January 20 |
Slava Krushkal (UVa) |
Engel relations and 4-manifolds
The 2-Engel relation, classically studied in group theory, will be used to analyze the topological surgery conjecture, a long-standing problem in 4-manifold 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 4-manifolds 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 4-manifolds and outline their construction. Along the way, we will define Casson handles and describe their use in cut-and-paste 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 well-understood. 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 2-dimensional 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 Deligne-Mumford 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 Newton-Okounkov 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 contact-type 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 Gromov-Witten invariants. This is joint work with Luis Diogo. |

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

April 17 |
Mark Powell (MPIM Bonn) |
Casson towers and slice knots
A Casson tower is constructed by thickening a 2-complex built from layers of immersed discs in a 4-manifold. 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 4-dimensional 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 h-cobordism techniques, originally from high dimensional topology, to 4-manifolds. 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 even-dimensional manifolds. The odd-dimensional case, pioneered by Vaisman, Wade and Poon, is much less understood. In this talk we report on an on-going project (joint with D. Grandini) aimed at developing a unified framework that provides generalized complex geometry concepts (such as type, pure spinors, T-duality etc.) with natural odd-dimensional 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 2-torsion, which we obtain using relations to chromatic graph homology theories and spectral sequence arguments. We also state non-existence results for odd torsion in certain gradings of Khovanov homology of semi-adequate knots and links. |