Department of Pure Math Alumni Symposium

Venue: MB 441

Time: 14:00-15:00

Title: Tropical regular cycles

Abstract: ?A tropical cycle is called regular if it supports a reduced 0-dimensional complete intersection, and for some cases the classification of regular cycles is already complete. In this talk, I will recall the construction of the tropical intersection product via two equivalent ways, introduce the existing classifications from Fink and Esterov-Gusev, and give some unified answers of the classification problem to lower dimensional regular tropical cycles. This talk is based on my preprint paper: Arxiv 2508.06694.

Personal introduction:?Linxuan Li?is?a second-year PhD student in the School of Mathematical Sciences at Queen Mary University of London, co-advised by Professors Alex Fink and Alex Esterov. His research interests lie in tropical geometry and algebraic combinatorics. In 2022, he graduated from the Department of Applied Mathematics at XJTLU under the supervision of Prof?Jinsong Xu. He was awarded funding from the China Scholarship Council (CSC).

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Speaker: Xiaomeng Xu

Venue: MB 441

Time: 15:30-16:30

Title: On $\ell_p$-Vietoris-Rips complexes?

Abstract:?We study the concepts of the? $\ell_p$-Vietoris-Rips simplicial set and the? $\ell_p$-Vietoris-Rips complex of a metric space, where 1 ¡Ü p ¡Ü ¡Þ. This theory?unifies two established theories: for p = ¡Þ, this is the classical theory of Vietoris-Rips complexes, and for p = 1, this corresponds to the blurred magnitude homology theory. We prove several results that are known for the Vietoris-Rips complex in the general case: (1) we prove a stability theorem for the corresponding version of the persistent homology; (2) we show that, for a compact Riemannian manifold and a sufficiently small scale parameter, all the ¡° $\ell_p -Vietoris-Rips spaces¡± are homotopy equivalent to the manifold; (3) we demonstrate that the? $\ell_p$-Vietoris-Rips spaces are invariant (up to homotopy) under taking the metric completion. Additionally, we show that the limit of the homology groups of the? $\ell_p$-Vietoris-Rips spaces, as the scale parameter tends to zero, does not depend on p; and that the homology groups of the? $\ell_p$-Vietoris-Rips spaces commute with filtered colimits of metric spaces.

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Personal introduction:?Xiaomeng Xu is?about to be a PhD student in the School of Mathematical Sciences at the University of Southampton, under the supervision of Prof. Stephen Theriault. He worked as an intern at Beijing Institute of Mathematical Sciences for one year and a half, working with Prof. Sergei O. Ivanov. Before that, He was a master student at the University of Oxford, supervised by Prof. Andr¨¦ Henriques. Even before, He was an undergraduate student at?XJTLU, where he won the Best Performance Prize in Final Year Project, supervised by Prof. Alastair Darby. He also achieved the University Academic Excellence award at XJTLU in 2020/21 and 2021/22. His research interests lie in algebraic topology and homotopy theory.