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 14:00
Venue : S17 #04-06 (SR1)
Speaker : Andreas Søjmark, London School of Economics

The lectures will be split into two parts (2 lectures of 2 hours each). In the first part, we will take a detailed look at the notion of interacting particle systems, formulated in terms of coupled stochastic differential equations (SDEs). Provided there is a suitable structure to the interactions, it may be possible to understand the macro-behaviour of such systems through so-called functional limit theorems for stochastic processes, akin to the way in which we study sequences of i.i.d. observations through the central limit theorem. Naturally, questions of convergence now become much more involved and the limiting objects, known as McKean—Vlasov SDEs, can have complicated nonlinearities. The main goal of the first part of the lectures is to gain a good understanding of the key intricacies pertaining to the convergence of interacting particle systems as the number of particles tends to infinity. In the second part of the lectures, we will then proceed to focus on the analysis of McKean—Vlasov systems, both in their own right and when a game component is added. In the latter case, we arrive at what is known as the theory of mean-field games. Throughout the lectures, the aim is to strike a good balance between theory and intuition, and we will maintain a strong focus on applications in finance and economics, especially relating to systemic risk modelling.

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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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Date/Time : 17 Feb 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Gabor Domokos, Budapest University of Technology and Economics

If we approximate natural fragments by convex polyhedra and count the respective numbers for faces, vertices and edges then, in most cases, we find averages remarkably close to 6,8,12, the values corresponding to the cube. Not only can this observation be translated into a simple Lemma about hyperplane convex mosaics, we also find that general convex mosaics may serve as particularly fitting models for natural fragmentation patterns. In 2D, this approach offers a complete, global catalog of natural tilings, ranging from atomic lattices to the geometry of tectonic plates. In 3D we can also build a similar catalog which suggests that the majority of natural fragments should indeed produce cubic averages. To verify our geophysical claims, we vetted field data from over 4000 natural fragments against computer simulations and found not only good agreement for the aforementioned cuboid averages, but we could also reproduce full distributions for many geophysical shape descriptors with remarkable accuracy.

The appearance of the cube (albeit in an averaged sense) in this context may remind one of Plato’s theory of the Element Earth. I will briefly comment on some purely geometric aspects of this connection.

This is joint work with Doug Jerolmack (U. Pennsylvania), Ferenc Kun (U. Debrecen) and János Török (Budapest U. of Technology and Economics).

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Date/Time : 16 Feb 2023 11:00
Venue : S17 #05-12 (SR4)
Speaker : Zeev Rudnick, Tel Aviv University

Selberg computed various statistics of the number of Riemann zeros up to a given height, obtaining what was later recognized to be the same answer as for the corresponding quantity for random unitary matrices with matching eigenvalue density. We examine a corresponding problem for the eigenvalues of the Laplacian on a “random” hyperbolic surface of large genus, and describe a new perspective from the theory of moduli spaces of Riemann surfaces, as developed by Mirzakhani. I will not assume familiarity with any of the notions above.

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Date/Time : 16 Feb 2023 16:00
Venue : S17 #04-06 (SR1)
Speaker : Javier Gómez-Serrano, Brown University

In this talk I will present results on singularity formation for the 3D isentropic compressible Euler and Navier-Stokes equations for ideal gases. These equations describe the motion of a compressible ideal gas, which is characterized by a parameter called the adiabatic constant. Finite time singularities for generic adiabatic constants were found in the recent breakthrough of Merle, Raphaël, Rodnianski and Szeftel. Our results allow us to drop the genericity assumption and construct smooth self-similar profiles for all values of the adiabatic constant. In particular, we will construct the first smooth self-similar profile for a monoatomic gas. Part of the proof is very delicate and requires a computer-assisted analysis. Joint work with Tristan Buckmaster and Gonzalo Cao-Labora.

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Date/Time : 16 Feb 2023 16:00
Venue : S16 #05-98
Speaker : Xia Kelin, Nanyang Technological University

Artificial intelligence (AI) based molecular data analysis has begun to gain momentum due to the great advancement in experimental data, computational power and learning models. However, a major issue that remains for all AI-based learning models is the efficient molecular representations and featurization. Here we propose advanced mathematics-based molecular representations and featurization (or feature engineering). Molecular structures and their interactions are represented as various simplicial complexes (Rips complex, Neighborhood complex, Dowker complex, and Hom-complex), hypergraphs, and Tor-algebra-based models. Molecular descriptors are systematically generated from various persistent invariants, including persistent homology, persistent Ricci curvature, persistent spectral, and persistent Tor-algebra. These features are combined with machine learning and deep learning models, including random forest, CNN, RNN, Transformer, BERT, and others. They have demonstrated great advantage over traditional models in drug design and material informatics.

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Date/Time : 16 Feb 2023 14:00
Venue : S17 #04-04 (SR3)
Speaker : Andreas Søjmark, London School of Economics

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https://nus-sg.zoom.us/j/86831047750?pwd=cjB6cmsvMmtpcm8zVTNvM0E4S09Ndz09

Meeting ID: 868 3104 7750
Passcode: 582832

In this talk, I will introduce a succinct one-dimensional mean-field model for the evolution of an epidemic over time. Each individual in a large population starts from some initial level of ‘shielding’, given by a nonnegative number, and this level then evolves over time according to diffusive dynamics driven by independent Brownian motions. If the level of shielding becomes too low, individuals may find themselves in ‘at-risk’ situations, captured by proximity to a lower moving boundary called the epidemic front. Specifically, the likelihood of infection depends not only on the intrinsic infectiousness of the disease but also on the local time accumulated along this epidemic front and on the current proportion of infected individuals. I will start the talk by giving a heuristic description of this modelling framework, highlighting how it compares to the classical SIR model. Next, I will discuss some of the key steps in the rigorous construction of the corresponding particle system, and I will then proceed to show how, in the mean-field limit, the particle system gives rise to a weak probabilistic formulation of a nonlinear moving boundary problem on the real line. Finally, I will end the talk by discussing how this approach may also be relevant for the modelling of financial contagion in large financial systems. The talk is based on joint work with Eliana Fausti (Oxford).

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