Introduction to Homotopy Theory- PART 1: UNIVERSAL CONSTRUCTIONS
The goal of this series is to develop homotopy theory from a categorical perspective, alongside the theory of model categories. We do this with the hope of eventually developing stable homotopy theory, a personal goal a passion of mine. I'm going to follow nLab's notes, but I hope to add t
From playlist Introduction to Homotopy Theory
Stable Homotopy Seminar, 8: The Stable Model Category of Spectra
We discuss the enrichment of spectra over spaces, and the compatibility of this enrichment with the model structure. Then we define the stable model structure by adding extra cofibrations to the levelwise model category of spectra, and restricting the weak equivalences to those maps which
From playlist Stable Homotopy Seminar
Stable Homotopy Seminar, 17: Universal Coefficient Theorem, Moore Spectra, and Limits
We finish constructing the universal coefficient spectral sequence, and look at some classical examples involving Moore spectra. As it turns out, it's really easy in stable homotopy theory to invert or localize at a prime. In particular, *rational* stable homotopy theory is completely alge
From playlist Stable Homotopy Seminar
Stable Homotopy Seminar, 2: Fiber and Cofiber Sequences
We review some unstable homotopy theory, especially the construction of fiber and cofiber sequences of spaces, and how they induce long exact sequences on homotopy and homology/cohomology. (There's a mistake pointed out by Jeff Carlson: when I take a CW-approximation at one point, I have
From playlist Stable Homotopy Seminar
Daniel Isaksen - 1/3 Motivic and Equivariant Stable Homotopy Groups
Notes: https://nextcloud.ihes.fr/index.php/s/F2BoSJ7zgfipRxP I will discuss a program for computing C2-equivariant, ℝ-motivic, ℂ-motivic, and classical stable homotopy groups, emphasizing the connections and relationships between the four homotopical contexts. The Adams spectral sequence
From playlist Summer School 2020: Motivic, Equivariant and Non-commutative Homotopy Theory
Dianel Isaksen - 3/3 Motivic and Equivariant Stable Homotopy Groups
Notes: https://nextcloud.ihes.fr/index.php/s/4N5kk6MNT5DMqfp I will discuss a program for computing C2-equivariant, ℝ-motivic, ℂ-motivic, and classical stable homotopy groups, emphasizing the connections and relationships between the four homotopical contexts. The Adams spectral sequence
From playlist Summer School 2020: Motivic, Equivariant and Non-commutative Homotopy Theory
Stable Homotopy Seminar, 3: The homotopy category of spectra
We discuss the Brown representability theorem, and give the Boardman-Vogt definition of the homotopy category of spectra. Examples include suspension spectra, Omega-spectra arising from cohomology theories, and Thom spectra. ~~~~~~~~~~~~~~~~======================~~~~~~~~~~~~~~~ This is
From playlist Stable Homotopy Seminar
Stable Homotopy Seminar, 6: Homotopy Groups of Spectra (D. Zack Garza)
In this episode, D. Zack Garza gives an overview of stable homotopy theory and the types of problems it was designed to solve. He defines the homotopy groups of a spectrum and computes them in the fundamental case of an Eilenberg-MacLane spectrum. ~~~~~~~~~~~~~~~~======================~~~
From playlist Stable Homotopy Seminar
Stable Homotopy Seminar, 7: Constructing Model Categories
A stroll through the recognition theorem for cofibrantly generated model categories, using it to construct (1) the Quillen/Serre model structure on topological spaces and (2) the levelwise model structure on spectra. The latter captures the idea that spectra are sequences of spaces, but no
From playlist Stable Homotopy Seminar
Why Does Persistent Homology Work in Applications? [Adam Onus]
In this video I explain what it means for persistent homology to be stable under perturbations and noise, how to quantify this stability in terms of bottleneck and Wasserstein distances, and use this to answer the question of why persistent homology is a good tool to use in application. T
From playlist Tutorial-a-thon 2021 Fall
Calista Bernard - Applications of twisted homology operations for E_n-algebras
An E_n-algebra is a space equipped with a multiplication that is commutative up to homotopy. Such spaces arise naturally in geometric topology, number theory, and mathematical physics; some examples include classifying spaces of braid groups, spaces of long knots, and classifying spaces of
From playlist École d'Été 2022 - Cohomology Geometry and Explicit Number Theory
Topology and arithmetic of some spaces of polynomials with some constraints - Oishee Banerjee
Short Talks by Postdoctoral Members Topic: Topology and arithmetic of some spaces of polynomials with some constraints Speaker: Oishee Banerjee Affiliation: Member, School of Mathematics Date: September 20, 2022
From playlist Mathematics
Stability conditions in symplectic topology – Ivan Smith – ICM2018
Geometry Invited Lecture 5.8 Stability conditions in symplectic topology Ivan Smith Abstract: We discuss potential (largely speculative) applications of Bridgeland’s theory of stability conditions to symplectic mapping class groups. ICM 2018 – International Congress of Mathematicians
From playlist Geometry
[BOURBAKI 2019] Homology of Hurwitz spaces and the Cohen–Lenstra (...)- Randal-Williams - 15/06/19
Oscar RANDAL-WILLIAMS Homology of Hurwitz spaces and the Cohen–Lenstra heuristic for function fields, after Ellenberg, Venkatesh, and Westerland Ellenberg, Venkatesh, and Westerland have established a weak form of the function field analogue of the Cohen–Lenstra heuristic, on the distrib
From playlist BOURBAKI - 2019
Ulrike Tillmann, Lecture II - 11 February 2015
Ulrike Tillmann (University of Oxford) - Lecture II http://www.crm.sns.it/course/4038/ Mapping class groups and diffeomorphism groups of manifolds play an important role in geometry and topology. We will discuss recent advances in the understanding of their homology exploring homotopy the
From playlist Algebraic topology, geometric and combinatorial group theory - 2015
Michael Lesnick (5/3/21): l_p-Metrics on Multiparameter Persistence Modules
Motivated both by theoretical and practical considerations in topological data analysis, we generalize the p-Wasserstein distance on barcodes to multi-parameter persistence modules. For each p ? [1,?], we in fact introduce two such generalizations d_I^p and d_M^p, such that d_I^? equals th
From playlist TDA: Tutte Institute & Western University - 2021
Connections between classical and motivic stable homotopy theory - Marc Levine
Marc Levine March 13, 2015 Workshop on Chow groups, motives and derived categories More videos on http://video.ias.edu
From playlist Mathematics
Asymptotic Bounded Cohomology and Uniform Stability of high-rank lattices - Bharatram Rangarajan
Arithmetic Groups Topic: Asymptotic Bounded Cohomology and Uniform Stability of high-rank lattices Speaker: Bharatram Rangarajan Affiliation: Hebrew University Date: March 16, 2022 In ongoing joint work with Glebsky, Lubotzky, and Monod, we construct an analog of bounded cohomology in an
From playlist Mathematics