Date/Time : 14 Aug 2024 16:00
Venue : S17 #06-11 (SR6)
Speaker : Felix Schremmer, The University of Hong Kong

This talk presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne-Lusztig varieties (ADLV). The primary objective is to investigate the nonemptiness pattern, dimension and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a mathematical proof of a newly identified problem concerning a certain lower bound of dimension. This is joint work with Bin Dong, Xuhua He, Pengfei Jin and Qingchao Yu, and the corresponding paper can be found on arXiv and the Peking Mathematical Journal.

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Date/Time : 12 Aug 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Daniel Bartl, University of Vienna

A major challenge in modern optimization problems is their instability, where small data perturbations can have disproportionately significant effects. We address this challenge in two settings. First, we focus on a distributionally robust approach, a popular method to counteract instability generically using optimal transport. Specifically, we demonstrate how this approach can systematically quantify the (in-)stability of a given optimization problem. In the second part, we focus on statistical robustness, dealing with data subject to outliers, such as those caused by adversarial corruption. We design robust estimators for optimization problems and risk measures.

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Date/Time : 08 Aug 2024 15:00
Venue : S17 #05-12 (SR4)
Speaker : Tirasan Khandhawit, Mahidol University

Seiberg-Witten invariants and monopole Floer homology have been important tools in the study of 3-manifolds and 4-manifolds. In this talk, I will give an overview of the stable homotopy refinement of these invariants. In particular, I will describe Bauer-Furuta invariants of 4-manifolds and Seiberg-Witten-Floer spectra of 3-manifolds. I will also discuss some applications of these refinements.

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Date/Time : 08 Aug 2024 13:45
Venue : S17 #06-11 (SR6)
Speaker : Binbin Xu, Nankai University

One generalization of flat structure on torus to oriented closed surface of genus greater than 1 is called translation surface structure. When a surface is equipped with such a structure, we call it a translation surface. Given a translation surface S, like on a flat torus, there is a globally well-defined notion of direction on S, and geodesics on S along a same direction are disjoint from each other, hence induces a flow on S. Such flows and their dynamical properties are important in the study of the symmetry group of S which is called Veech group. In particular, when the flow on S along one direction has only periodic regular orbits, we call this direction a periodic direction of S. In a joint-work with Anja Randecker, using the close relation between interval exchange transformations and flows on surfaces, we obtain a full classification of translation surfaces of genus 2 with one singularity according to the size of their periodic directions sets. In this talk, I will first give some necessary background, then I will explain our result and give the rough idea of the proof.

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Date/Time : 07 Aug 2024 17:00
Venue : S17 #04-04 (SR3)
Speaker : Zhang Jing, University of Toronto

We expose an organizing framework to study higher dimensional infinitary combinatorics based on \v{C}ech cohomology, originating from works by Barry Mitchell, Barbara Osofsky and others. Key combinatorial notions include n-coherence and n-triviality for sequences of functions. We will use some recent vanishing and non-vanishing results to demonstrate “aleph_n is incompact for (n+1)-dimensional combinatorics” and “aleph_{omega+1} can be compact for n-dimensional combinatorics for all n”. Time permitting, we will also discuss the possibility of generalizing classical 2-dimensional properties like being special or being Suslin to higher dimensions. The talk will be purely combinatorial. Joint work with Jeffrey Bergfalk and Chris Lambie-Hanson.

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Date/Time : 07 Aug 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Yongnam Lee, KAIST, Korea

In this talk, I will discuss the positivity of the tangent bundle of smooth projective surfaces. The first part of the talk will be focused on the classification of non-ruled surfaces with pseudo-effective tangent bundles, which is a joint work with Jia Jia and Goulei Zhong. The second part will be focused on weak del Pezzo surfaces with big tangent bundles. The bigness of the tangent bundle is expected to be determined through the configuration of (−2)-curves. Our approach is to produce irreducible effective divisors on the projectivized tangent bundle, called the total dual VMRT, associated to a conic fibration on weak del Pezzo surfaces.

This part is a joint work with Hosung Kim and Jeong-Seop Kim.

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Date/Time : 01 Aug 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Armin Schikorra, University of Pittsburgh

n-Laplace systems with antisymmetric potential are known to govern geometric equations such as n-harmonic maps between manifolds and generalized prescribed H-surface equations. Due to the nonlinearity of the leading order n-Laplace and the criticality of the equation they are very difficult to treat.

I will discuss some progress we obtained, combining stability methods by Iwaniec and nonlinear potential theory for vectorial equations by Kuusi-Mingione.

Joint work with Dorian Martino

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Date/Time : 31 Jul 2024 14:00
Venue : S17 #04-05 (SR2)
Speaker : Jan M. Swart, The Czech Academy of Sciences

In a classical game two players, Alice and Bob, take turns to play n moves each. Alice starts. For each move each player has two options, 1 and 2. The outcome is determined by the exact sequences of moves played by each player. Prior to the game, a winner is assigned to each of the 2^{2n} possible outcomes in an i.i.d. fashion, where p is the probability that Bob is the winner for a given outcome. Then it is known that there exists a value 0<p_c<1 such that the probability that Bob has a winning strategy for large n tends to one if p>p_c and to zero if p<p_c. We study a modification of this game for which the outcome is determined by the exact sequence of moves played by Alice as before, but in the case of Bob all that matters is how often he has played move 1. We show that also in this case, there exists a sharp threshold p’_c that determines which player has with large probability a winning strategy in the limit as n tends to infinity.

Joint work with Anja Sturm and Natalia Cardona-Tob\’on (G\”ottingen).

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Date/Time : 31 Jul 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : George Barmpalias, Chinese Academy of Sciences

I will discuss current challenges and progress in algorithmic randomness, focusing on Chaitin’s halting probability, almost everywhere domination and measures of relative randomness. I will offer conjectures, partial results and benchmark problems toward solving the main questions.

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