||John Baldwin (Boston College)
||A Bordered Monopole Floer Theory
I’ll discuss work-in-progress with Jon Bloom toward constructing a gauge theoretic analogue of the Fukaya category and monopole Floer theoretic invariants of bordered 3-manifolds. Our construction associates an A-infinity category to a surface, an A-infinity functor to a bordered 3-manifold, and an A-infinity natural transformation to a 4-dimensional cobordism of bordered 3-manifolds. In this talk, I'll discuss the (2+1)-dimensional aspects of the theory and describe a finite set of bordered handlebodies which "generate" our category.
||Michael Freedman (Microsoft Station Q)
||"Over-gauging a field"
Abstract: I will explore what happens to U(1) gauge theory if we allow the vector potential to explore a narrow cone around su(1) within su(1,1). There is surprising decoupling between curvature and characteristic classes leading to nontrivial principal SU(1,1) bundles with flat connections. This suggest a novel Lagrangian generalizing the Ginsberg- Landau formulation of superconductivity. In this context, the familiar tradeoff between curvature, ||F_A||^2 , and covariant derivative,||D_A (phi)||^2, becomes more complicated: Energy can be also be shifted to a Beltrami term or stored in topology.
A pseudo-meridian of a knot k in the three-sphere is a killer of the group of k that is not a meridian. We will see that any pseudomer of k is not the automorphic image of a meridian and can never be peripheral. We will also look into the meaning and existence of pseudomers, give several examples, and pose a number of questions and conjectures. (This is joint work with D. S. Silver and S. G. Williams.)
||Ben Cooper (University of Zurich)
||Steenrod structures on quantum groups
Abstract: I will discuss Steenrod structures on algebras related to knot homology theories (work with A. Beliakova and H. Queffelec)
||Allison Gilmore (UCLA)
||On the knot Floer cube of resolutions
Most knot homology theories were originally defined algebraically using a cube of resolutions approach. The commonalities in these algebraic definitions contributed to the discovery of spectral sequences relating the various theories. Knot Floer homology has a cube of resolutions construction (due to Ozsvath and Szabo), and should lie at the end of a spectral sequence from HOMFLY-PT homology (according to a conjecture of Dunfield, Gukov, Rasmussen), but so far these facts have not come together. In this talk, we describe the similarities between the knot Floer and HOMFLY-PT cubes of resolutions, then account for the main difference: the appearance of a mysterious ``non-local ideal'' in the knot Floer construction. Our main result expresses this non-local ideal in terms of algebraic objects that appear the HOMFLY-PT construction. We will sketch a proof via Grobner basis techniques, then explore potential applications to the spectral sequence conjecture.
||Shea Vela-Vick (LSU)
||Transverse knots, infinite cyclic covers and Heegaard Floer homology
In recent years, Heegaard Floer theory has proven an invaluable tool for studying contact manifolds and the Legendrian and transverse knots they contain. I plan to discuss a method for defining a variant of Heegaard Floer theory for infinite cyclic covers of transverse knots in the standard contact 3-sphere. This invariant generalizes one defined in joint work with Baldwin and Vertesi for transverse knots braided about open book decompositions. In this talk, I will discuss how our invariant is constructed and present some basic properties. This is joint work with Tye Lidman and Sucharit Sarkar.
||Michael Abel (UNC Chapel Hill)
||Virtual crossings and filtrations in HOMFLY-PT homology
Soergel and Rouquier categorified the Hecke algebra by complexes of special bimodules called Soergel bimodules. HOMFLY-PT homology can be realized as the homology of these complexes after replacing the bimodules with their Hochschild homology. Soergel bimodules also have a filtration by what are known as standard bimodules, which can be used to represent virtual crossings in this categorification. We can use this filtration to represent Soergel bimodules as mapping cones of virtual crossings after passing to the correct category. I will review the construction of HOMFLY-PT homology and then I will present joint work with L. Rozansky which shows that this resolution into virtual crossings gives rise to both a more diagrammatic calculus of HOMFLY-PT homology and a filtered triply-graded homology theory which is a link invariant. We can use this to define a new fourth grading on HOMFLY-PT homology.
||Bulent Tosun (UVa)
||Contact surgery and contact invariant
One of the fundamental problems in 3-dimensional contact geometry is the existence of a (tight) contact structure on a fixed 3-manifold. One is, in particular, interested in explicit construction of such structures. In my talk, I will first review known results which hopefully will motivate the notion of contact surgery, then will explain main result that answers the fundamental problem above for some class of three manifolds.
||Adam Knapp (American U)
||Cotangent bundles of open 4-manifolds
We show that, if X_1 and X_2 are two open homeomorphic
smooth 4-manifolds, then the cotangent bundles of X_1 and X_2 are symplectomorphic with their natural symplectic structures.
||Paul Melvin (Bryn Mawr College)
||Exotic 2-spheres in 4-manifolds
It is a well known principle in 4-dimensional topology that homeomorphic smooth simply-connected 4-manifolds become diffeomorphic after "stabilizing" (meaning connected summing with a product of two 2-spheres) sufficiently many times. Many explicit examples are known for which exactly one stabilization is required, but no examples have been shown to require more than one. Perhaps all these exotic smooth structures "dissolve" quickly.
An analogue of this principle holds for 2-spheres embedded in a smooth, simply-connected 4-manifold, namely, any two that are topologically isotopic become smoothly isotopic after stabilizing the manifold sufficiently many times. Until now, however, no bounds on the number of stabilizations needed have been found. In this talk, we will explicitly describe pairs of topologically isotopic knots that are not smoothly isotopic, but become so after one stabilization. This is joint work with Dave Auckly, Hee Jung Kim and Danny Ruberman.
||Distortion and thickness in Euclidean spaces
Distortion and thickness are measures of geometric complexity of embeddings of manifolds and simplicial complexes into Euclidean spaces. They have been studied in geometry, analysis, combinatorics and CS. I will survey recent results in the subject and I will discuss new qualitatively different estimates when the ambient dimension is twice the dimension of the complex.
||Thomas Mark (UVa)
||Stein fillings and a relative symplectic isotopy problem
I will describe recent progress on the problem of classifying Stein fillings of 3-manifolds with contact structures supported by planar open books. In particular I will show how this problem can be reduced to a relative symplectic isotopy problem for a disjoint union of disks in a blowup of the 4-ball having given boundary, and outline a solution to this problem. As a consequence, we will see that any planar contact structure admits only finitely many Stein fillings, up to symplectic deformation. This is joint work with Chris Wendl.
||Grassmannians and Khovanov homology
||Wild Embeddings in Manifolds
A survey of some classical techniques and results related to nonstandard (or “wild”) embeddings into manifolds of standard spaces such as Cantor sets, cells and spheres. This talk is intended for non-experts and is appropriate for students having some familiarity with standard results in topology.