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.

Babytop Seminar

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

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.

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.

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.

03.13.07: No Seminar -- Conference at Johns Hopkins.

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.

03.27.07: No Seminar -- Spring Break.

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).

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.

04.17.07: No Seminar -- Patriot's Day.

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.

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.

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.

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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