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This page was the home of the Xtraordinary cohomology theory & K-theory Collective Discussion group* for the fall 2013 semester.
All talks at Berkeley were on Berkeley time (i.e. they started 10 minutes late).
2:30-3:30 -- 383B Sloan Hall -- K-theory and Kalculus
We'll talk about Waldhausen's algebraic K-theory, which gives us a set of rather deep invariants of topological spaces. Unfortunately, these invariants are really, really hard to calculate. I'll give a brief overview of how different forms of calculus lets us approximate K-theory, what we know about these approximations, and what we may know soon.
4:00-5:00 -- 383N Sloan Hall -- Determinantal K-theory and a few applications
Chromatic homotopy theory is an attempt to divide and conquer algebraic topology into a sequence of what we'd first assumed to be "easier" categories. These categories turn out to be very strangely behaved -- and furthermore appear to be equipped with intriguing and exciting connections to number theory. To give an appreciation for the subject, I'll describe the most basic of these strange behaviors, then I'll describe an ongoing project which addresses a small part of the "chromatic splitting conjecture".
references: Lecture notes.
1:00-2:00 -- 383N Sloan Hall -- Recurrences in Thom spectra
Real projective space is an absurdly important space in homotopy theory. We will review many of the ways in which it is special and show in particular that its cellular structure contains information that makes homotopy theorists drool. Then, we will suggest a program for computing some of it.
references: Lecture notes.
3:05-3:55 -- Littlefield 104 -- Scanning and duality
Scanning is a local-to-global principle in algebraic topology, usually used to study stability properties of e.g. labeled configuration spaces on manifolds. We'll describe McDuff's original scanning map from a modern point of view. Time permitting, we'll explain how scanning generalizes to spaces other than manifolds, and discuss its relationship with Verdier duality.
11:00-12:00 -- Étale Dold--Thom
We shall either give an actual proof or sketch a proposed proof of the analogue of the classic Dold--Thom theorem on the infinite symmetric power in the étale topology.
1:00-2:30(ish) -- Every love story is a GHOsT story: Goerss--Hopkins obstruction theory for ∞-categories
Goerss--Hopkins obstruction theory is a tool for obtaining structured ring spectra from purely algebraic data, originally conceived as the main ingredient in the construction of tmf as an E∞-ring spectrum. However, while the story is extremely beautiful, it is also absurdly intricate. Part of this is because the real mathematical ideas at its core are quite deep, but a good deal of the complexity arises from an overwhelming amount of model-categorical technicalities.
In this talk, I will present a generalization of Goerss--Hopkins obstruction theory for presentable ∞-categories. At this level of abstraction, the entire story becomes...well, perhaps not tautological, but certainly a whole lot clearer. It takes a while to get there, though; running straight through my beamer slides (without anyone asking questions, of course) took me 75 minutes. So, this will be a two-part talk with a short break in the middle (for which I will provide cookies). This all may sound a bit daunting, but I give you my word as a gentleman and a topologist that you will come out with a much better understanding of Goerss–Hopkins obstruction theory than you did going in. Unless you're Mike Hopkins: then I make no promises.
* the xkcd group -- making xkcd not stand for nothing since 2010!