09/08/11 Lizhen Qin (Purdue).
On
Moduli Spacnes and CW Structures from Morse Theory
Abstract: This
talk will focus on various results concerning the moduli spaces of
broken flow lines associated with a Morse function. We study the
compactness of flow lines, manifold structures of certain compactified
moduli spaces, orientation formulas, and CW structures on the
underlying manifold.
09/15/11 Ben Ward (Purdue).
Cyclic
$A_\infty$ Algebras and Deligne's Conjecture
Abstract:
First we describe a class of homotopy Frobenius algebras via cyclic
operads which we call cyclic $A_\infty$ algebras. We then define
a suitable new combinatorial operad which acts on the Hochschild
cochains of such an algebra in a manner which encodes the homotopy BV
structure. Moreover we show that this operad is equivalent to the
cellular chains of a certain topological (quasi)-operad of CW complexes
whose constituent spaces form a homotopy associative version of the
Cacti operad of Voronov. These cellular chains thus constitute a
chain model for the framed little disks operad, proving a cyclic
$A_\infty$ version of Deligne's conjecture.
09/22/11 Ben McReynolds (Purdue).
Geometric variants of topological bound problems
Abstract: In 1982, Hamrick and Royster proved that every compact
flat n-manifold (topologically) bounds. Motivated by work of Gromov,
Farrell and Zdravkovska conjectured that one can solve this bounding
problem with a manifold that admits a complete negatively curved metric
in it's interior. One candidate class of manifolds are finite volume,
noncompact hyperbolic n-manifolds. These manifolds have cusp ends that
are topologically flat (n-1)-manifolds and so produce examples of
manifolds with flat boundaries. I will discuss my work on the geometric
classification of these ends. The main technical tools employed are
results that resolve various types of singularities under passage to
some finite degree covering space.
10/06/11 Justin Young (IU).
Brace Bar-Cobar Duality
Abstract:
After providing motivation for studying the E_2 algebra structure
on the cochain complex of a space, we will consider the category of dg
E_2
algebras. We will show that the classical bar-cobar duality between dg
E_1 algebras
and dg coalgebras can be enhanced to a duality between dg E_2 algebras
and dg Hopf algebras. Over a field, this induces an isomorphism of
homotopy categories
between E_2 algebras and Hopf algebras. If time permits, I will discuss
work in progress
attempting to apply this duality to prove a conjecture of Mandell that
with
connectivity and dimension restrictions, and after inverting
sufficiently
many primes, the cochain complex of a space is equivalent as an E_2
algebra
to a commutative algebra.
10/20/11 Jim McClure (Purdue). Verdier
Duality and the Cap Product
Abstract:
It is well-known that there are two ways to prove Poincare duality,by
using Verdier duality or by using the cap product. In this talk
we show that the Poincare duality isomorphisms obtained by these two
methods are the same. As a byproduct we show that the proof using
Verdier duality can be done
more simply without Verdier duality.
11/03/11 Angelica Osorno (U of Chicago). 2-vector spaces, 2-representations and
2-characters
Abstract: In this talk we will describe
an attempt to categorify certain notions in linear algebra and
representation theory. We will define the notions of 2-vector space and
2-representation of a group. We will then define the 2-character of a
2-representation, which is a function on commuting pairs of elements in
G, and derive expressions for the 2-character of a direct sum, tensor
product, and induced 2-representations.
We will then establish a
connection with the character theory developed by Hopkins, Kuhn and
Ravenel, and state some conjectures of how this
connection can be made more concrete.
Cancelled. Peter Zograf
(Steklov and Simons
Center Stony Brook). Large genus
asymptotics of intersection numbers on moduli spaces of algebraic curves.
Abstract: The aim of the
talk is to review the latest developments in understanding the large
genus behavior of intersection numbers of tautological classes on
moduli spaces of (pointed) algebraic curves, with the main emphasis on
the Weil-Petersson volumes (after a recent work by M. Mirzakhani and
the speaker).
11/17/11. Noah Snynder (Columbia). Local Topological
Field Theory and Fusion Categories
Abstract: A
fusion category is a category that looks like the category of
representations of a finite group: it has a tensor product, duals, is
semisimple, and has finitely many simple objects. A somewhat mysterious
fact about fusion categories (generalizing a theorem of Radford's about
Hopf algebras) is that the quadruple dual functor is canonically
isomorphic to the identity functor. The goal of this talk is to
explain this mystery by showing that it follows directly from the Dirac
belt trick.
The main technique in this proof is the construction of a local
topological field theory attached to any fusion category.
Topological field theories are invariants of manifolds which can
be computed by cutting along codimension 1 boundaries. Local
topological field theories allow cutting along lower codimension
boundaries. Since manifolds with corners can be glued together in
many different ways, this can be formalized using the language of
n-categories. Using Lurie's cobordism hypothesis, we describe
local field theories with values in the 3-category of tensor
categories. In addition to a transparent topological proof of
Radford's theorem, I will also explain the occurrence of the
technical notion of "spherical" in Turaev-Viro TFTs: any fusion
category gives a 3-framed TFT, while a spherical structure gives an
SO(3) homotopy fixed point and thus an oriented field theory.
