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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Date/Time : 29 Aug 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Tran Chieu Minh, National University of Singapore

In a recent work, we show that if A is an open subset of SO(3,R) with sufficiently small normalized Haar measure, then mu(A^2) > 3.99 mu(A).

This was conjectured by Emmanuel Breuillard and Ben Green around 15 years ago in the context of getting optimal bounds and finding continuous counterparts of product theorems by Helfgott, Pyber-Szabo, and Breuillard-Green-Tao. The result is also related to the Brunn-Minkowski inequalities from convex geometry, the Kunze-Stein phenomenon from harmonic analysis, and the Pillay conjectures from model theory.

In this talk, I will explain these connections and discuss some ideas from the proof, which uses nonstandard analysis and neostable group theory.

(The talk is based on joint work with Yifan Jing and Ruixiang Zhang)

URL: http://www.comp.nus.edu.sg/~fstephan/logicseminar.html

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Date/Time : 28 Aug 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Gao Ziyang, Leibniz Universitat Hannover

It is a fundamental question in mathematics to find rational solutions to a given system of polynomials, and in modern language this question translates into finding rational points in algebraic varieties. This question is already very deep for algebraic curves defined over Q. An intrinsic natural number associated with the curve, called its genus, plays an important role in studying the rational points on the curve. In 1983, Faltings proved the famous Mordell Conjecture (proposed in 1922), which asserts that any curve of genus at least 2 has only finitely many rational points. Thus the problem for curves of genus at least 2 can be divided into several grades: finiteness, bound, uniform bound, effectiveness. An answer to each grade requires a better understanding of the distribution of the rational points.

In my talk, I will explain the historical and recent developments of this problem according to the different grades. Another important topic on studying points on curves is the torsion packets. This topic goes beyond rational points. I will also discuss briefly about it in my talk.

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Date/Time : 25 Aug 2023 14:00
Venue : S17 #04-04 (SR3)
Speaker : Zhang Lei, National University of Singapore

In this talk, I will report our progress on the joint project with Chen Wan (Rutgers University—Newark). In this project, we investigate a family of strongly tempered spherical varieties and conjecture that they share analogous global and local properties with well-known GGP models. Furthermore, over local fields, we give a general epsilon dichotomy conjecture for strongly tempered spherical subgroups without type N root.

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Date/Time : 24 Aug 2023 14:00
Venue : S17 #04-06 (SR1) (Colloquium (joint with IMS))
Speaker : Yujiro Kawamata, University of Tokyo

We will describe infinitesimal deformations of complex manifolds to the direction of something having possibly non-commutative structure sheaves by using Hochschild cohomology. We will also describe global non-commutative deformations of some surfaces.

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Date/Time : 23 Aug 2023 15:00
Venue : S17 #05-12 (SR4)
Speaker : William Johnson, Fudan University

Model theory is the study of “definable sets” in mathematical structures like posets, graphs, groups, rings, and fields. In some structures, the definable sets are very tame and well-behaved, while in other structures they are wild. Many of the model-theoretically tame structures are “finite dimensional”, in the sense that there is a good dimension theory for definable sets. There are several model-theoretic ways to define “dimension”. In this talk, we will mostly be interested in Morley rank and dp-rank. These notions of “dimension” or “rank” are intuitively analogous to notions like topological dimension and Hausdorff dimension in geometry, but they are defined purely based on the combinatorics of definable sets.

The driving question of this talk is the following: Which structures from algebra (e.g., groups, rings, and fields) are “finite-dimensional” (e.g., finite Morley rank, finite dp-rank)? In this talk, we will survey the classic work of Macintyre on fields and rings of finite Morley rank, the conjecture of Cherlin and Zilber on
groups of finite Morley rank, and my own work in recent years on fields and rings of finite dp-rank, including a recent unpublished result, joint with Yatir Halevi and Christian d’Elbee on rings of dp-rank 1.

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Date/Time : 23 Aug 2023 13:00
Venue : S17 #04-04 (SR3)
Speaker : Lim Jeck, California Institute of Technology

An old problem in discrete geometry, originating with Kupitz, asks whether there is a fixed natural number k such that every finite set of points in the plane has a line through at least two of its points where the number of points on either side of this line differ by at most k. We give a negative answer to a natural variant of this problem, showing that for every natural number k there exists a finite set of points in the plane together with a pseudoline arrangement such that each pseudoline contains at least two points and there is a pseudoline through any pair of points where the number of points on either side of each pseudoline differ by at least k.

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Date/Time : 22 Aug 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Shahar Mendelson, Australian National University

Consider an isotropic measure \mu on R^d (i.e., centred and whose covariance is the identity) and let X_1,…,X_m be independent, selected according to \mu. If \Gamma=m^{-1/2} \sum_{i=1}^m <X_i,->e_i is the random operator whose rows are X_i/\sqrt{m}, how does the random set \Gamma S^{d-1} typically look like? For example, if the extremal singular values of \Gamma are close to 1 then \Gamma S^{d-1} is “well approximated” by a d-dimensional section of S^{m-1} and vice-versa. But is it possible to give a more accurate description of that set?

I will show that under minimal assumptions on \mu, with high probability and uniformly in t \in S^{d-1}, each vector \Gamma t inherits the structure of the one-dimensional marginal <X,t> in a strong sense. If time permits I will also present a few generalisations of this fact – for an arbitrary indexing set.

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Date/Time : 17 Aug 2023 10:30
Venue : S17 #05-11 (SR5)
Speaker : Sayan Das, Columbia University

Sticky Brownian Motions are a family of correlated Brownian motions that have a tendency to stick together. They can be realized as the scaling limit of random walks in random environments and can also be viewed as random motions in a continuum-random environment. In this talk, I will present some results related to the quenched density of the motion of a particle in this continuum-random environment. In particular, I will show that under moderate deviation regime (a particular regime in between large deviation regime and diffusive regime), this quenched density after rescaling weakly converges to the solution of the stochastic heat equation with multiplicative space-time white noise. I will explain how our results shed light on the extreme behavior of Sticky Brownian Motions. This is a joint work with Hindy Drillick and Shalin Parekh.

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