Department of Mathematics - Geometry Seminar

Geometry
Fall 2008

Regular time and location: Tuesdays, 2-3pm, 326 Kerchof Hall

September 16	Thomas Mark (UVa) Introduction to handle calculus Abstract: Any manifold can be described in terms of a "handle decomposition." In low dimensions, such a description gives a nice visual representation of a 3- or 4-dimensional manifold in terms of diagrams known as "Heegaard diagrams" (for dimension 3) or "Kirby diagrams" (for dimension 4). Such diagrams and the rules for manipulating them--"Kirby calculus"--have been central in many of the "positive" results in low-dimensional topology: they constitute one of the few tools for proving the existence of diffeomorphisms between 4-manifolds. This lecture is the first in what may be an intermittent series geared mainly for students with interest in geometric topology. We won't have any particular result as a goal, but rather the description of the techniques provided by handle calculus and some illustrations of their use. In this first lecture, I hope to present the basics of handle calculus and exhibit some not-otherwise-obvious diffeomorphisms using them.
September 23	Thomas Mark (UVa) Introduction to handle calculus (continued)
September 30	Slava Krushkal (UVa) Title: The spectral gap, Cheeger constant, and its relatives
October 7	Thomas Mark (UVa) Title: Handle calculus part 3
October 14	(Reading holiday)
October 21	Thomas Mark (UVa) Title: Dissolving 4-manifolds without 1- or 3-handles
October 28	Slava Krushkal (UVa) Title: Introduction to Khovanov homology Abstract: This lecture is part of an ongoing series geared towards students with interest in geometric topology. I will talk about the Jones polynomial of knots and links, and I will describe a basic construction underlying its categorification.
November 4	Ralf Gramlich (U. Birmingham, Technische Universitat Darmstadt) Title: Finiteness properties of some S-arithmetic groups acting on twin buildings
November 11	John Baldwin (Princeton) Title: Khovanov homology, open books, and tight contact structures Abstract: I will discuss a construction, modeled on Khovanov homology, which associates to a surface, S, and a product of Dehn twists, Φ, a group Kh(S, Φ). The group Kh(S,Φ) may sometimes be used to combinatorially determine whether the contact structure compatible with the open book (S, Φ) is tight or non-fillable. This construction generalizes the relationship between the reduced Khovanov homology of a link and the Heegaard Floer homology of its branched double cover.
November 18
November 25	Thanksgiving Break
December 4* (Note special day)	Jeremy Van Horn-Morris (University of Quebec at Montreal) Title: Monoids and Contact Structures Abstract: The study of contact structures on 3-manifolds changed radically by Giroux's rather surprising correspondence with open book decompositions. Abstractly, an open book can be thought of as a pair $(\Sigma, \phi)$ where $\Sigma$ is a compact bordered surface and $\phi$ is an automorphism of $\Sigma$. Work by Goodman, Honda-Kazez-Mati\'{c}, and Giroux and Baldwin showed that one could extract rather strong geometric information directly from the monodromy $\phi$, and that this data was encoded by certain monoids in the mapping class group of $\Sigma$. We cement this analogy and show that most properties of interest to a contact geometer have a corresponding monoid in the mapping class group.

January 20
January 27	Slava Krushkal (UVa) Title: Polynomial invariants of links and graphs on surfaces
February 3
February 10
February 17	Matthew Hogancamp (UVa) Title: On Khovanov's categorification of the Jones polynomial
February 24	Sean Droms (UVa) Title: Elementary surgeries on torus knots
March 10
March 17	Dan Margalit (Tufts University) Title: Braid groups and spin structures on surfaces Abstract: I will discuss some work in progress with Tara Brendle regarding the symmetric mapping class group, the spin subgroup of the mapping class group, the braid groups, and the Torelli groups.
March 24	Matthew Hogancamp (UVa) Title: Rasmussen's proof of the Milnor conjecture
March 31	Matthew Hogancamp (UVa) Title: Rasmussen's proof of the Milnor conjecture (continued)
April 7	Tom Mark (UVa) Title: Monodromy relations and rational blowdowns of Lefschetz fibrations
April 14	Robert McEwen (UVa) Title: A proof that the complex of trees interpolating between n vertices is n - 4 spherical Abstract: The complex of (at least tri-valent) trees that interpolate between n vertices is known to be n - 4 spherical by, among others, a result of Karen Vogtmann. This complex can also be described as the complex with vertices the two-block partitions with block size at least two of the set {1, 2, ..., n} and k simplices given by a collection of k+1 compatible partitions. Using this description, I will introduce a filtration on the complex and show that each stage has the desired spherical property. Time permitting, I will show the connection between the pieces of the filtration and search for a subfiltration of A(n,k), Auter space of graphs with fundamental group isomorphic to the free group on n generators and degree equal to k.
April 21	Elisenda Grigsby (Columbia) Title: On Khovanov- and Heegaard Floer-type homology theories Abstract: Khovanov and Heegaard Floer homology, two theories inspired by ideas in physics, have transformed the landscape of low-dimensional topology in the past decade. The philosophies underlying the theories' constructions are quite different, yet there are intriguing connections between the two. In this talk, I will focus on one such connection: a relationship between a reduced version of Khovanov homology and a relative version of Heegaard Floer homology recently developed by Andras Juhasz. This relationship can be used to prove that Khovanov's categorification of the reduced n--colored Jones polynomial detects the unknot when n>1; furthermore, the relationship, in its most general form, satisfies nice naturality properties with respect to standard TQFT-type operations like cutting and stacking. This is joint work with Stephan Wehrli.
April 28	Stefan Witzel (Technische Universität Darmstadt) Title: Finiteness properties of unitary forms of Kac-Moody groups Abstract: Topological finiteness properties of a group are generalizations of being finitely generated or being finitely presented. The unitary form K of a Kac-Moody group G is the centralizer of a certain involutory automorphism of G. We investigate finiteness properties of K by showing that a geometry on which K acts cocompactly with finite stabilizers is highly connected. The methods used here are a variation of those used to show analogous results for the Borel group of G.

*indicates different day, time, and/or location

Contact: Thomas Mark (tmark at virginia.edu)