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
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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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Date/Time : 08 Feb 2023 17:00
Venue : via Zoom
Speaker : Will Johnson, Fudan University

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

Is every NIP integral domain a henselian local ring? In this talk, I will give evidence for this conjecture, sketching why it holds in the positive characteristic case and the finite dp-rank case. I will also discuss how this “generalized henselianity conjecture” is related to the standard conjectures on NIP fields and valued fields. For example, the henselianity conjecture for NIP valued fields implies the generalized henselianity conjecture for Noetherian NIP integral domains. Lastly I will discuss how this conjecture fits into the general problem of classifying NIP fields and NIP commutative rings. For example, it implies that NIP fields with definable field topologies must be ample/large in the sense of Pop, and NIP commutative rings must be finite products of henselian local rings.

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Date/Time : 08 Feb 2023 15:30
Venue : S17 #04-04 (SR3)
Speaker : Fabrizio Bianchi, University of Lille

We show that every Hölder observable satisfies the Central Limit Theorem (CLT) with respect to the measure of maximal entropy of every complex Hénon map. The proof is based on a general criterion to ensure the validity of the CLT, valid for abstract measurable dynamical systems. This is a joint work with Tien-Cuong Dinh.

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Date/Time : 07 Feb 2023 10:00
Venue : S17 #04-06 (SR1)
Speaker : Zhang Zhe, Georgia Institute of Technology

The modern big data infrastructure has drastically transformed the computation landscape of operations research. Novel decision-making strategies are imperative for handling the sheer volume and the exceptional dimension of the data we are generating. To store and process such data, advanced computing networks, from cloud computing to edge computing, is ubiquitous. Nevertheless, the fundamental objective of operations research remains unchanged: to provide robust and optimal decision-making tools to ensure safe and consistent outcomes. Risk-averse optimization is an essential framework to achieve this goal. However, while distributed optimization without risk consideration has been thoroughly studied, our understanding of distributed risk-averse optimization is limited. In this talk, we will discuss our recent progress that helps to shed light on the subject. Specifically, for a distributed network with a computationally powerful central server, we introduce novel distributed optimization methods to solve risk-averse optimization with the same communication complexity as that of risk-neutral optimization. Extensions to the decentralized network setting and the fully stochastic setting will also be discussed.

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Date/Time : 02 Feb 2023 17:00
Venue : S17 #04-06 (SR1)
Speaker : Jan Sbierski, University of Edinburgh

The strong cosmic censorship conjecture roughly states that Einstein’s theory of general relativity is generically a deterministic theory. Using the language of PDEs this can be paraphrased as stating that generically global uniqueness holds for the initial value problem for the Einstein equations. I will start out this talk by giving a non-technical introduction to this conjecture restricted to the interior of rotating black holes. Moreover, I will discuss the analytic as well as the geometric aspects of the conjecture and outline its current state.

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