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Geometry and Topology 2018 Spring

Spring 2018
Time & Location: All talks are on Thursday in Gibson Hall 308 at 12:30 PM unless otherwise noted.
Organizer: Slawomir Kwasik

January 25
Organizational Meeting

Mentor Stafa - Tulane University
Abstract: TBA

February 1
Knot concordance in homology spheres

Jenifer Hom - Georgia Tech
The knot concordance group C consists of knots in S^3 modulo knots that bound smooth disks in B^4, with the operation induced by connected sum. We consider various generalizations of the knot concordance group, and compare these to the classical case. This is joint work with Adam Levine and Tye Lidman.

Fri. February 2
Special date and time
Representation stability, homological stability, and commuting matrices.

Daniel Ramras - Institution: Indiana University – Purdue University Indianapolis
Spaces of commuting matrices have received considerable attention in the last 20 years, starting with conjectures of Witten  regarding their connected components. The rational homology of these spaces can be described quite explicitly in terms of classical Weyl group invariants. These descriptions expose a surprising stability pattern, and I'll discuss work in progress (joint with Mentor Stafa) regarding applications of representation stability (in the sense of Church and Farb) to this phenomenon.  No knowledge of representation stability will be assumed.
Location:  Gibson Hall 308
Time:  9:00 AM

February 8
Fixed point indices and 2-complexes

Michael Kelly - Loyola University
Given a self-map of a compact, connected topological space we consider the problem of determining upper and lower bounds for the fixed point indices of the map.  To obtain bounds one needs to restrict attention to the class of spaces considered and also the class of self-maps.  Motivated by an elementary result in the case of a 1-dimensional complex this talk will focus attention to the setting of 2-complexes.  Some past results and related examples will be presented, leading to some current joint work with D. L. Goncalves (U. Sao Paulo, Brasil).

May 3
Graph complexes, formality, and configuration space integrals for braids

Robin Koytcheff - University of Louisiana, Lafayette
In joint work with Rafal Komendarczyk and Ismar Volic, we study the space of braids, that is, the loop space of the configuration space of points in a Euclidean space.  We relate two different integration-based approaches to its cohomology, both encoded by complexes of graphs.  On the one hand, we can restrict Bott-Taubes configuration space integrals for the space of long links to the subspace of braids.  On the other hand, there are integrals for configuration spaces themselves, used in Kontsevich’s proof of the formality of the little disks operad.  Combining the latter integrals with the bar construction and Chen’s iterated integrals yields classes in the space of braids, extending a result of Kohno.  We show that these two integration constructions are compatible by relating their respective graph complexes.  As one consequence, we get that the cohomology of the space of long links surjects onto the cohomology of the space of braids.

May 8
The $3$-dimensional Sol-manifolds and the automorphims of the fundamental group

Daciberg Goncalves - University of Sao Paulo
We describe in detail the family of the 3-manifolds which admits geometry Sol.  This description is topological in the sense that it is given in terms of bundles.  We provide some recent results about these manifolds which are connected with fixed points theory.  Then using a presentation for the fundamental group $G$ of the manifold, we describe a procedure to determine the group of  the automorphisms of $G$. These classification of such manifolds is close related with the group $Gl(2,Z)$. We describe several properties of the group $Gl(2,Z)$ that are used to provide the results for many such matrices. For the particular case of torus bundle, which is simpler than the general case, we obtain that $Aut(G)$ is the middle term of the  short exact sequence $$ 1 \arrow{r}  \Aut_0(G) \arrow{r}  \Aut(G) \arrow{r}{\psi}   Z_2 \arrow{r}  1$$,  where   $ \Aut_0 (G)\cong   ((Z\oplusZ)\rtimes\limits_{M_0} Z)\rtimes_{\phi} Z_2$.
Location: Stanley Thomas 316
Time: 12:30