Activities Fall 2013

Tuesday activities take place  between 15:15-17:00. When at KTH, then in room 3418, and when at SU, then in room 306. Divergence from this is marked with a "*" and completed by, if needed, neccessary information. 

• December 10. 15:15-17:00, room 306 SU. Sean Tilson, Definitions and computations of factorization homology.
  Abstract:  After recalling the axioms for a homology theory, we will present a derived functor definition of factorization homology. Using this, as well as the axioms, we will compute the factorization homology of an n-dimensional manifold with coefficients in a free $E_n$-algebra as well as the factorization homology of the $S^1$. Next we will begin discussing factorizations algebra in order to give a Cech style construction of a homology theory.

• December 3. 15:15-17:00, room 3418 KTH. Philip Hackney, Modules and extensions. 
  Abstract: If a manifold M decomposes as R ∪ L, glued along a submanifold N×ℝ, then F_R(A) and F_L(A) are modules over the E^1 algebra F_{N×ℝ}(A). The excision axiom for factorization homology says that the map from the derived tensor product F_R(A) ⊗ F_L(A) to F_M(A) is an equivalence. We will attempt to make precise what this statement says; in particular, we talk about algebras over ∞-operads, modules over such, and why the chain complexes from the first sentence give examples.
• November 26. 15:15-17:00, room 306 SU. We have a guest lecture: Richard Hepworth, Homology of Algebras with Structure.
  Abstract: It is often the case that if an associative algebra admits some kind of "extra structure" then its Hochschild homology also admits some "extra structure".  For example the Hochschild homology of a commutative algebra is itself a commutative algebra.  Recent work of Nathalie Wahl and Craig Westerland investigates this question of "extra structure" in detail, giving several general results and working out many interesting examples.  My talk will explain how to generalise this work to new contexts such as cyclic homology, topological Hochschild homology and topological chiral homology.  The main tool is enriched category theory (which I won't assume any knowledge of).  I will then explain the cyclic case in some detail. This is joint work with Jeff Giansiracusa. 

• November 19, 15:15-17:00, room 3418 KTH. Wojtek's second attampt of making sence of the third section of the  paper, with clarifications and explenations by Anssi.
• November 5, 15:15-17:00, room 3418 KTH. Anssi Lahtinen, symmetric monoidal ∞-categories.
  Abstract: Building on Alexander Berglund's talk on infinity-categories on October 29, I will discuss the definition of symmetric monoidal infinity-categories. References: [G], section 9; [Gr]; [L], beginning of Chapter 2.
  
  [G] G. Ginot, Notes on factorization algebras, factorization homology and applications
  [Gr] M. Groth, A short course on infinity-categories  
  [L] J. Lurie, Higher algebra

• October 29, 15:15-17:00, room 306 SU. Alexander Berglund will tell us about Infinity-categories. Abstract: I will give an introduction to infinity-categories (quasi-categories) following the first sections of [L] and Appendix 9.1 of [G]. Our goal is to understand constructions that are relevant for the definition of factorization homology, such as the homotopy coherent nerve, overcategories, and homotopy pullbacks. (We will see how far we get, doing all this may require more than one lecture.) [G] G.Ginot, Notes on factorization algebras, factorization homology and applications, http://arxiv.org/abs/1307.5213. [L] J.Lurie, Higher Topos Theory, http://www.math.harvard.edu/~lurie/

• October 22, 15:15-17:00, room 3418 KTH. Sean Tilson will tell us about E_n algebras and their role in algebraic topology classically, e.g., iterated loop spaces, recognition principle, homology operations etc. Our hope is to understand why Disk_n^{fr}-algebras are 'the same' as E_n-algebras.

• October 8, 15:15-17:00, room 3418 KTH. This is the first meeting in a reading seminar on factorization homology (aka topological chiral homology). Alexander Berglund will discuss the categories of n-manifolds and E_n-algebras on which factorization homology is defined, following, roughly, §3.1 and §9.2 of Grégory Ginot, Notes on factorization algebras, factorization homology and applications,  arxiv.org/abs/1307.5213

• October 1, 15:45-17:00, room 306 SU. We'll have three little presentations each between 10-20 minutes. Alexander: factorization homology, Wojtek: Waldhausen K-theory, Tilman: equivariant stable homotopy. Then we'll discuss, vote, and decide on a topic for our reading seminar for the following weeks. It'll be one of these three or etale homotopy, which Magnus has already given us a nice introductionon September 24.
• September 24, 15:15-17:00, room 3418 KTH, Magnus Carlson: Étale homotopy, pro-categories and obstruction theory.

• September 17, 15:15-17:00, room 306 SU. Philip Hackney, SU will give a brief introduction to colored props and their ilk. We then propose ways of encoding the notion of up-to-homotopy prop.  

• September 10, 15:15-17:00, room 3418 KTH.  Anssi Lahtinen :  String Topology

Here is a list of suggestions that were discussed on October 1 for  our Tuesday activities. We chose factorization homology.

Factorization homology: this can be viewed as an outgrowth of Lurie's work on the classification of topological field theories and his notion of "topological chiral homology". One could maybe use lecture notes by Gregory Ginot (attached).

Equivariant stable homotopy theory: towards understanding Hill-Hopkins-Ravenel, slice filtration etc

Etale homotopy theory

Algebraic K-theory of spaces: A(X) and the connection to stable pseudo-isotopy. Following Waldhausen, and perhaps the new book by Jahren-Rognes-Waldhausen.

Galois theory of Rognes