Paul Lessard (CU-Boulder) From Invariants, to functors, to homotopy, to model categories, and back to homotopy: part 2
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Eilenberg and Mac Lane developed or discovered category theory in the context of algebraic topology; the very utility of topological invariants such as homology, co-homology, and the homotopy groups is contingent on their functoriality. These particular invariants are moreover homotopy invariants. This property can be rephrased as those invariants factoring through the homotopy category, where homotopic maps are identified. We try and reverse this observation. Can we, starting with a functor, interpret it as a homotopy invariant by finding the right notion of homotopy? What data do we need to do so, and how universal are these constructions?
In part one we will motivate and define Quillen's concept of a model category, and in part two we will outline how such a structure begets a homotopy category.
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In the first talk, I will introduce some of the concepts which play a central role in homotopy theory. I will give an introduction to the concepts of stable homotopy groups, spectra, generalized cohomology theories and localization. I will also explain how formal group laws enter the subject. For the majority of the talk, I will only assume that the audience is comfortable with the concepts of topological spaces and their cohomology.
The second talk will be focused on chromatic homotopy theory. The philosophy of chromatic homotopy theory is that the stable homotopy groups of the sphere $S$ can be re-assembled from the homotopy groups of a family of spectra $L_nS$. Roughly, $L_nS$ is the $n$-th chromatic layer of $S$. Each layer corresponds to the height of a formal group law. The homotopy groups of the chromatic layers are related to the cohomology of the group of automorphisms of the corresponding formal group laws. In this talk, I will give an overview of this computational program. I will also talk about the role played by elliptic curves and their deformation theory at chromatic level $2$. I will assume that the audience has heard of spectral sequences.
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In the first part of the talk, we will recall the basic theory of impartial games. We’ll learn how to win at familiar games like NIM and precision bowling. Then, we’ll apply this knowledge to study games in which two players alternately pick elements of a given group until the picked elements generate the group.
X
Eilenberg and Mac Lane developed or discovered category theory in the context of algebraic topology; the very utility of topological invariants such as homology, co-homology, and the homotopy groups is contingent on their functoriality. These particular invariants are moreover homotopy invariants. This property can be rephrased as those invariants factoring through the homotopy category, where homotopic maps are identified. We try and reverse this observation. Can we, starting with a functor, interpret it as a homotopy invariant by finding the right notion of homotopy? What data do we need to do so, and how universal are these constructions?
In part one we will motivate and define Quillen's concept of a model category, and in part two we will outline how such a structure begets a homotopy category.