Multiplicative Irreducibility of Shifted Multiplicative Subgroups; Fisher Zeros and Phase Transitions on the Sierpiński Gasket

讲座时间：6月11日15:00-17:00

讲座地点：长安校区网安大楼A-1222室

主讲人1：Semin Yoo

主讲人简介：Semin Yoo博士于2021年获得罗切斯特大学博士学位。她曾任韩国高等研究院（KIAS）研究员，自2024年4月起任韩国基础科学研究院（IBS）离散数学组（DIMAG）高级研究员。她的研究主要集中于组合数学领域，已在CanadianJournal of Mathematics、Proceedings of the American Mathematical Society、Mathematika、European Journal of Combinatorics等期刊发表论文。

讲座题目：Multiplicative Irreducibility of Shifted Multiplicative Subgroups

讲座内容：A central theme in additive combinatorics is the interplay between addition and multiplication. Roughly speaking, sets with strong multiplicative structure are not expected to exhibit rich additive structure, and vice versa. In a recent breakthrough, Kalmynin proved conjectures of Lev--Sonn and Sárközy on additive decompositions of multiplicative subgroups in prime fields. Motivated by this work, we study multiplicative analogues of these questions. We show that, under a certain additional condition, a shifted multiplicative subgroup cannot be written as a product set, and that it also cannot be written as a ratio set unconditionally. In this talk, I will discuss these results and the main ideas of the proofs, which combine Stepanov's method with techniques involving symmetric polynomials.

This talk is based on joint work with Seoyoung Kim (University of Basel) and Chi Hoi Yip (Georgia Institute of Technology).

主讲人2：Jorge Olivares Viñales

主讲人简介：Jorge Olivares Viñales博士于2023年获得罗切斯特大学博士学位。自2024年起，他在复旦大学上海数学中心担任博士后研究员。他的研究主要集中于动力系统，并与一维动力系统、热力学形式、层级图上的统计力学以及复动力系统等方向相关。他已在Journalof the London Mathematical Society、Bulletin of the London Mathematical Society、Ergodic Theory and Dynamical Systems等期刊发表论文。

讲座题目：Fisher Zeros and Phase Transitions on the Sierpiński Gasket

讲座内容：The Sierpiński gasket is a classical self-similar graph whose recursive structure makes it a natural object in both combinatorics and statistical mechanics. In this talk, I will discuss the ferromagnetic Ising model on finite approximations of the Sierpiński gasket and the limiting distribution of its Fisher zeros.

The main idea is to use the combinatorial self-similarity of the gasket to obtain exact recursive formulas for the partition functions. These recursions lead to a renormalization map, and the dynamics of this map describes the asymptotic behavior of the zeros. From this point of view, the absence of a positive-temperature phase transition on the Sierpiński gasket can be understood through the geometry of the associated dynamical system.