Fall 2020 Talks

Day	Speaker	Title
08/31	William	Koszul algebras In 1970, Priddy introduced the notion of a Koszul algebra. In short, if an augmented k-algebra A is Koszul, then there is a small model for the cobar complex of A suitable for computing its cohomology algebra. I’ll explain some ways of understanding this story, illustrated with some explicit examples.
09/14	Doron	Finite Spaces and Finite Models When we try to model simplicial complexes using posets, finite spaces arise as a natural bridge between these two categories. In this talk, I will describe the theory of these spaces and the nature of this correspondence, and discuss the resulting theory of finite models.
09/21	Antonio	Straightening and unstraightening For any quasicategory X with objects w,x, Hom(w,x) is a Kan complex. Given any edge, f:x→y in X, we can construct a functor f⁎:Hom(w,x)→Hom(w,y) but this construction is such that for a pair of composable edges, f:x→y and g:y→z, the lifts g⁎f⁎ and (gf)⁎ are not necessarily equal, only homotopic. Consequently, Hom(w,–) does not induce a functor from X into the category of Kan complexes. We will show how we can ‘straighten’ a homotopy coherent functor (i.e. a left fibration) such as Hom(w,–) into an actual functor between infinity categories.
10/12	Sai	Picard groups in homotopy theory A very classical invariant of a commutative ring R is the Picard group. In this talk I will introduce some “interesting” Picard groups in homotopy theory and some techniques to compute them.
10/19	Brian	An Introduction to Algebraic K-Theory In this expository talk I’d like to give a brief introduction to algebraic K-theory. We’ll start with an short historical overview. We’ll then go into one of the many successful definitions for higher algebraic K-theory, that given by Waldhausen. If time permits, I’ll give a slightly more conceptual perspective on Waldhausen K-theory.
10/26	Haoyuan	Introduction to simplicial sets This is an introduction to simplicial sets. I will first talk about the definition of simplicial sets and also some examples of simplicial sets. Then I will introduce the Kan complex and some of its properties. If time permits, I will talk a little bit about the homotopy groups of Kan complex.
11/02	Joseph	Unfolding Orbifold Theory Orbifolds, since their inception in the 1950’s have taken on many different mathematical definitions despite their simple underlying concept. This talk will start with the conceptual definition and end with a definition based on equivariant sheaves, all without making any unmotivated changes to the definitions along the way.
11/09	Tsutomu	Equivariant formal group laws Formal group laws arise in homotopy theory through complex oriented cohomology theories. They are particularly interesting for finite characteristic and led to the creation of many “designer spectra” in stable homotopy theory. In this talk I will introduce equivariant formal group laws for abelian compact Lie groups and highlight some known structural results. I will also talk about some key differences and difficulties compared to the non-equivariant setting.
11/16	Liz	Brown-Gitler Spectra and Their Applications Brown-Gitler spectra are a class of spectra whose cohomology form certain submodules of the Steenrod algebra. These spectra were originally constructed to study immersions of manifolds, but they have turned out to have many other applications in homotopy theory. In this talk, we’ll define Brown-Gitler spectra and discuss some applications. In particular, we’ll focus on their use in investigating bo- and Adams ℓ-summand cooperations.
12/07	Sam	A very brief survey of some forms of Day convolution Day convolution is a very useful tool in many places, and at its most basic level endows a hom between two monoidal objects in some 2-categorical gadgets with a monoidal structure of its own subject to some conditions. We review Day convolution in 1-category theory, make some tiny observations, and then observe what happens in a formal category theoretic setting. Afterwards, we move on to the (∞, 1)-categorical case and take a quick peek at what happens for a very certain class of double ∞-categories. We’ll see some applications including operads and enriched categories in the Gepner-Haugseng/Hinich sense, following Haugseng. Afterwards if there is time we will look at some more applications and end on a few questions.