Date/Time : 09 Nov 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Debsoumya Chakraborti, Institute for Basic Science

Abstract:  There have been extensive studies on the following question: given k graphs G_1,… , G_k over a common vertex set of size n, what conditions on G_i ensure a rainbow copy of H, i.e., a copy of H with at most one edge from each G_i? In this talk, we will survey some results on this topic.

Joos and Kim showed that for k=n, the condition that each G_i has minimum degree at least n/2 confirms a rainbow Hamiltonian cycle. This generalizes the classical Dirac’s theorem. In this talk, we will discuss our recent rainbow generalizations of some results on embedding Hamiltonian cycles in tournaments. This part of the talk will be based on a joint work with Jaehoon Kim, Hyunwoo Lee, and Jaehyeon Seo.

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Date/Time : 09 Nov 2022 17:15
Venue : S17 #04-05 (SR2)
Speaker : Benjamin T Castle, Ben Gurion University of the Negev

Abstract: Motivated by recent progress toward Zilber’s Restricted Trichotomy Conjecture, we study reducts of the complex field up to interdefinability over parameters. Precisely, we will consider structures of the form (C, P_1,…,P_n), where C is set of complex numbers and the P_i are polynomial maps of potentially different arities.   Somewhat surprisingly, our main result states that almost all such structures (in a precise sense) are interdefinable. The proof uses a mix of tools from geometric stability theory, combinatorics, and algebraic geometry.

This is joint work with Chieu-Minh Tran.

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Date/Time : 04 Nov 2022 15:00
Venue : via Zoom
Speaker : Ting-Kam Leonard Wong, University of Toronto

Join Zoom Meeting
https://nus-sg.zoom.us/j/83007497441?pwd=cmJmSEpuTWRUVHhJc3JJZ2h3dFNZdz09

Meeting ID: 830 0749 7441
Passcode: 087943

Information geometry is the geometric study of statistical manifolds which are spaces of probability or nonnegative measures. It originated from statistical inference and has been applied in diverse fields including statistical physics and machine learning. This short course is an introduction to information geometry and some of its recent developments. Mathematical prerequisites such as differential geometry and convex analysis will be introduced as needed. In the first part, we will introduce statistical manifolds, divergence functionals and the induced dualistic geometry. In the second part, we will discuss some relations between information geometry and optimal transport.

Further update : Information and updates about this course (including current and future Zoom links) will also be available on the webpage https://ujangangopadhyay.github.io/information_geometry.html

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Date/Time : 04 Nov 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Li Te, National University of Singapore

Hydrodynamic stability at high Reynolds number is a central topic in fluid mechanics, which is related to turbulence. The stability of general flows and their transition are challenging problems. We develop a systematic method to establish some sharp resolvent estimates for the linearized operator around some typical shear flows and rotating flows. One of the main difficulties is that the linearized operator is non-self-adjoint and non-local such that the spectral analysis becomes very complicated. Based on the resolvent estimates, we first show the sharp enhanced dissipation rate of the solution for the linearized system. We also derive the space-time estimates of the nonlinear part using the resolvent estimates. Combining this, we obtain the nonlinear transition threshold of some typical shear flows and rotating flows.

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Date/Time : 04 Nov 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Fang Xinyu, National University of Singapore

Abstract: A conjecture by Cohen and Lenstra (1984) predicts a distribution for the class group of a random quadratic field. This was one of the first motivating examples for the probability theory of random groups that arise in number theory. In this talk we will give a brief overview of the theory, and introduce some major results concerning the moment problem as well as universality. Then we will focus on some universality results for random abelian groups that arise as cokernels of random matrices, possibly with additional structure (such as being symmetric, Hermitian or alternating). We show that a method used to compute a formula for the distribution of cokernels of symmetric matrices also works in the case of Hermitian ones, when they are drawn from additive Haar measure.

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Date/Time : 03 Nov 2022 10:00
Venue : S17 #05-11 (SR5)
Speaker : Michael Choi, National University of Singapore

Join Zoom Meeting
https://nus-sg.zoom.us/j/9775606531?pwd=cVZ6eTRYS3ZzVUlFNXRuSlhMVjJSUT09

Meeting ID: 977 560 6531
Passcode: 123321

This talk centers around a technique that we call landscape modification. The core idea is that the Hamiltonian function is suitably modified in a way for rapid mixing while maintaining proximity with the original target distribution. We first present model-independent results that give rapid mixing and tunneling in the low-temperature regime. Building upon these results, we investigate the effect of landscape modification on some prototypical statistical physics models including the Ising model on various deterministic and random graphs as well as Derrida’s random energy model. This talk highlights a novel use of the geometry and structure of the landscape, in particular the global minimum value or its estimate, to the design of accelerated samplers or optimizers.

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Date/Time : 02 Nov 2022 17:00
Venue : S17 #04-05 (SR2)
Speaker : Wu Guohua, Nanyang Technological University

Abstract: Many constructions in rings involve applications of various axioms, which guarantee the existence of wanted objects. In this talk, I will present examples of such constructions, and show how these axioms are applied and really needed.

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Date/Time : 02 Nov 2022 16:30
Venue : via Zoom
Speaker : Ting-Kam Leonard Wong, University of Toronto

Join Zoom Meeting
https://nus-sg.zoom.us/j/83007497441?pwd=cmJmSEpuTWRUVHhJc3JJZ2h3dFNZdz09

Meeting ID: 830 0749 7441
Passcode: 087943

Information geometry is the geometric study of statistical manifolds which are spaces of probability or nonnegative measures. It originated from statistical inference and has been applied in diverse fields including statistical physics and machine learning. This short course is an introduction to information geometry and some of its recent developments. Mathematical prerequisites such as differential geometry and convex analysis will be introduced as needed. In the first part, we will introduce statistical manifolds, divergence functionals and the induced dualistic geometry. In the second part, we will discuss some relations between information geometry and optimal transport.

Further update : Information and updates about this course (including current and future Zoom links) will also be available on the webpage https://ujangangopadhyay.github.io/information_geometry.html

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Date/Time : 02 Nov 2022 14:00
Venue : via Zoom
Speaker : Zilin Jiang, Arizona State University

Join Zoom Meeting
https://nus-sg.zoom.us/j/81639604748?pwd=UXZXTnhwZHRmQTFrZEs0VEtWYjRVUT09

Meeting ID: 816 3960 4748
Passcode: 23571113

A set of unit vectors in a Euclidean space is called a spherical two-distance set if the pairwise inner products of these vectors assume only two values α and β. It is known that the maximum size of a spherical two-distance grows quadratically as the dimension of the Euclidean space grows. However when the values α and β are held fixed, a very intricate behavior of the maximum size emerges. Building on our recent resolution in the equiangular case, that is β = -α, we make a plausible conjecture which connects this behavior with spectral theory of signed graphs in the regime β < 0 ≤ α, and we confirm this conjecture when α + 2β < 0 or (1 – α)/(β-α) < 2.0198. Joint work with Alexandr Polyanskii, Jonathan Tidor, Yuan Yao, Shengtong Zhang and Yufei Zhao.

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