Babytop Seminar

The seminar will meet Tuesdays at 4:00pm in 2-131 unless otherwise noted.

02.13.07: Ricardo Andrade (MIT). Definitions of algebraic K-theory.
This talk will be primarily about Waldhausen's definition of K-theory. Nonetheless, using Waldhausen's setup we will be able to obtain some comparison results with other standard definitions of algebraic K-theory.
02.20.07: Martin Frankland (MIT). LS-category in homotopy theory.
Historically, the Lusternik-Schnirelmann category was first developed for analytical purposes, related to critical point theory and dynamical systems. However, major results by Tudor Ganea in the 1960's made it very relevant to topology, especially homotopy theory. We will give an overview of LS-category: basic properties, equivalent definitions, and some tools to estimate it. Time permitting, we will see how it relates to constructions in homotopy theory, such as localization of spaces, rational homotopy, and Hopf invariants.
02.27.07: Sam Isaacson (Harvard). Koszul duality for operads.
Quillen showed in the paper "Rational Homotopy Theory" that the rational homotopy category is equivalent to both the category of one-connected differential graded (DG) Lie algebras over Q and two-connected DG cocommutative coalgebras. I'll talk about the relationship between these categories from an operadic perspective due to Ginzburg and Kapranov.
03.20.07: Jennifer French (MIT). Pants and Copants.
Khovanov used 2-dimensional topological quantum field theories to categorify the Jones polynomial, giving a homology theory for knots and links whose Euler characteristic recovers the Jones polynomial. I will define TQFTs and construct the state model of the Jones polynomial in order to construct Khovanov homology; I intend to discuss further categorifications of the theory found in Bar--Natan's exposition using the language of Lauda and Pfieffer and applications in other areas of geometry.
04.03.07: Samik Basu (Harvard). Smith-Toda complexes.
A Smith-Toda complex is a finite p-local spectrum whose BP homology is isomorphic to BP_*/(p,v_1, v_2,.. v_k). These are closely related to the construction of Greek letter elements in the Adams spectral sequence. I will talk about some non existence results of Smith Toda complexes using calculations in real K theory and EO_(p-1).
04.10.07: Matt Gelvin (MIT). An introduction to Hopkins-Kuhn-Ravenel character theory.
Undeterred by its recent defeat by Floer homology, HKR character theory turns to Babytop to have its voice heard. This basic overview will begin with a reminder of a few results from the classical character theory of finite groups that HKR aims to generalize. In particular, we will give an analogue of Artin's theorem stating that the character ring of a finite group is rationally determined by its cyclic subgroups. We will go on to give a short description of the computation of the Morava K(n)-theory Euler characteristic of BG in terms of commuting n-tuples of p-power elements of G. Time permitting, there may be a vague description of what generalized HKR characters are at the end.
04.24.07: Tyler Lawson (MIT). Hochschild homology and the transfer.
I will talk about Hochschild homology HH(A) of an algebra A, the "Morita invariance" of this construction, and how one uses it to construct a transfer morphism HH(B) -> HH(A) when B is a "nice" A-algebra. I'll finish by talking a little bit about topological Hochschild homology and how one can similarly define a transfer under some analogous circumstances.
05.01.07: Angelica Osorno (MIT). The Representation Ring and K-theory.
Following Matt's trend, I will give a talk that could have been part of Juvitop if it had been about HKR. We will prove that the completion of the representation ring of a finite group G is isomorphic to the K-theory of BG. This and Artin's theorem were partially the motivation for HKR character theory.
05.08.07: Justin Noel (University of Chicago). HKR Character Theory and Power Operations.
In Babytop this Tuesday, I will give the next installment in the block of lectures on HKR character theory. I will describe how generalized characters are used in building power operations for the Lubin-Tate theories. I will review some general facts about power operations and generalized character theory in the process.
05.15.07: Teena Gerhardt (MIT). BABYTOP JEOPARDY!.
In the final meeting of Babytop for the year, we will be playing BABYTOP JEOPARDY! We will test the collective knowledge of the group with questions on topology, topologists, and all things topological. Esteemed Babytop alumni Andre Henriques, Chris Douglas, Vigleik Angeltveit, and Mike Hill have written challenging questions in categories such as "E-infinity ring spectra" and "Haynes Miller." There will be food. There will be fun. There may even be prizes.

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