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Place: N902
Title: Some topics in applied topology and geometry
Abstract
Place: N902
Title: On the May Spectral Sequence at the Prime 2
Abstract: The May spectral sequence is one of the first effective methods to compute the cohomology of the Steenrod algebra. The E2 page is the cohomology of the associated graded algebra of the Steenrod algebra. In addition to a conjecture of May about all of the indecomposables of E2, we are going to state a conjecture about all of the relations. We also conjecture that this E2 page is nilpotent free. We will show that these conjectures all hold in a big subalgebra of E2 which covers a large range of dimensions.
Place: N902
Title: Twisted Milnor hypersurface and its genus
Abstract: In this talk, I would like to give a simple introduction about the twisted Milnor hypersurface. This provides a large family of spin (string) manifolds with non zero genus, such as A hat genus, alpha invariant, Witten genus and so on. This is a joint with with Zhi Lv, Fei Han and Jingfang Lian.
Place: N902
Title: Homotopy decompositions of manifolds (0): warm up
Abstract: In the next several weeks, I will talk about several recent works on the homotopy decomposition of manifolds with applications.
The plan is roughly as follows
(0) Warm up: homotopy groups of fibered Calabi-Yau threefolds;
(1) Loop homotopy of $6$-manifolds over $4$-manifolds;
(2) A digression: Suspension homotopy of simply connected $6$-manifolds;
(3) (with Stephen Theriualt) Exponential growth in the rational homology of free loop spaces;
(4) (with Stephen Theriault) Loop decomposition of $(2n-2)$-connected of $(4n-1)$-manifolds.
Place: N208
Title: Tate cohomology for real oriented cohomology theories
Abstract: In this talk, we present Tate blue shift phenomena for real oriented cohomology theories and norms of them. The ?/p-Tate cohomology spectrum of the n’th Johnson–Wilson theory splits as a wedge of (n-1)’th Johnson–Wilson theories (after completion). We construct a C_2-equivariant lifting of this splitting for Real Johnson–Wilson theories. The C_2-fixed points of this splitting is a higher height analogue to Davis and Mahowald’s splitting of the Tate cohomology spectrum of ko as a wedge of H?. If time allows, I will discuss the normed case. This talk is based on a sequence of joint projects with Hood Chatham, Vitaly Lorman and James D. Quigley.
Place: N208
Title: Covering types of spaces
Abstract:
Place: N208
Title: Characteristic classes of semi-complex vector bundles (II)
Abstract:
Place: N208
Title: Characteristic classes of semi-complex vector bundles (I)
Abstract:
Place: N211
Title: Choosing points on cubic plane curves
Abstract: It is a classical topic to study structures of certain special points on complex smooth cubic plane curves, for example, the 9 flex points and the 27 sextactic points. We consider the following topological question asked by Farb: Is it true that the known algebraic structures give all the possible ways to continuously choose n distinct points on every smooth cubic plane curve, for each given positive integer n? This work is joint with Ishan Banerjee.
Place: N208
Title: Mapping class groups of projective like planes
Abstract:
Place: N208
题目：持续同调和UMAP背后的数学
摘要：拓扑数据分析(Topological data analysis, TDA)是一门历经15年快速发展的学科，至今已经成为非常活跃的研究方向。它充分应用了代数拓扑中的同调论和同伦论以及诸如离散莫尔斯理论、图论、组合拓扑、动力系统乃至同调代数、范畴论和层论等种种经典的拓扑和代数工具来分析数据点集的形状、演化和分类。它的核心概念–持续同调(persistent homology)及条形码(barcode)已经在生物大分子结构(macro-biomolecular topology)的研究中取得了重要的进展，同时在数学神经科学(mathematical neuroscience)，脑研究(brain study)，传感器网络(sensor network)和材料学等众多领域也有很多崭新应用。最近，许多研究者正在尝试将机器学习的方法与TDA结合起来。这个报告将简要介绍TDA的发展，持续同调的概念和一种最新的机器学习算法背后的数学。