Babytop Seminar

The seminar will meet Tuesdays at 4:00pm in 2-146 unless otherwise noted.

04.22.08: Student Holiday.

04.15.08: No babytop this week..

The category $Ch(R)$ of unbounded chain complexes of modules over a
ring $R$ is a nice algebraic example of a model category. The
homotopy category of $Ch(R)$ is the derived category of $R$, and
homotopy classes of maps can be used to compute Ext groups for
$R$-modules.
I will present the standard cofibrantly generated model structure on
$Ch(R)$, describe its fibrant and cofibrant objects, and talk about
ties to homological algebra.

Time permitting, I'll introduce the concept of a projective class on an abelian category $A$ and discuss a generalized existence question: when does the category of chain complexes of objects in $A$ have a model category structure that reflects the relative homological algebra associated to a given projective class?

$\Pi$-algebras are algebraic objects that behave like the collection of homotopy groups of a pointed space. However, not all $\Pi$-algebras arise that way. In this talk, I will introduce this notion, and try to tackly the question: Given a $\Pi$-algebra $A$, can it be realized topologically, i.e. is there a space $X$such that $A$ is isomorphic to $\pi_*x$?

The key idea is to consider the moduli space of all such topological realizations $X$. Then, building it one stage at a time, the problem can be reduced to an obstruction-theoretic one.

03.25.08: Spring Break.

Let $S$ be a Sylow $p$-subgroup of a finite group $G$; elementary transfer
arguments show that the mod-$p$ cohomology of $BG$ is a subring of the
mod-$p$ cohomology of $BS$. More is true: The classical theorem of
Cartan-Eilenberg identifies $H^*(BG)$ as the subring of "$p$-stable"
elements of $H^*(BS)$.

We seek a sort of generalization of this phenomenon. Instead of looking at a cohomology from the eyes of the prime $p$, we consider instead the $p$-completion of our classifying space, $BG^\wedge_p$ (thought of as the $BG$ seen from the eyes of $p$). Martino and Priddy asked the following question: Given two finite groups $G$ and $H$, when do we have a homotopy equivalence of their $p$-completed classifying spaces? They conjectured that this is the case if and only if their "$p$-local data" is the same, and proved the only if implication. Later, the reverse implication was proved by Oliver using the work of Broto-Levi-Oliver (and the Classification of Finite Simple Groups).

The goal of this talk is to explain what is meant by "$p$-local data," which will lead naturally to the work of BLO. Interested parties will finally get to hear what these "saturated fusion systems," "centric linking systems," and "$p$-local finite groups" that I've been jabbering about are, at least as concepts. Rigor will depend on time constraints and my mood.

Last semester I talked about some definitions of higher categories.
Not satisfied, I will make a renewed more serious attempt: this time we
will explore less ad hoc approaches to relevant categories of shapes
(e.g. the category $\Delta$).

It has been observed that certain geometrically interesting moduli spaces
such as configuration spaces have the structure of an operad. I will
describe this structure. If time permits, I will also describe how the
Deligne-Mumford compactification of the moduli spaces of curves forms an
operad.

The category of dendroidal sets is an extension of that of simplicial sets, which encodes information about the homotopy theory of operads in the same ways as simplicial sets encode information about categories. The talk will be a summary of Ieke Moerdijk's talks in Barcelona.

Triangulated categories, developed by Puppe and Verdier in the 1960's,
are additive categories with an invertible suspension functor and a
class of cofiber sequences satisfying a few axioms. They are meant to
model stable homotopy theories (e.g. spectra, the homotopy category of
chain complexes of R-modules, stable modules over a Frobenius
algebra). Recently, Muro, Schwede and Strickland exhibited an example
of a triangulated category which admits no maps to or from any full
subcategory of the homotopy category of a stable model category. I'll
talk about their construction and give an idea of the proof.

02.12.08: Taken over by the Grown-ups Topology Seminar.

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