January 20 
Slava Krushkal (UVa) 
Engel relations and 4manifolds
The 2Engel relation, classically studied in group theory, will be used to analyze the topological surgery conjecture, a longstanding problem in 4manifold topology.

January 27 
Ina Petkova (Rice U) 
Combinatorial tangle Floer homology
In joint work with Vera Vertesi, we extend the functoriality in Heegaard Floer homology by defining a Heegaard Floer invariant for tangles which satisfies a nice gluing formula. We will discuss the construction of this combinatorial
invariant for tangles in S^3, D^3, and I x S^2. The special case of S^3 gives back a stabilized version of knot Floer homology.

February 10 
Julia Bennett (UT Austin) 
Large R^4's in Stein surfaces
While many open 4manifolds are known to admit uncountably many diffeomorphism classes of smooth structures, it is still unknown if this behavior can be expected in general. The goal of this talk is to introduce a family of "large"
R^4's with the property that each is contained in some compact Stein surface. We will discuss why these new R^4's are relevant to understanding smoothing theory on more general open 4manifolds and outline their construction.
Along the way, we will define Casson handles and describe their use in cutandpaste arguments.

February 17 
Bulent Tosun (UVA) 
Tight small Seifert fibered manifolds
The existence problem of a tight contact structure on a fixed three manifold is hard and still widely open. The classification problem is even less wellunderstood. In this talk, we will focus on the classification of tight contact structures
on (certain) Seifert fibered manifolds on which the existence problem was settled recently by Lisca and Stipsicz.

February 24 
Elizabeth Denne (Washington & Lee) 
Ribbonlength for knot diagrams
Abstract: Knots and links are modeled as ribbons immersed in the plane. This is a 2dimensional analogue of thick knots and the ropelength problem. This talk will introduce the idea of a flat ribbon link and give examples of 'tight'
ribbonlength knots. It turns out there are some surprising technicalities involved  moving from 3 to 2 dimensions does not necessarily simplify the mathematics.

March 3 
Chris Manon (GMU) 
Toric geometry of moduli spaces of principal bundles on a curve
For $C$ a smooth projective curve, and $G$ a simple, simply connected complex group, let $M_C(G)$ be the moduli space of semistable $G$principal bundles on $C$. As the curve $C$ moves in the moduli $\mathcal{M}_g$ of smooth curves,
the spaces $M_C(G)$ are known to define a flat family of schemes, and this family can be extended to the DeligneMumford compactification $\bar{\mathcal{M}}_g$. We describe the geometry of the fibers of this family which appear
at the stable boundary, in particular we discuss a recent result which shows that the fibers over maximally singular curves contain an important and ubiquitous moduli space, the free group character variety $\mathcal{X}(F_g,
G),$ as a dense, open subspace. The latter is a moduli space of representations of the free group $F_g$ in $G$, and naturally appears as an object of interest in Teichm\"uller theory, the theory of geometric structures, and the
theory of Higgs bundles. For $G = SL_2(\C)$ and $SL_3(\C)$ we describe maximal rank valuations on the coordinate rings of these spaces, and how the associated NewtonOkounkov polyhedra can be used to study the geometry of both $\mathcal{X}(F_g, G)$ and $M_C(G).$

March 17 
James Conway (Georgia Tech) 
Tight Surgeries on Knots in Overtwisted Contact Manifolds
Most work on surgeries in contact manifolds has focused upon determining the situations where tightness is preserved. We will discuss an approach to this problem from the reverse angle: when negative surgery on a fibred knot in an overtwisted
contact manifold produces a tight one. We will examine the various phenomena that occur, and discuss an approach to characterising them via Heegaard Floer homology.

March 24 
Samuel Lisi (U Mississippi) 
Symplectic Homology for Affine Algebraic Varieties
Symplectic homology is a version of Floer homology defined for symplectic manifolds with contacttype boundaries. This invariant detects a number of interesting properties of Weinstein domains, and is very rich in many examples; for instance,
the symplectic homology of a cotangent bundle is isomorphic to the homology of the loop space (by Viterbo; Abbondandolo and Schwarz). I will provide some background and motivation for studying symplectic homology, and will discuss the
computation of symplectic homology for certain affine algebraic varieties in terms of relative GromovWitten invariants. This is joint work with Luis Diogo.

April 7 
Adam Knapp (American Univ) 
Towards a SeibergWitten stable homotopy type for 3manifolds with boundary
I will report on progress in defining an invariant of 3manifolds with boundary in the spirit of C. Manolescu's stable homotopy type for rational homology 3spheres.

April 17 
Mark Powell (MPIM Bonn) 
Casson towers and slice knots
A Casson tower is constructed by thickening a 2complex built from layers of immersed discs in a 4manifold. A Casson tower has a height and an attaching circle, which is the boundary of its base disc. A higher tower, namely a tower with more
layers of immersed discs, is a better approximation to an embedded disc. Casson towers featured prominently in Freedman's original proof of the 4dimensional topological Poincaré conjecture. In particular Freedman showed that an infinite
Casson tower, which is called a Casson handle, contains an embedded disc. The ability to embed discs is the key to being able to apply surgery and hcobordism techniques, originally from high dimensional topology, to 4manifolds. I will
explain what a Casson tower is and present embedding results, from work with Jae Choon Cha, on Casson towers of height 4, 3 and 2. The height 2 result in particular can be used to find a new family of topologically slice knots. I will explain
why we think these slice knots are interesting.

April 21 
Marco Aldi (VCU) 
Odd dimensional generalized geometry
Generalized geometry is the study of the geometry of the generalized tangent bundle (direct sum of the tangent and cotangent bundle) of a manifold. Partly because of the connection with physics, most of the literature on generalized geometry specializes
to evendimensional manifolds. The odddimensional case, pioneered by Vaisman, Wade and Poon, is much less understood. In this talk we report on an ongoing project (joint with D. Grandini) aimed at developing a unified framework that provides
generalized complex geometry concepts (such as type, pure spinors, Tduality etc.) with natural odddimensional counterparts.

April 28 
Radmila Sazdanovic (NC State) 
Relations between chromatic and Khovanov homology
We analyze the algebraic structure of Khovanov homology and related chromatic homology theories, with a focus on understanding torsion. Although computations hint at the abundance of torsion, describing and understanding it is no easy task.
We will discuss current results on 2torsion, which we obtain using relations to chromatic graph homology theories and spectral sequence arguments. We also state nonexistence results for odd torsion in certain gradings of Khovanov homology
of semiadequate knots and links.
