Fall 2012
Time and place: Thursdays 4:30-5:30 in MATH 731
Organizer: Ralph Kaufmann. Contact organizer.

Abstracts

02/16/12 Stephan Stolz (Notre Dame). Traces in Monoidal Categories.
Abstract: In joint work with Peter Teichner we show that an endomorphism $$f$$ of an object $$c$$ in a braided monoidal category $$\sf C$$ has a trace $$\operatorname{tr}(f)\in \sf C(1,1)$$ (the endomorphisms of the monoidal unit in $$\sf C$$), provided $$f$$ is trace class and $$c$$ has the approximation  property. These are classical results for the category of topological vector spaces. Our key observation is that  trace class and approximation property can be formulated in any braided monoidal category. Our statement is a generalization of the classical result that an endomorphism of a dualizable object has a trace in $$\sf C(1,1)$$ (e.g., a vector space is dualizable if and only if it is finite dimensional).

02/23/12 Mike Mandell (Indiana U). Quillen Cohomology of Operadic Algebras and Obstruction Theory.
Abstract: Quillen defined homology in terms of abelianization.  For operadic algebras Quillen homology is the derived functor of indecomposibles, and the bar duality (or derived Koszul duality'') construction provides a model for the Quillen homology.  The $$k$$-invariants of Postnikov towers for algebras over an operad $$\cal O$$ lie in the Quillen cohomology groups, and this leads to an obstruction theory for $$\cal O$$-algebra structures and $$\cal O$$-algebra maps. An application is a proof that the ring spectrum BP has an $$E_4$$ multiplication and that it is unique up to automorphism. (Joint work with Maria Basterra.)

02/28/12 Tom Church (Stanford). Representation Theory and Homological Stability.
Abstract: Homological stability is the remarkable phenomenon where for certain sequences $$X_n$$ of groups or spaces -- for example $$SL(n,Z)$$, the braid group $$B_n$$, or the moduli space $$M_n$$ of genus $$n$$ curves -- it turns out that the homology groups $$H_i(X_n)$$ do not depend on n once $$n$$ is large enough.  But for many natural analogous sequences, from pure braid groups to congruence matrix groups to Torelli groups, homological stability fails horribly.  In these cases the rank of $$H_i(X_n)$$ blows up to infinity, and in the latter two cases almost nothing was known about $$H_i(X_n)$$;  indeed it's possible there is no nice "closed form" for the answers.
Representation stability is a notion which takes into account the action of certain symmetries to meaningfully talk about "the stable homology of the pure braid group" or "the stable homology of the Torelli group", even though the homology never stabilizes. In this talk I will explain our broad picture of representation stability and describe a number of connections to other areas of math.
In particular, I will consider various sequences of integers $$a_n$$ arising in topology (e.g. Betti numbers of spaces on configurations of points, of n-pointed curves, of matrices of rank at most n, etc.) and in algebra/combinatorics (e.g. dimensions of spaces of harmonic polynomials, of coinvariant algebras, of free Lie algebras, etc.), and explain how to use representation stability to prove that for each of these sequences (and many more) there is a polynomial $$P(n)$$ with $$a_n = P(n)$$ for all $$n$$ big enough. Joint work with Benson Farb and Jordan Ellenberg.

03/01/12  Jeremy Van Horn-Morris (AIM/Stanford)  Lefschetz fibrations and open books - using topology to understand symplectic manifolds.

Abstract: Lefschetz fibrations are a classical tool used to study algebraic varieties. More recently, they have become a tool used to study symplectic manifolds and their contact boundaries. We'll discuss the well understood and powerful correlation in dimensions 3 and 4 -- a situation that reduces to highly nontrivial combinatorial data involving mappping class group factorizations -- before moving to higher dimensions.

03/08/12 Sean Lawton. (UTPA) Topology of  Character Varieties of Abelian Groups.
Abstract: Let K be an identity-split compact Lie group (which includes all connected compact Lie groups), G be its complexi cation, and F be any finitely generated Abelian group. We prove that the conjugation orbit space Hom(F,K)/K is a strong deformation retract of the GIT conjugation orbit space Hom(F,G)//G. As a corollary, we determine necessary and sufficient conditions for Hom(F,G)//G to be irreducible when G is connected and semisimple, and F is free Abelian. This is joint work with C. Florentino.

03/29/12 Andrew Toms (Purdue) Homotopy groups of  rank-banded matrices
Abstract:  Motivated by a question in C*-algebra theory, we consider when the homotopy groups of rank-banded self-adjoint complex matrices vanish.  We will explain a sufficiently wide band with respect to s implies the vanishing of the s-homotopy group.