Date/Time : 08 Mar 2023 16:00
Venue : S17 #04-05 (SR2)
Speaker : Jurgen Jost, Max Planck Institute for Mathematics in the Sciences

Hypergraphs naturally emerge as models in many fields, ranging from chemical reaction systems to collaboration networks. Their mathematical theory, however, was underdeveloped until recently. I shall introduce some new ideas and concepts that we have developed, describe some of the interesting new mathematical developments and show some novel phenomena in pattern formation.

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Date/Time : 03 Mar 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Jun-Muk Hwang, Institute for Basic Science, Korea

When an analytic equation has a formal power series solution, does it have a convergent power series solution?

It has been well-known that this question has an affirmative answer if we work in local setting. The question is much subtler if we work in global setting. A typical example is Grauert’s formal prinicple for compact complex submanifolds in complex manifolds. We give an introductory overview of the results on the formal principle for rational curves in complex manifolds.

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Date/Time : 02 Mar 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Wu Yifei, Tianjin University

In recent years, more and more attention has been paid to the low regularity numerical study based on the practical needs. In this talk, some Fourier integrators are proposed for solving the KdV equation and the nonlinear Schrodinger equation. The designation of the scheme is based on the exponential-type integration, Splitting methods and the Phase-Space analysis of the nonlinear dynamics. By the rigorous analysis, the new schemes provide the first-order or second-order accuracy in Sobolev spaces for rough data, and reduce the regularity requirement of existing methods so far for optimal convergence. Moreover, the conservation laws of the numerical solutions are considered.

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Date/Time : 02 Mar 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Luo Yi, National University of Singapore

In a recent work by Tasho Kaletha, local langlands conjectures were extended to a certain class of disconnected groups. Moreover, a local Langlands correspondence for disconnected tori was established, where the main ingredient is Tate-Nakayama duality in Galois hypercohomology. In this talk, we review and reformulate Kaletha’s construction. Using these local data, we show an automorphic multiplicity formula for disconnected tori over a number field.

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Date/Time : 01 Mar 2023 15:00
Venue : via Zoom
Speaker : Shukun Wu, California Institute of Technology

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https://nus-sg.zoom.us/j/81046462775?pwd=aDhaYnhQOTgybzdYS21yajlQdnBmZz09

Meeting ID: 810 4646 2775
Passcode: 23571113

The Kakeya conjecture suggests a fundamental phenomenon in Euclidean space: (thin) tubes with different directions are essentially disjoint. The problem also arises naturally in other contexts, (e.g. Fourier analysis, geometric measure theory), and it often serves as a main enemy—tool. In this talk, I will briefly survey the conjecture and some techniques people developed to attack it. Along the way, I will also discuss some recent progress and applications.

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Date/Time : 01 Mar 2023 17:00
Venue : via Zoom
Speaker : Linus Richter, Victoria University of Wellington

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https://nus-sg.zoom.us/j/83049258042?pwd=UWViaWNvTFUrdFdhOHJCdEVydnVkdz09

Meeting ID: 830 4925 8042
Passcode: 1729=x3+y3

A recent “point-to-set principle” of Jack Lutz and Neil Lutz characterises the Hausdorff dimension of any subset of Euclidean space in terms of the Kolmogorov complexity of its individual points.
Sets with particular fractal properties can now be constructed point-by-point, by coding “enough” information into each point, bit-by-bit.

After introducing the point-to-set principle, I will present a new result in fractal geometry: under V=L (the axiom of constructiblity), I will outline the construction of co-analytic sets of Euclidean space which fail Marstrand’s Projection Theorem, a classical result in fractal geometry concerning the dimension of orthogonal projections of analytic plane sets onto lines.

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Date/Time : 23 Feb 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Vikas Krishnamurthy, Indian Institute of Technology Hyderabad

Point vortices are solutions of the Euler equation with a singular (point) distribution of vorticity. Point vortex equilibria have been studied in connection with various experimentally observed equilibrium configurations, and are also related to areas of mathematical physics. In this talk we will first discuss the background and importance of studying point vortex equilibria. We then describe a large class of new exact solutions of the two-dimensional steady incompressible Euler equation, called Liouville chains. They consist of point vortices embedded in a non-constant background vorticity field, which is described by the Liouville PDE, such that the whole arrangement is stationary. Liouville chains can be constructed as iterated solutions starting from a simple purely point vortex equilibrium (without a background field). Each iteration forms one link in the chain and can terminate after one step, a finite number of steps, or go on indefinitely. We describe several examples illustrating all these properties.

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Date/Time : 22 Feb 2023 15:00
Venue : S17 #04-05 (SR2)
Speaker : Tan Yan Fu, Vincent, National University of Singapore

Join Zoom Meeting : https://nus-sg.zoom.us/s/9839098564

In this seminar, I will discuss two works of mine post-tenure, both in the study of multi-armed bandits (MABs), a popular model in online decision making.

In the first problem, we consider the cascading bandits model in the regret minimization setting. There are two major classes of learning algorithms for minimizing the regret in MABs, namely the upper confidence bound (UCB) and Thompson sampling (TS) algorithms. On one hand, the analysis of TS usually involves additional analytical challenges compared to UCB. On the other hand, the empirical performance of TS is usually better than UCB. Consequently, for MAB problems, it is desirable to derive a theoretical guarantee for the performance of TS, but this was unavailable for the cascading bandits model prior to our work. We prove a theoretical performance guarantee for a TS-based algorithm TS-Cascade that matches that of UCB.

In the second problem, we consider the best arm identification problem in linear MABs in the so-called fixed-horizon setting. By leveraging properties of the G-optimal design and incorporating it into the arm allocation rule, we design a parameter-free algorithm, Optimal Design-based Linear Best Arm Identification (OD-LinBAI). We provide a theoretical analysis of the failure probability of OD-LinBAI. Instead of all the optimality gaps, the performance of OD-LinBAI depends only on the gaps of the top d arms, where d is the effective dimension of the linear bandit instance. Complementarily, we present a minimax lower bound for this problem. The upper and lower bounds show that OD-LinBAI is minimax optimal up to constant multiplicative factors in the exponent, which is a significant improvement over existing methods (e.g., BayesGap, Peace, LinearExploration, and GSE), and settles the question of ascertaining the difficulty of learning the best arm in the fixed-horizon setting.

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Date/Time : 21 Feb 2023 16:15
Venue : S17 #04-06 (SR1)
Speaker : Eliana Fausti, University of Oxford

Nonlinear filtering consists in finding the best estimate for the true value of a system when only incomplete, noisy observations are available. Since the filtering equations are often very high dimensional, and not tractable from a practical perspective, many applications must rely on approximation methods. Understanding the error of several such approximations remains an open problem. In this talk we will focus on the case where the underlying process is a continuous-time finite-state Markov chain. Leveraging tools from information geometry, we will show how to obtain a contraction result for the filtering equations, strengthening now-classical results on the stability of filters. This allows us to characterize the error of approximate filters, and to construct a low-dimensional geometric approximation with a well-behaved, hopefully small, error.  (Joint work with Sam Cohen, Oxford).

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