江蘇省應用數學（中國礦業大學）中心系列學術報告
短期課程報告題目：Coset complexes in finite groups
報告人：孟沆洋
個人簡曆：孟沆洋，上海大學理學院數學系副教授，西班牙瓦倫西亞大學基礎數學博士，主要研究領域為有限群論及其表示。相關結果發表于Trans. Amer. Math. Soc., J. Lond. Math. Soc.，Proc. Amer. Math.Soc.，J. Algebra等雜志。 獲上海市2020年度揚帆計劃項目、2020年度上海市引智人才XSC計劃，2021年獲 Baer Prize特别提名獎，主持國家自然科學基金青年項目一項，面上項目一項。
邀請人：張馳
報告 1：Posets and order complexes
報告時間：2024年11月25日（周一）上午9：30-10：30
報告地點：博1-A202
報告摘要：In this discourse, we will delve into the concept of partially ordered sets, commonly known as posets, and their corresponding order complexes. The study of posets is fundamental in various fields such as combinatorics, algebra, and theoretical computer science. We will then explore how these order complexes can be realized as geometric simplicial complexes in Euclidean space. This realization is crucial as it bridges the gap between combinatorial structures and topological spaces, enabling us to study the topological properties of posets through the lens of algebraic topology.
報告2： Order complexes of p-subgroups and Quillen’s conjecture
報告時間：2024年11月25日（周一）上午10：40-11：40
報告地點：博1-A202
報告摘要：In this talk, we will touch upon the famous Quillen's Conjecture, which is a deep and influential conjecture in algebraic topology and finite group theory. Although it remains unproven in general, it has inspired a wealth of research and has been confirmed in various special cases.
報告3： Order complexes of proper cosets in finite groups
報告時間：2024年11月25日（周一）下午16:00-17:00
報告地點：伟德bvA310
報告摘要：In this talk, we will show some topological properties of proper coset posets in finite groups. Let G be a finite group and X be a subgroup of G. Denote by C_X(G) the set of all cosets Hx in G with X ≤ H < G. We will show that C_X(G) is non-contractible if G is solvable or N_G(X) contains a Sylow 2-subgroup and a Sylow 3-subgroup of G. This result follows J. Shareshian and R. Woodroofe’s work in Adv.Math(2016). We also give some divisibility properties of the Euler characteristic of C_X(G) when X is a p-group, which follows K. S. Brown’s classical result in J.Algebra (2000).