# Topology Seminar 2016--2017

Speaker: Don Stanley (U.Regina)

Date/Time: April 6th, 11am--11:50am

Title: An Algebraic Generalized Blakers-Massey Theorem

Abstract: We give an algebraic version of the Blakers-Massey Theorem for n-cubes. The proof uses the bar construction and also give an alternative prove of the traditional topological version for n-cubes. This is joint work with Yang Hu.

Speaker: Arnaud Songhafouo (U.Regina)

Date/Time: April 6th, 1:30pm--2:20pm

Title: Classification of very good homogeneous functors

Abstract:  Let M be a manifold, and O(M) be the poset of open subsets of M. A homogeneous functor of degree k from O(M) to K-vector spaces is called very good if it sends isotopy equivalences to isomorphisms. We show that the category of such functors is equivalent to the category of linear representations of the fundamental group of Conf(k, M) over K (here Conf(k, M) stands for the configuration space of k points in M). This is a joint work with Don Stanley.

Speaker: Victor Turchin (Kansas State)

Date/Time: April 7th, 10am--10:50am

Title: Spaces of long embeddings and mapping spaces of the little discs operads

Abstract:  In my talk I will discuss two recent works of mine: one with J. Ducoulombier and another one with B. Fresse and T. Willwacher. In the first one we show that for n-m>2, the space of embeddings of discs fixed on the boundary and taken modulo immersions is the (m+1)st loop space of the derived mapping space Oper^h(B_m,B_n) of the little discs operads. This result was proved earlier by Boavida de Brita and Weiss. Our approach works for more general mapping spaces. (The same result was announced by Dwyer and Hess several years ago, but their paper never appeared. Our method is very different from theirs.)  The second work describes the rational homotopy type of Oper^h(B_m,B_n), n-m>2.

Speaker: Pascal Lambrechts (Louvain la neuve)

Date/Time: April 7th, 12:30pm--1:20pm

Title: Cosimplicial models for manifold calculus.

Abstract: Manifold calculus (or Goodwillie-Weiss calculus) is a way to associate to some good contravariant functor F:{m-Manifolds}--->TOP (or some other category than TOP in which we can do homotopy) a sequence of approximations T_k(F):{m-Manifolds}--->TOP        for k=1,2,...,infty. This theory works very well for the functor Emb(-,W) of embeddings in a fixed manifold in which case  T_inftyEmb(-,W) is equivalent to Emb(-,W), when dim(M)>m+2, by a deep theorem of Goodwillie-Klein. In this talk I will explain how to   construct an explicit cosimplicial model of T_infty(F(M)) starting from a simplicial model of M. This generalizes Sinha's cosimplicial models of space of knots to any functor, in particular to embedding spaces of any manifold in another manifold, from a simplicial model of the source. This yield to cute small models for T_inftyF(M). Joiint work with Pedro Boavida, Daniel Pryor and Arnaud Songhafouo.

Speaker: Robin Koytcheff (U.Mass Amherst and soon U.Louisiana-Lafayette)

Date/Time: April 7th, 1:30pm--2:20pm

Title: Configuration space integrals and integer-valued cohomology classes in spaces of knots and links

Abstract:  Configuration space integrals generalize the Gauss linking integral.  They can be used to construct all Vassiliev invariants of knots and links, as well as nontrivial real-valued “Vassiliev classes” in the cohomology of spaces of knots and links.  I will explain how these integrals can be reinterpreted topologically to recover an integer lattice among the real-valued Vassiliev classes.  This work also provides constructions of mod-p classes which need not be mod-p reductions of classes in this integer lattice.

Speaker: Tom Baird (Memorial, Newfoundland)

Title: Kirwan surjectivity in real symplectic geometry and moduli spaces of vector bundles over a real curve

Abstract:  In the early 80s, Kirwan proved a relationship between the equivariant cohomology of a Hamiltonian action on a symplectic manifold, and the cohomology of its symplectic quotient. I present a version of this relationship for symplectic manifolds equipped with an anti-symplectic involution, relating the cohomology of corresponding fixed point Lagrangian submanifolds. I then apply this result to study the topology of moduli spaces of vector bundles and a real algebraic curve, in the style of Atiyah-Bott.

Date / Time: October 21st, 1:30pm

Location: CLE C108

Speaker: Bala Krishnamoorthy (Washington State University, Vancouver)

Title: Linear Programming in Geometric Measure Theory

Abstract: We present results on two problems related to shapes in geometric
measure theory (GMT) that employ techniques from algebraic topology
and linear programming. Currents represent generalized surfaces in
GMT, and were introduced to study area minimizing surfaces and other
related problems. The flat norm provides a natural distance in the
space of currents, and works by decomposing a d-dimensional current
into d- and (the boundary of) (d+1)-dimensional pieces. A natural
question about currents is the following. If the input is an integral
current, i.e., a current with integer multiplicities, can its flat
norm decomposition be integral as well?  Surprisingly, the answer is
not known in general. On the other hand, for the discretization of the
flat norm on a finite simplicial complex, the analogous statement is
true for d-chains in a (d+1)-complex. This result is implied by the
boundary matrix of the simplicial complex being totally unimodular,
guaranteeing integer solutions for an associated integer linear
program. We develop an analysis framework that extends the result in
the simplicial setting to that for d-currents in (d+1)-dimensional
space, provided a suitable triangulation result holds. We also prove
this result holds in 2D. In the second problem, we consider a notion
of average shape defined as the median shape of currents using the
flat norm distance. In the corresponding simplicial version of the
problem, the median chain of a set of input chains in a finite
simplicial complex is computed using linear programming.

Date / Time: Friday October 14th, 1:30pm

Location: CLE C108