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
complexication, 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.