# Seminars & Colloquia

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We introduce a new method to derive a lower bound of the mean curvature for the solutions of inverse mean curvature flow. This estimate for IMCF, an expanding curvature flow, corresponds to interior curvature estimates of Ecker-Huisken and Caffarelli-Nirenberg-Spruck for shrinking curvature flow and elliptic problem. It yields the smoothness of solutions and the existence of the flow for complete non-compact convex initial hypersurfaces.

In recent years, community detection has been an active research area in various fields including machine learning and statistics. While a plethora of works has been published over the past few years, most of the existing methods depend on a predetermined number of communities. Given the situation, determining the proper number of communities is directly related to the performance of these methods. Currently, there does not exist a golden rule for choosing the ideal number, and people usually rely on their background knowledge of the domain to make their choices. To address this issue, we propose a community detection method that also adaptively finds the number of the underlying communities. Central to our method is fused l1 penalty applied on an induced graph from the given data.

There have been many exciting developments on the arithmetic of Shimura varieties in the recent decade, which have contributed to better understandings of the (still largely conjectural) Langlands programme. In this talk, I’ll try to give an overview of recent developments in mod p reduction of Shimura varieties (in both good reduction and bad reduction cases), and conclude the talk by introducing my joint work with Paul Hamacher on Mantovan’s formula (a cohomological formula conjecturally encoding the local-global compatibility of Langlands correspondences).

The classical Grunwald–Wang theorem is an example of a local–global principle stating that except in some special cases which are precisely determined, an element m in a number field K is an a-th power in K if and only if it is an a-th power in the completion K℘ of K for all but finitely many primes ℘ of K. In this talk, based on the analogue between the power map and the Carlitz module, I introduce an analogue of the Grundwald-Wang theorem in the Cartlitz module setting.

The original version of Waring's problem asks whether, for every positive integer n, there exists M(n) such that every non-negative integer is of the form a_1^n + ......+ a_{M(n)}^n, where the $a_i$ are non-negative integer. Since 1909, Hilbert proved that such a bound exists. In this talk, I introduce an analogue of Waring's problem for an algebraic group G, which is a generalization of the original Waring's Problem in the algebraic-group setting. (Joint work with Michael Larsen)

Groups of piecewise projective homeomorphisms provide elegant examples of groups that are non amenable, yet do not contain non abelian free subgroups. In this talk I will present a survey of these groups and discuss their striking properties. I will discuss properties such as (non)amenability, finiteness properties, normal subgroup structure, actions by various degrees of regularity and Tarski numbers.

We show that for any connected sum of lens spaces L there exists a connected sum of lens spaces X such that X is rational homology cobordant to L and if Y is rational homology cobordant to X, then there is an injection from H_1(L; Z) to H_1(Y; Z). Moreover, as a connected sum of lens spaces, X is uniquely determined up to orientation preserving diffeomorphism. As an application, we show that the natural map from the Z/pZ homology cobordism group to the rational homology cobordism group has large cokernel, for each prime p. This is joint work with Paolo Aceto and Daniele Celoria.

The Swendsen-Wang dynamics is an MCMC sampler of the Ising/Potts model, which recolors many vertices at once, as opposed to the classical single-site Glauber dynamics. Although widely used in practice due to efficiency, the mixing time of the Swendsen-Wang dynamics is far from being well-understood, mainly because of its non-local behavior. In this talk, we prove cutoff phenomenon for the Swendsen-Wang dynamics on the lattice at high enough temperatures, meaning that the Markov chain exhibits a sharp transition from “unmixed” to “well-mixed.” The proof combines two earlier methods of proving cutoff, the update support [Lubetzky-Sly ’13] and information percolation [Lubetzky-Sly ’16], to establish cutoff in a non-local dynamics. Joint work with Allan Sly.