Time: Wednesdays 4:30  6:00 PM, unless otherwise noted.
Location: CL 251
In this learning seminar, we will study different models in which one can do homotopy theory, starting from Quillen's seminal work on model categories. We will discuss examples and applications, such as computing homotopy (co)limits.
The lecture series will be interspersed with guest research talks, supported by the Pacific Institute for the Mathematical Sciences (PIMS).
Date  Speaker  Title and abstract  References 

September 11  Martin Frankland  Introduction to model categories In classical homotopy theory, one can look at continuous maps up to homotopy and consider two topological spaces as essentially the same if they are homotopy equivalent. Model categories, introduced by Quillen, allow us to make similar homotopical constructions in a much broader context. I will present the definition and some basic properties. I will then focus on the examples of topological spaces and chain complexes. 
W.G. Dwyer and J. Spaliński, Homotopy theories and model categories, Handbook of algebraic topology (1995). 
September 18  Martin Frankland  Model categories, part 2 I will describe the homotopy category of a model category and how it forms a localization. I will then say more about the model structures on spaces, and come to the example of chain complexes. 

September 25  Martin Frankland  Model categories, part 3 I will say more about the model structures on topological spaces and on chain complexes. We will see that homological algebra is a form of homotopical algebra. 

October 2  Larry So  The equivalence between simplicial sets and topological spaces I will talk about the model structure on simplicial sets and the equivalence between the homotopy theory of simplicial sets and topological spaces. 
P.G. Goerss and J.F. Jardine, Simplicial homotopy theory (1999), Chapter I. 
October 9  Paul Songhafouo Tsopméné  Simplicial categories and the hammock localization Given a category C and a class W of "weak equivalences" in C, one can construct a new category $C[W^{1}]$ which as the same objects as C and is obtained from C by formally inverting the maps of W. As shown by DwyerKan, the category $C[W^{1}]$ reflects just one aspect of a much richer object, the simplicial localization LC, which is a simplicial category. Because it is difficult to get a hold on the homotopy type of the simplicial set LC(X, Y), for X, Y in C, DwyerKan considered a homotopy variation of LC, the hammock localization $L^H C$. In this talk I will recall the notion of simplicial category, and explain the construction of the hammock localization. I will also go over some properties of this localization. 
W.G. Dwyer and D.M. Kan, Calculating simplicial localizations, J. Pure Appl. Algebra 18 (1980). 
October 16  No seminar.  
October 23 5:00  6:00 PM 
Martin Frankland  Homotopy automorphisms Automorphism groups arise in many contexts. For instance, the automorphism group of the set of n elements is the symmetric group $\Sigma_n$ on n elements. The automorphism group of an ndimensional vector space is the general linear group $GL(n)$, consisting of invertible $n \times n$ matrices. In a homotopical context, what should the analogous notion be? Taking the automorphism group in the homotopy category loses too much information. In this talk, I will explain how the hammock localization can be used to construct a space of homotopy automorphisms with good properties. 
W.G. Dwyer and D.M. Kan, Function complexes in homotopical algebra, Topology 19 (1980). 
October 30  Paul Songhafouo Tsopméné  Simplicial model categories We have talked about model categories, we have talked about simplicial categories. Now we must tackle the two monsters together. This talk will be about the interactions of the simplicial and the model structure on the category of simplicial objects in a model category. 
P.G. Goerss and J.F. Jardine, Simplicial homotopy theory (1999), Chapter II. 
November 6  Fall Break. No seminar.  
November 13  Don Stanley  Homotopy colimits We give a gentle introduction to homotopy colimits. 

November 20  Arnaud Ngopnang  Model category of chain complexes We are going to describe and prove a model category structure on the category of nonnegatively graded chain complexes. 

November 27  Paul Songhafouo Tsopméné  Homotopy limits and colimits I will present some explicit formulae for homotopy limits and colimits. 

December 4  Larry So  Introduction to quasicategories In this talk I will introduce quasicategories, which are simplicial sets with the weak Kan property and generalize the notion of ordinary categories. 
M. Groth, A short course on ∞categories. 
December 11 4:30  5:30 PM Room CL 251 (usual room) 
Apurva Nakade (University of Western Ontario)  An application of Gromov's hprinciple to manifold calculus I will talk about applying manifold calculus to study the Lagrangian embeddings in a symplectic manifold. Using Gromov's hprinciple for directed embeddings, we will prove that the analytic approximation of the Lagrangian embeddings functor is the totally real embeddings functor. 
For more information, contact Martin Frankland.