Fall 2019 Talks

Day	Speaker	Title
09/09	William	Algebraic theories and homotopy theory In this talk, I will motivate and introduce algebraic theories as a category-theoretic approach to finite product theories. I will then talk about a well-behaved notion of an infinitary algebraic theory, and the introduction of homotopy-theoretic structure, which can be used to define notions of homology and cohomology for the models of an algebraic theory. This is the first of two talks; the second will use these ideas to produce applications in stable homotopy theory.
09/23	Liz	Splitting BP⟨1⟩∧BP⟨1⟩ at odd primes The Adams Spectral Sequence is a tool for approximating π∗X, where X is a connective spectrum. If E is a ring spectrum satisfying certain properties, then we can define an E-based Adams spectral sequence converging to π∗X̂, where X̂ is the E-completion of X. When E∗E is flat over E∗, the E2-page of the spectral sequence can be described as ExtE∗E(E∗,E∗X). But if E∗E is not flat over E∗, then there is no such description. Instead, we must study E∧E to understand the spectral sequence. The Brown-Peterson spectra BP⟨n⟩ are an example of such spectra. One approach is to split the product E∧E into more manageable pieces. When n=1, we can construct a splitting BP⟨1⟩∧BP⟨1⟩ as ∨∞k=0Σ2k(p−1)BP⟨1⟩1∧B(k), where B(k) is the kth integral Brown-Gitler spectrum. We give a sketch of Kane’s construction of this splitting for odd primes.
09/30	Brian	The Recognition Principle for Infinite Loop Spaces In this expository talk, I’d like to discuss the infinite loop space recognition principle. In particular, I’d like to examine Boardman-Vogt’s infinite loop space machine from a modern view point.
10/07	Joseph	Motivating Higher Toposes: Geometric Characteristic Classes This will be part one of two talks aimed at motivating higher topos theory from physics. In this talk we will give a brief history of classical obstructions for manifolds, then suddenly find ourselves naturally requiring tools from higher topos theory. In the end, we shall see how working with simplicial sheaves on Manifolds allows us to define (but more importantly compute) geometric characteristic classes.
10/14	Joseph	Motivating Higher Toposes: Higher Bundle Theory In this (self-contained) talk, I will begin with a quick recap of the motivation for higher bundle theory from the first talk. I will then say a few words about Toposes, and proceed to spend the majority of the talk attempting to develop a general theory of higher bundles. Along the way, we will see how the necessary properties for this development (almost) force higher topos structure. (Technical details will be sacrificed for intuitive clarity. No particular model of higher categories will be imposed.)
10/21	Daniel	Topological Data Analysis: Theory and Applications There are two algorithms which form the backbone of many applications of modern topological data analysis: the Mapper algorithm (Singh, Memoli, Carlsson), and persistent homology (Zomorodian, Carlsson). In this talk I’ll introduce both algorithms, talk about the homotopy theory behind them, and give an application of each.
10/28	Heyi	Global homotopy groups and global functors The 0th equivariant homotopy group of an orthogonal G-spectrum defines a G-Mackey functor with restriction and transfer functors out of the the orbit category of G and when G is finite, the interactions between restrictions and transfers is given by a double coset formula. In the global setting, we define an orthogonal spectrum as a functor out of inner product spaces and will see that its 0th global homotopy group defines a “global functor” out of the global Burnside category so that restriction generalizes naturally for arbitrary continuous maps of groups and transfer along inclusion of closed subgroups. In this sense, the global functors generalize G-Mackey functors by allowing G to vary. This talk follows the treatment in Schwede’s book Global homotopy theory.
11/04	Tsutomu	Atiyah-Segal completion theorem In this talk I will walk through Atiyah and Segal’s “Equivariant K theory and completion”. The main result is a nice proof of the so-called Atiyah-Segal completion theorem, which relates the equivariant K theory of a G-space with the K theory of its homotopy orbit. Towards the end, I will also discuss the algebraic analogue of this result.
11/11	Sai	Introduction to Picard groups Picard groups have been classically studied in commutative algebra and algebraic geometry. These can be suitably generalised to define picard groups of E∞ ring spectra. In this situation a natural question to ask is “How is the picard group of R related to the picard group of R∗?”. In this expository talk, I will explain some methods which have been used to answer the above question.
11/18	Abhra	Splitting induced from transfer Associated to a finite covering map is a map in homology and cohomology, known as transfer, which goes in the opposite direction. This map usually doesn’t come from a map at the level of spaces but interestingly enough it does when we look at its version in spectra. This often leads to interesting splittings in spectra after appropriate localisation. In this talk, I will explain how one gets this map. Then I will concentrate on one such transfer and the splitting induced from it. I will show a few pictures of algebraic objects associated to this transfer which will illustrate this splitting and then end by observing some facts about the splitting from those pictures.
12/02	Ningchuan	The Gross-Hopkins duality In this talk, we introduce the Gross-Hopkins duality in chromatic homotopy theory, which relates the the Spanier-Whitehead duality and the Brown-Comenetz duality in the K(n)-local category. We will mainly focus on the underlying algebraic geometry of the duality phenomena and work out some explicit examples at height 1.
12/09	Ningchuan	The Gross-Hopkins duality (part 2) Following last week’s talk, we will sketch the proof of the Gross-Hopkins duality following Neil Strickland’s paper. In addition, we will explain how the Gross-Hopkins period map is related to the duality phenomenon.