Date/Time : 07 Feb 2024 16:00
Venue : S17 #05-11 (SR5)
Speaker : Didac Martinez-Granado, University of Luxembourg and National University of Singapore

We define a notion of hyperplane for a hyperbolic group G and study G-invariant measures on the space of hyperplanes of its Gromov boundary, which we call “co-geodesic currents´´. Co-geodesic currents are induced by many classical objects such as geodesic currents for surface groups, certain cocompact actions of hyperbolic groups on CAT(0) cube complexes, quasi-fuchsian representations on a surface group, etc. Moreover, there is a natural intersection pairing between co-geodesic currents and the more classical geodesic currents, introduced by Bonahon, i.e. G-invariant measures on the square of the boundary of G. This pairing generalizes Bonahon’s intersection number when G is a surface group. Furthermore, every co-geodesic current \mu induces natural dual pseudo-metric space, in the sense that the intersection of \mu with the current determined by the conjugacy class of an element g in G recovers the stable length of g in the pseudo-metric space. This is joint work in progress with Eduardo Reyes.

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Date/Time : 06 Feb 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Dong Bin, Peking University

This presentation delves into the recent advancements in applying artificial intelligence (AI) to mathematical exploration. Initially, we will outline the current state of integrating AI methodologies, including machine learning, deep learning, and neural networks, into mathematical research, thereby fostering enhanced approaches and innovative concepts. Subsequently, we will highlight some preliminary results from ongoing research in this intersecting domain. Concluding the discussion, we will anticipate the future trajectory of research at the intersection of AI and mathematics, focusing on the potential of AI techniques to tackle new challenges and uncover novel problems in mathematical studies.

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Date/Time : 06 Feb 2024 10:00
Venue : S17 #04-06 (SR1)
Speaker : Ian Gleason, Universitat Bonn

The local Langlands correspondence, proposed by Langlands as the non-abelian analogue of local class field theory for non-Archimedean local fields, has been established for various reductive groups over the years. However, a construction in full generality was lacking until the recent proposal by Fargues and Scholze, who introduced semi-simplified Langlands parameters for every reductive group. Their construction employs Scholze’s theory of diamonds, a modern approach to p-adic analytic geometry. This talk begins by revisiting historical developments in number theory that led to the formulation of the local Langlands correspondence. Subsequently, we explore the relevance of p-adic analytic geometry to the study of the local Langlands program and the impact these innovative methods have on our comprehension of the local Langlands correspondence.

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Date/Time : 05 Feb 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Xiaoping Wang, The Chinese University of Hong Kong, Shenzhen

Topology optimization, which aims to find the optimal physical structure that maximizes mechanical performance,is vital in engineering design applications in aerospace, mechanical, and civil engineering. We introduce a deep generative model, based on diffusion models, to address the structure optimization problem. Combine with the threshold dynamics method, we present a successful frame for the topology optimization.

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Date/Time : 02 Feb 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Dianbo Liu, National University of Singapore

Our team at National university of Singapore utilizes newest discovery from cognitive sciences especially consciousness related knowledge to inspire next generation of machine learning and apply the methods to understand biomedical sciences. In this talk we focus on using method inspired by compositional thoughts in human brain to discover causal relationships within Gene Regulatory Networks (GRNs). Causal discovery in GRNs is a challenging problem for multiple reasons including the existence of cyclic feedback loops and uncertainty that yields diverse possible causal structures. Previous works in this area either ignore cyclic dynamics (assume acyclic structure) or struggle with scalability. We introduce Swift-DynGFN as a novel framework that enhances causal structure learning in GRNs while addressing scalability concerns. Specifically, Swift-DynGFN exploits gene-wise independence to boost parallelization and to lower computational cost. Experiments on real single-cell RNA velocity and synthetic GRN datasets showcase the advancement in learning causal structure in GRNs and scalability in larger systems.

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Date/Time : 01 Feb 2024 15:00
Venue : S17 #04-06 (SR1)
Speaker : Zhang Jiacheng, University of California, Berkeley

In this talk, we are going to talk about recent development on Stackelberg equilibrium and Mean Field Game. Stackelberg game has attracted recent interest modeling an asymmetric, general sum game between leaders and followers.

Firstly, we consider a discrete-time Stackelberg mean field game (SMFG) with one leader and an infinite number of homogeneous followers via optimization lense.We focus on a robust setting where the followers have bounded rationality and only aim at achieving some ε-Nash equilibrium (NE) while the leader wants to maximize her objective in the worst case scenario among all ε-NEs. We study the sensitivity analysis when the leader only has access to some perturbed model. We show that the value function is unstable with respect to the perturbation of the model and the leader needs to be more pessimistic and solve a relaxed problem to obtain a near optimal solution. The analysis is done by formulating this SMFG problem into an explicit minimax optimization problem and studying the sensitivity property using the optimization framework.

Second, we investigate the existence of an optimal policy to monitor a mean field system of agents managing a risky project under moral hazard with accidents modeled by L ́evy processes magnified by the law of the project. We provide a general method to find both a mean field equilibrium for the agents and the optimal compensation policy under general, sufficient and necessary assumptions on all the parameters. We formalize the problem as a bilevel optimization with the probabilistic version of a mean field game which can be reduced to a controlled McKean-Vlasov SDE with jumps. We apply our results to an optimal energy demand-response problem with a crowd of consumers subjected to power cut/shortage when the variability of the energy consumption is too high under endogenous or exogenous strains. In this example, we get an explicit solution to the mean field game and to the McKean-Vlasov equation with jumps.

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Date/Time : 01 Feb 2024 10:00
Venue : S17 #04-06 (SR1)
Speaker : Wang Jialin, National University of Singapore

Fix a finite group G and an algebraically closed field F of characteristic p. For an FG-module M, the complexity of M is the rate of growth of a minimal projective resolution of M which is a cohomological invariant of M. The rank varieties introduced by Carlson are defined for modules for elementary abelian p-groups and can be extended to modules for G by looking at the restriction to elementary abelian subgroups of G. Moreover, the dimension of the rank variety gives the complexity of the module. In this talk, I will discuss some basic properties of rank varieties and complexities and then review some known results on complexities of some simple modules for symmetric groups.

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Date/Time : 01 Feb 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Jeff York Ye, National University of Singapore

The decomposition of tensor products of finite-dimensional irreducible representations of simple Lie groups is a long-studied problem. I will discuss some known results describing the multiplicity using canonical bases for quantum groups and tropicalization. Then I will discuss the related saturation conjecture, which is proved in type A by Knutson and Tao and open for general types.

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Date/Time : 31 Jan 2024 15:00
Venue : S17 #04-05 (SR2)
Speaker : Matteo Mucciconi, University of Warwick

The deformed polynuclear growth is a growth process that generalizes the polynuclear growth studied in the context of KPZ universality class. In this talk, I will discuss the mathematical derivation of large time large deviations for the height function. Rare events, as functions of the time t, display distinct decay rates based on whether the height function grows significantly larger (upper tail) or smaller (lower tail) than the expected value. Upper tails exhibit an exponential decay with rate function which we determine explicitly. Conversely, the lower tails experience a more rapid decay and the rate function is given in terms of a variational problem.

Our analysis relies on two inputs. The first is a connection between the height function h and an important measure on the set of integer partitions, the Poissonized Plancherel measure, which stems from nontrivial applications of the celebrated Robinson-Schensted-Knuth correspondence. The second ingredient is the derivation of a priori convexity bounds for the rate function, which combines combinatorial and coupling arguments.

This is a joint work with S.Das (Chicago) and Y.Liao (Wisconsin-Madison).

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