Date/Time : 08 Sep 2023 16:30
Venue : S17 #04-06 (SR1)
Speaker : Zhu Chengbo, National University of Singapore

I will explain how to count irreducible representations with a given infinitesimal character and a given bound of the complex associated variety. The main conceptual tool for the counting is coherent families of representations. For the category of Harish-Chandra modules, the concept was developed by Schmid, Zuckerman, Speh-Vogan, and Vogan. Putting together all coherent families (based on a translate of the weight lattice), one obtains a representation of a certain integral Weyl group. The key to the counting lies in understanding cell structures of this representation (called Harish-Chandra cells), and this is where Kazhdan-Lusztig theory, more specifically the theory of primitive ideals (as developed by Joseph and Barbasch-Vogan), double cells and special representations (in the sense of Lusztig), comes into play. The end result is a counting inequality on irreducible representations, which is shown to be an equality in important cases (e.g. all classical groups). This is joint work with Dan Barbasch, Jia-Jun Ma and Binyong Sun, and may be considered as Part I of our project to construct and classify special unipotent representations of all classical groups (with unitarity as a direct consequence).

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Date/Time : 08 Sep 2023 14:45
Venue : S17 #05-11 (SR5)
Speaker : Atsushi Ichino, Kyoto University

The Bertolini-Darmon-Prasanna formula is a p-adic analog of the Gross-Zagier formula, and relates the special value of the p-adic L-function with the p-adic Abel-Jacobi image of some cycle. In this formula, the local root number at infinity is assumed to be -1. In this talk, we discuss the case when this sign is +1. This is joint work in progress with Kartik Prasanna.

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Date/Time : 08 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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Date/Time : 07 Sep 2023 15:30
Venue : S17 #05-11 (SR5)
Speaker : Wei Darren, National University of Singapore

Two flows are said to be Kakutani equivalent if one is isomorphic to the other after time change, or equivalently if there are Poincare sections for the flows so that the respective induced maps are isomorphic to each other. Ratner showed that if G=SL(2,R) and \Gamma is a lattice in G, and if U is a one parameter unipotent subgroup in G then the U action on G equipped with Haar measure is loosely Bernoulli, i.e. Kakutani equivalent to a circle rotation. Thus any two such systems SL(2,R)/\Gamma are Kakutani equivalent to each other. On the other hand, Ratner showed that if G’=SL(2,R) x SL(2,R) and \Gamma’ is a reducible lattice, and U’ is the diagonally embedded one parameter unipotent subgroup in G’, then (G’/\Gamma’, U’, m) is not loosely Bernoulli. We show that in fact in this case and many other situations one cannot have Kakutani equivalence between such systems unless they are actually isomorphic. This is a joint work with Elon Lindenstrauss.

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Date/Time : 07 Sep 2023 10:00 - 12:00
Venue : I4-01-03 (Innovation 4.0 Building)
Speaker : Jean Bernard Lasserre, Laboratory for Analysis and Architecture of Systems

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Date/Time : 06 Sep 2023 13:00
Venue : S17 #05-11 (SR5)
Speaker : Yunkun Zhou, Stanford University

Degeneracy plays an important role in understanding Turán- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a k-uniform hypergraph as the degeneracy of its 1-skeleton (i.e., the graph formed by replacing every k-edge by a k-clique). We prove that skeletal degeneracy controls hypergraph Turán and Ramsey numbers in a similar manner to (graphical) degeneracy.

Specifically, we show that k-uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erdős conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Turán number of a k-uniform k-partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process’ introduced by Lee in his resolution of the Burr-Erdős conjecture.

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Date/Time : 05 Sep 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Sun Mengzhou,

Compared with end extensions, much little is known about cofinal extensions for models of fragments of PA. In particular, it is not even known known whether the elementarity of cofinal extensions can fail at some specific level. In this talk, I will try to give a complete answer. I will present a systemetic way to `compress’ the truth of M into the second-order structure of a definable cut, and as a consequence, a correspondence theorem between the first-order theory of M and the second-order theory of the cut. Through this method I will construct several models with special cofinal extension properties.

I will also show that every countable model of arithmetic fail to satisfy PA admits a non-elementary cofinal extension. It provides a model-theoretic characterization for PA in terms of cofinal extensions.

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Date/Time : 31 Aug 2023 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ofer Zeitouni, Weizmann Institute and NYU Courant

The extreme value theory for Gaussian logarithmically correlated fields has emerged in the last decade as a powerful tool in the analysis of interface models, quantum gravity, random matrices and in a myriad of other applications.

The two dimensional Gaussian free field (and its discrete analogue) is an important motivating example of such a field. In this lecture, I will describe the relation and differences between the extreme value theory for i.i.d. variables and that for Gaussian logarithmically correlated fields, and discuss a sample of non-Gaussian examples.

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Date/Time : 30 Aug 2023 13:00
Venue : S17 #05-11 (SR5)
Speaker : Max Wenqiang Xu, Stanford University

Multiplicative energy of a set A is defined to be the total number of solutions to the equation: $m_1m_2=n_1n_2$ where the m_i’s and n_i’s are in the set A.

The first application is to use this quantity to study the product set of set A. Yunkun Zhou and I managed to solve some conjectures of Elekes-Ruzsa by studying this quantity.

The second application is to use this quantity to prove central limit theorems for a random model of arithmetic functions. The random model nowadays is called random multiplicative functions. This part is mainly based on joint work with K. Soundararajan.

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