowski spacetime with minimal decays and nonlinear stability of Kerr black holes with small angular momentum. This accompanies Christodoulou’s nonlinear displacement memory effect and the spin memory effect. The connections and differences to these effects are also addressed. This is a joint work with Xinliang An and Dawei Shen.

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Date/Time : 13 Mar 2024 15:30
Venue : S17 #05-12 (SR4)
Speaker : Yasushi Yamashita, Chuo University

The Riley slice of Schottky space is a family of Kleinian groups generated by two parabolic elements. It was studied in detail by Keen and Series. In this talk, we consider families of Kleinian groups generated by two elliptic elements, and show some computer generated pictures related to the groups.

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