Sabin Cautis Knot homology using algebraic geometry
I will describe an algebraic geometer's perspective on
knot invariants. This includes a geometric construction of knot invariants such
as Khovanov homology by looking at certain flag-like spaces.
Jian He Symplectic field theory of subcritical Stein manifolds
In this talk we will discuss how to define symplectic invariants for Stein manifolds. Under the assumption $c_{1}=0$, the cylindrical contact homology and the rational contact homology algebra of the boundary of a Stein domain is determined. The computation of these invariants also has applications for closed manifolds, in particular a monotone Kahler manifold with a subcritical polarization is uniruled.
Jesse Johnson Bounding the stable genus of Heegaard splittings from below.
I will describe a way to construct for each positive integer k a 3-manifold with Heegaard splittings of genus 2k-1 and 2k such that the smallest common stabilization of these Heegaard splittings has genus at least 3k-1.
Jonathon Keiter Layered Triangulations and Genus Two Heegaard Splittings
This talk first presents a visual way to see the layered triangulation of one-vertex triangulated 3-manifold formed from the layering onto the one tetrahedron triangulation of the solid torus. Interesting patterns of symmetry can be shown. In the layered triangulations of a genus two handlebody, different obstacles exist in trying to determine properties of the triangulation. I will present sufficient conditions that the resulting triangulation of a genus two Heegaard splitting formed from layering is 0-efficient, one with the vertex linking 2-sphere as the only normal sphere.
Adam Knapp Monopoles and Montesinos Twins
"Montesinos Twins" are a configuration of 2-spheres in S^4 originally defined by J. Montesinos. We define a invariant of Montesinos twins related to the Fintushel-Stern knot surgery on E(2) and give a "skein relation" which computes this invariant in some cases.
Scott Morrison Khovanov homology as a 4-manifold invariant
(Joint work with Kevin Walker.) I'll describe how Khovanov homology forms a "lasagna algebra". This is a 2+2 dimensional analogue of the 1+1 dimensional notion of a planar algebra; it's a good axiomatisation of what you might otherwise call "a braided tensor 2-category with duals". I'll then explain how this lasagna algebra structure allows us to define the Khovanov homology of a link in the boundary of an arbitrary 4-manifold. Hopefully I'll have time to sketch a calculation, as well as to explain why our definition isn't the right one!
Kei Nakamura Fox re-embedding and Bing submanifolds
Let M be an orientable closed connected 3-manifold, and Y be a connected compact 3-manifold. We show that the following two conditions are equivalent: (i) Y can be embedded in M so that the closure of the complement of the image of Y is a union of handlebodies; and (ii) Y can be embedded in M so that every embedded closed loop in M can be isotoped to lie within the image of Y. Our result can be regarded as a common generalization of Fox's reimbedding theorem (1948) and Bing's characterization of 3-sphere (1958), as well as more recent results of Hass and Thompson (1989) and Kobayashi and Nishi (1994).
Alexandra Pettet Minimality of the well-rounded retract
The well-rounded retract of SL_n(Z), first described by Soule and Ash, is an equivariant deformation retract of the associated symmetric space, having minimal dimension, the virtual cohomlogical dimension of SL_n(Z). We prove that the well-rounded retract is minimal in the sense that it contains no proper, closed, contractible, invariant subsets. This is joint work with Juan Souto.
Shawn Rafalski Immersions of Hyperbolic Turnovers in 3-Orbifolds
Take two copies of a hyperbolic triangle with interior angles pi/p, pi/q and pi/r, where p, q and r are integers, and identify these two triangles together in the natural way along their boundaries. The result is a 2-dimensional orbifold called a hyperbolic turnover. In this talk, we will see that mapping a turnover by an immersion (which is not an embedding) into a hyperbolic 3-orbifold places strong restrictions on, among other things, the volume of the 3-orbifold. Along the way, we will also observe that hyperbolic turnovers in 3-orbifolds exhibit phenomena very reminiscent of the results of some well-known theorems on certain surfaces in 3-manifolds.
Kasra Rafi Convexity of lengths along a Teichmuller geodesic
We will show that both the extremal length and the hyperbolic length of a simple closed curve along a Teichmuller geodesic are quasi-convex functions of time. We conclude that a round ball in Teichmuller space is quasi-convex. It is notable that these lengths are not in general convex functions. (Joint work with Lenzhen.)
Lawrence Roberts On Heegaard-Floer homology and fibrations
We will describe an ongoing program for computing Heegaard-Floer invariants of fibered three-manifolds, fibered knots, and other fibrations.
Saul Schleimer Compressed words in hyperbolic groups
Suppose that G is a Gromov hyperbolic group. Then the "compressed word problem" in G has a polynomial-time solution. (When G is a free group this result is due to Markus Lohrey.) As a consequence, the word problems in Aut(G) and Out(G) are also polynomial-time. Since surface groups are hyperbolic, this gives a new solution to the word problem in the mapping class group.
Jeremy Van Horn-Morris Fibered Transverse Links
The Bennequin-Eliashberg bound constrains the self-linking number of a transverse link in a tight contact manifold. For fibered links, sharpness in this bound is related to Giroux's theory of open book decompositions on contact manifolds and gives rather strong information regarding the classification of such transverse links. This is joint work with John Etnyre.
Ben Webster Springer fibers and disoriented knot homology (joint w/ Catharina Stroppel)
We describe a geometric construction, using certain Springer fibers, of a modification of Khovanov's arc algebra. This modified arc algebra can be thought of as a "disoriented" analogue of the arc algebra, in the sense of the Morrison-Walker disoriented TQFT construction of knot homology. While still preliminary, this result suggests a connection between the Seidel-Smith symplectic knot homology and disoriented Khovanov homology.
Mike Williams Lens space surgeries on tunnel number one knots
I will discuss the conjectural picture for obtaining lens spaces by Dehn surgery on knots in the 3-sphere. In particular, it is conjectured that all lens space surgeries arise from John Berge's double primitive construction; this conjecture is known as the Berge Conjecture. Berge's construction is an example of a surface slope surgery. I will discuss other surface slope surgery constructions and their relationships to the Berge Conjecure for tunnel number one knots.
Hao Wu Matrix factorization and colored MOY graphs
I will introduce matrix factorizations associated to colored MOY graphs, which naturally generalize the matrix factorizations used by Khovanov and Rozansky. I will also discuss generalizations of several key properties of Khovanov and Rozansky's matrix factorizations. These results indicate that it may be possible to generalize the Khovanov-Rozansky homology to a categorification of the (colored) MOY link invariant.