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 : 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
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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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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 : 15 Feb 2023 16:00
Venue : via Zoom
Speaker : Akshat Mudgal, University of Oxford

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Meeting ID: 810 4646 2775
Passcode: 23571113

Let d be a fixed positive integer and let A be a finite set of n elements in R^d. A classical result of Freiman then suggests that if A is not contained in a translate of some hyperplane, then

|A+A| >= (d+1)n – d(d+1)/2,

where A+A = {a + b : a,b in A}. This is known to be sharp by setting A to be a union of arithmetic progressions. Naturally, one may ask whether a corresponding result holds for the difference set A-A = { a – b : a,b in A}. This forms a conjecture of Uhrin from 1980, which was subsequently raised again by Ruzsa in 1994. While the d=1,2 and d=3 cases of this problem were solved by Freiman-Heppes-Uhrin (1989) and Stanchsecu (1998) respectively, the d> 3 case was open till quite recently. In this talk, we will survey some of the recent results in this direction, including our own work which shows that whenever d>3 and A is not lying in some affine hyperplane, we have

|A-A| >= (2d – 2 + 1/(d-1))n – O(n^{1-c}),

for some constant c = c(d) >0. The main term here is sharp, thus resolving the conjecture of Uhrin and Ruzsa up to a polynomially small error term.

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Date/Time : 15 Feb 2023 17:00
Venue : S17 #05-11 (SR5)
Speaker : David Belanger, National University of Singapore

We present a system of functionals that can serve as Skolem functions, so that any arithmetical formula can be rewritten in terms of them, in a model of BSigma_n + not ISigma_n. A distinguishing feature of our construction is that each functional is coded as a natural number within the model, and there is an upper bound on the codes. The method has a number of interesting applications.

This is joint work with Chong, Li, Wong and Yang.

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Date/Time : 14 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 : 10 Feb 2023 14:00
Venue : S17 #04-06 (SR1)
Speaker : Håkan Hedenmalm, KTH Royal Institute of Technology

Fourier series is a classical method to expand periodic functions on the line, with roots in the theory of heat conduction. If we want to expand non-periodic functions on the line, we are told to use the Fourier transform instead and accept that the result is a distribution. In 2011, Hedenmalm and Montes-Rodriguez suggested an alternative expansion. It now turns out that this expansion exists uniquely and that it connects with the Fourier interpolation of Radchenko and Viazovska (2019). We will explain how the hyperbolic Fourier series is very natural and corresponds to a “low temperature” expansion where automorphic forms and theta functions are examples from a high temperature regime where nonuniqueness holds.

This reports on joint work (with Viazovska, Radchenko, Bakan, Montes-Rodriguez).

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Date/Time : 09 Feb 2023 16:00
Venue : via Zoom
Speaker : Jing Yifan, University of Oxford

The celebrated Brunn-Minkowski inequality states that for compact subsets A and B of R^d, m(A+B)^{1/d} \geq m(A)^{1/d} + m(B)^{1/d}, where m() is the Lebesgue measure. It is a cornerstone in convex geometry and plays an important role in other areas of mathematics, particularly combinatorics. In this talk, I will present a generalization of the Brunn-Minkowski inequality in non-abelian groups. More refined phenomena for compact and discrete groups will also be discussed. If time permits I will mention some applications of these results.

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