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.

Add to calendar :

Date/Time : 01 Nov 2022 16:00
Venue : via Zoom
Speaker : Piyush Srivastava, Tata Institute of Fundamental Research

Join Zoom Meeting
https://nus-sg.zoom.us/j/83199111482?pwd=L09Yd054dDNTbDBpRkNtc0tmQnk5Zz09

Meeting ID: 831 9911 1482
Passcode: 569991

Abstract: This series of lectures will survey various applications to algorithms of a collection of techniques that are broadly referred to as “geometry of polynomials”. Specific examples include:

1) Real stable polynomials and constructions of Ramanujan expanders
2) Lee-Yang theory and deterministic “approximate counting” algorithms”
3) Lorentzian or “strongly log concave” polynomials and applications to sampling

Add to calendar :

Date/Time : 31 Oct 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

Add to calendar :

Date/Time : 28 Oct 2022 15:00
Venue : S17 #04-06 (SR1)
Speaker : Ting-Kam Leonard Wong, University of Toronto

Abstract: Macroscopic properties of equity markets present interesting problems in portfolio management and stochastic modeling. If we think of stocks as stars in a galaxy, a macroscopic property would be how the galaxy spirals. We begin by presenting some empirical observations about the capital distribution curve, market diversity and intrinsic volatility, and discuss why these macroscopic properties are difficult to capture probabilistically. One way to exploit these properties is to use functionally generated portfolios in stochastic portfolio theory.

Financially, they capture market volatility by systematic rebalancing. Mathematically, they are generalized gradient fields of exponentially concave functions, i.e., functions whose exponentials are concave. We show that these portfolios solve an optimal transport problem whose cost function is related to the Dirichlet distribution. Finally, motivated by online portfolio selection, we consider construction of prior (probability) distributions on spaces of concave functions and present some interesting limit theorems.

Add to calendar :

Date/Time : 28 Oct 2022 16:00
Venue : S17 #04-06 (SR1)
Speaker : Gu Yejun, Institute of High Performance Computing, A*STAR

Abstract:  The mechanical properties of metallic materials are strongly dependent on their microstructural features (e.g. morphology and distribution of dislocations, grains, and other defects). Thus it necessitates a comprehensive understanding of the microstructure evolutions in materails during deformation. I will present some attempts of applying mathematical methods to address this problem. The presentation consists of two parts: one is the fast algorithms for solving partial differential equations, in order to precisely describe the microstructure evolutions; the other is the probabilistic description and statistical analysis, which are employed to quantify the relationship between the microstructures and mechanical properties.

Add to calendar :

Date/Time : 28 Oct 2022 14:00
Venue : S17 #05-12 (SR4)
Speaker : Koh Shiwei Isaac, National University of Singapore

Abstract:   The concept of cellularity, defined by Graham and Lehrer, has proven extremely useful in the fundamental problem of classifying the irreducible representations of a given finite-dimensional algebra. By definition, a cellular algebra comes with a finite chain of ideals whose subquotients, called cells, are isomorphic to a matrix ring over a Noetherian domain k with a twisted multiplication. Examples of cellular algebras include many finite-dimensional Hecke algebras and Temperely Lieb algebras.

Koenig and Xi have generalised this concept to algebras over a Noetherian domain k of not necessarily finite dimension, by introducing the notion of an affine cellular algebra. We keep the same idea of having a filtration by a finite chain of ideals, but the ideals are now allowed to be of infinite dimension. A cell is now allowed to be isomorphic to a matrix ring (of finite rank) over a quotient of a polynomial ring over k, with a twisted multiplication. An example of an affine cellular algebra is the extended affine Hecke algebra of type A.

In this talk, we will introduce the aforementioned concepts and discuss some possible work to be done in the future

Add to calendar :

Date/Time : 27 Oct 2022 15:00
Venue : via Zoom
Speaker : Ting-Kam Leonard Wong, University of Toronto

Join Zoom Meeting
https://nus-sg.zoom.us/j/81251294825?pwd=aVNqdEs4R0J0c3lMTnRDVmxOWGlZZz09

Meeting ID: 812 5129 4825
Passcode: 898472

Abstract: 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

Add to calendar :

Date/Time : 27 Oct 2022 15:30
Venue : S17-05-11 (SR 5)
Speaker : Takashi Kishimoto, Saitama University

Abstract: We address the following questions: ”(1) How to construct a Mori fiber space (Mfs) X containing the affine space Ank as an open subset ?” and ”(2) In which Mfs as in (1), the translations on Ank can be extended to automorphisms on X, namely X can be an equivariant completion of the vector group Gna,k ?” As for (1), we will present a certain machinery to construct Mfs over P1k, which contains the affine space, with birationally rigid Fano varieties being general fibers. Note however that such completions X are never equivariant ones, more precisely, the unipotent radical of Aut0(X) is at most of dimension one. Actually, it turns out that there are few equivariant completions. Then we will focus on forms defined over any field of characteristic zero of quintic del Pezzo varieties. The main result lies in clarifying which ones among them can be equivariant completions and determining their automorphism groups. The talk is based on two joint works, one of which is with Adrien Dubouloz and Karol Palka, and the other of which is with Adrien Dubouloz and Pedro Montero.

Add to calendar :

Date/Time : 26 Oct 2022 14:00
Venue : S17 #04-06 (SR1)
Speaker : Cai Zhenning, National University of Singapore

Abstract: The gas kinetic theory is usually employed to solve rarefied gas dynamics. The fundamental mathematical model for the gas kinetic theory is the Boltzmann equation, which describes the fluid states using the distribution function defined over the joint position-velocity space. Due to its high dimensionality and a complicated integral operator, straightforward discretization of the Boltzmann equation will lead to high computational cost. Therefore, we need either highly effective numerical schemes or reduced kinetic models to make simulations feasible. In this talk, I will introduce our improvement on the classical Fourier method to preserve all physical properties including the conservation laws and the entropy dissipation. Also, we will discuss methods for kinetic model reductions including the simplification of the collision operator and the moment methods. Our simplified collision operator includes a parameter that can be adjusted to balance the computational cost and the accuracy, and this parameter can be tuned adaptively during the simulation of the dynamics. As for the moment methods, we study the linear Boltzmann equation and introduce the regularized 13-moment equations with stable boundary conditions, which can be applied to solve slow-moving gases with moderate rarefaction effects.

Add to calendar :