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 : 31 Jan 2024 17:00
Venue : S17 #04-05 (SR2)
Speaker : Yu Liang, Nanjing University

Geometric measure theory relates effectivity notions to dimensions and measures like the Hausdorff dimension. The talk gives further links to the Axiom of Determinacy over ZF (it is not consistent with ZFC) and how these influence the geometry of the finite-dimensional Euclidian Space and its subsets. The talk explains the theorems of Besicovitch and Davis, of father and son Lutz and of Slaman; these theorems are related to recent results in the field including those by the speaker.

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Date/Time : 31 Jan 2024 14:00
Venue : S17 #06-11 (SR6)
Speaker : Han Qing, University of Notre Dame

A version of the singular Yamabe problem in bounded domains yields complete conformal metrics with negative constant scalar curvatures. In this talk, we study whether these metrics have negative Ricci curvatures. We present two classes of results. First, we prove that the Ricci curvature is indeed negative for the complete conformal metrics on convex domains. Second, we demonstrate the Ricci curvature splits in sign if a component of the boundary of the domain is close to a lower dimensional set.

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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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Date/Time : 29 Jan 2024 10:00
Venue : S17 #04-06 (SR1)
Speaker : Jimmy He, Massachusetts Institute of Technology

Random growth models in 1+1 dimension capture the behavior of interfaces evolving in the presence of noise. These models are expected to exhibit universal behavior, but we are still far from proving such results even in relatively simple models. A key development which has led to recent progress is the discovery of exact formulas for certain models with rich algebraic structure, leading to asymptotic results. I will survey some of these results, with a focus on models where a single boundary wall is present, as well as applications to rates of convergence for a Markov chain.

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Date/Time : 26 Jan 2024 15:00
Venue : S17 #06-11 (SR6)
Speaker : Tong Zhang, East China Normal University

It is known by Chen-Chen-Jiang that every complex smooth threefold X of general type with p_g(X) >= 11 satisfies the Noether inequality Vol(X) >= 4/3 p_g(X) – 10/3. If X attains the equality, it is said to be on the Noether line. In this talk, I will discuss the explicit structure of threefolds on the Noether line, and talk about what we know so far about their moduli space. This is a joint work in progress with S. Coughlan, Y. Hu, and R. Pignatelli.

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Date/Time : 25 Jan 2024 16:00
Venue : S17 #05-12 (SR4)
Speaker : Yifeng Liu, IASM, Zhejiang University

In this talk, I will formulate the anticyclotomic Iwasawa main conjecture for Rankin-Selberg products of automorphic representations of hermitian type, and state our results. I will then explain the main ingredients of the proof: an Ihara type lemma, Abel-Jacobi image of basic locus, and a bipartite Euler system over the Iwasawa algebra. This is a joint work with Yichao Tian and Liang Xiao.

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Date/Time : 25 Jan 2024 16:00
Venue : S17 #04-06 (SR1)
Speaker : Sungwoon Kim, Jeju National University

Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper, we relax Sullivan’s axioms and introduce a notion of meandering hyperbolicity for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain nonhyperbolic groups, such as actions of uniform lattices in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust, and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions. This is a joint work with Jaejeong Lee and Misha Kapovich.

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Date/Time : 25 Jan 2024 15:00
Venue : S16-06-118, Seminar Room
Speaker : Zhou Fan, Yale University

In statistical and probabilistic models, “mean-field” behavior arises when the aggregate effect of many variables is well-approximated by an average or effective field. Ideas from the analysis of mean-field models in statistical physics have been influential in our understanding of various problems of high-dimensional statistics in recent years. In this talk, I will argue that statistical applications can conversely drive the basic study of mean-field phenomena.

As a motivating example, I will discuss an application of high-dimensional mean-field theory to a nonparametric empirical Bayes procedure for PCA. I will then describe several developments of the theory beyond i.i.d. models, including extensions of Approximate Message Passing algorithms and analyses of spin glass models where the couplings matrices are orthogonally invariant in law.

Joint work w/ Yufan Li, Subhabrata Sen, Chang Su, Tianhao Wang, Yihong Wu, Xinyi Zhong.

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