Timetable : here

Mini-courses (December 4-7, 2023, Quy Nhon University)

• Grégory Ginot: Introduction to persistence homology and its applications

Abstract: The course will present some ideas of algebraic topology applied to data science; more precisely persistence homology theory. Roughly speaking, this is the idea of applying simplicial complexes and simplicial homology to provide computer friendly invariants of filtered spaces and data sets. We will explain the basic construction, the notion of barcodes (= the invariants of filtered spaces) and of distance between them; focusing on constructions from data analysis

• Neil Strickland: Ambidexterity

Abstract and Lecture notes: https://strickland1.org/vietnam/

• Paolo Salvatore: Introduction to operads

Abstract: Operads describe operations with several inputs and play a major role in modern mathematics. They are relevant in algebra, geometry, and mathematical physics. We will study several examples, and in particular the E_n operads, that describe higher commutativity coherence. We will introduce Koszul duality for operads in algebra and stable homotopy theory and will mention a recent result of Koszul self-duality of the stable E_n operads obtained with Michael Ching.

Invited talks at the Workshop (December 8, 2023, ICISE)

• Lorenzo Guerra: Symmetric groups and the cohomology rings of extended symmetric powers

Abstract: The k-th extended symmetric powers construction D_k(X) = (E(Σ_k)×(X^k))/(Σ_k) consists of the homotopy quotient of the cartesian power X^k of a topological space X by the action of the symmetric group Σ_k that permutes its k factors. The spaces D_k(X) (and their spectral analogs) have been widely used in algebraic topology, in connection with Steenrod operations, the Adams spectral sequence, Nishida’s nilpotent elements theorem, etc.
In this talk I will present a new approach to the cohomology of D_k(X) that clarifies the underlying geometry and makes its ring structure accessible. I will focus on the cohomology with coefficients in F_2, the field with two elements, and only briefly sketch the case of fields with characteristic different from 2. I will first discuss the algebraic properties of H^∗(D_k(X);F2), which arise from the interplay of a Hopf ring structure and divided powers operations. I will then describe the structural morphisms and the generating cohomology classes geometrically via the Fox-Neuwirth models of configuration spaces.
Finally, I will present a combinatorial description of the cup product and I will deduce a characterization of the cohomology of the E_∞-algebra and the ∞-loop space freely generated by a topological space X as rings. The content of this talk stems from a joint work with Paolo Salvatore and Dev Sinha.

• Huynh Mui: TBA
• Jean Lannes: On the Quillen map for the mod2 cohomology of certain finite groups

Abstract: https://drive.google.com/file/d/1xbmLtW3hnDzlUJAkmAMxwXHpHpbPYIS7/view?usp=sharing

• Nguyen Dang Ho Hai: Invariants of polynomials modulo Frebenius powers

Abstract: We consider the action of the general linear group GL_n(F_q) on the quotient ring Q:=S/I where S is the polynomial ring over F_q in n variables x_1,...,x_n and I the ideal generated by the q^m power of the x_i's. For each parabolic subgroup P of GL_n(F_q), a conjecture due to Lewis, Reiner and Stanton (2017) predicts a precise formula for the Poincaré series of the graded vector space of P-invariants in the quotient Q. In this talk, we explain how to prove this conjecture for the Borel subgroup. This is a joint work with L. M. Ha and N. V. Nghia.

• Nguyen Duc Nga: The Margolis homology of the invariants under the Sylow subgroup of the general linear group

Abstract: https://drive.google.com/file/d/1_680DxLxHUCeX3hDIABkg4ipINlJQSX6/view?usp=sharing

• Sarah Whitehouse: Model structures and spectral sequences

Abstract: Model categories provide the classical abstract setting for homotopy theory, allowing study of different notions of equivalence. I'll survey recent work with various coauthors on model category structures related to spectral sequences. For a category with associated functorial spectral sequences, one can consider a hierarchy of notions of equivalence, given by morphisms inducing an isomorphism at a fixed stage of the associated spectral sequence. I'll discuss model structures with these weak equivalences for filtered complexes, for bicomplexes and for multicomplexes. More recent work shifts the focus to the category of spectral sequences itself. One can endow this with a somewhat weaker structure and work in progress takes a linear pre-sheaf approach to further understand the homotopy theory. The talk will range over joint work with subsets of: Joana Cirici, Daniela Egas Santander, Xin Fu, Ai Guan, Muriel Livernet and Stephanie Ziegenhagen.