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 : 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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Date/Time : 13 Mar 2024 16:30
Venue : S17 #05-11 (SR5)
Speaker : Lin Bo, National University of Singapore

The committor function is a central object for quantifying the transitions between metastable states of dynamical systems. Recently, a number of computational methods based on deep neural networks have been developed for computing the high-dimensional committor function. The success of the methods relies on sampling adequate data for the transition, which still is a challenging task for complex systems at low temperatures. This talk will present a deep learning method with two novel adaptive sampling schemes. In the two schemes, the data are generated actively from a modified potential where the bias potential is constructed from the learned committor function. We theoretically show that the data in the two schemes are uniformly distributed along the transition tube. The efficiency of the method is illustrated on high-dimensional systems including the alanine dipeptide and a dimer system immersed in the water box.

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Date/Time : 13 Mar 2024 16:00
Venue : S16 #03-09
Speaker : Hampus Nyberg, KTH Stockholm

Deep zero problems consider holomorphic functions satisfying local properties at a small number of points. We take a look at one such problem about uniqueness in the Bargmann-Fock space and show that it reduces down to a question about products such as $\frac{\sin{1}}{\sin{2}}\frac{\sin{3}}{\sin{4}}\frac{\sin{5}}{\sin{6}}…$.

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Date/Time : 07 Mar 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Ziyang Gao, Leibniz University Hannover

The goal of this talk is to propose a possible framework to study quadratic relations among holomorphic periods of CM abelian varieties. We define, for each Shimura variety and a CM point, a bi-$\overline{\mathbb{Q}}$-structure on the tangent space. Then we explain that in the case of the Siegel moduli variety, the numbers comparing the two $\overline{\mathbb{Q}}$-structures are precisely the products of the holomorphic periods of the CM abelian varieties parametrized by the CM point (up to $2\pi i$). Next we propose a hyperbolic analytic subspace conjecture, which is the analogue of Wüstholz’s analytic subgroup theorem in this context, and explain why it implies the desired consequence on the quadratic relations among these holomorphic CM periods. This is joint work with Emmanuel Ullmo and (partially with) Andrei Yafaev.

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