David Bachman

Title: Topologically minimal surfaces in 3-manifolds

Abstract: Topologically minimal surfaces are the topological analogue of geometrically minimal surfaces. Such surfaces generalize well known classes, such as incompressible, strongly irreducible (or weakly incompressible), and critical surfaces. Applications include problems dealing with stabilization, amalgamation, and isotopy of Heegaard splittings and bridge spheres for knots. In this talk we will review the basic definitions and discuss both existing and potential applications of this new theory.

Title: The Hyper-Reals and the Thurston boundary of Teichmuller Space.

Abstract: Most of the talk will be a gentle introduction to the hyper-reals: an ordered field which contains infinitesimals. This material is of wide applicability in many areas of mathematics. We will then briefly discuss using these ideas in geometry and topology. As an application we will explain how to view the Thurston boundary of Teichmuller space in terms of non-standard hyperbolic structures.

Title: Sutured manifolds revisited

Title: Homology cylinders in knot theory

Abstract: This is a joint work with Takuya Sakasai. Two concepts, sutured manifolds and homology cylinders, treating cobordisms between surfaces are compared. The former ones defined by Gabai are useful to study knots and 3-dimensional manifolds, and the latter are in an important position in the recent theory of the mapping class group, homology cobordisms of surfaces and finite-type invariants. We study a relationship between them by considering which knot has a homology cylinder as a complementary sutured manifold that is a sutured manifold obtained from a knot complement. As the answer to it, homological fibered knots are introduced. They are characterized by their Alexander polynomials and genera. Then we use some invariants of homology cylinders to give applications such as fibering obstructions, Reidemeister torsions and handle numbers of homological fibered knots.

Title: On Generalizing Property R

Abstract: The Property R Conjecture for knots, proved by Gabai in the 1980's, has a natural generalization to links, provided that we are allowed to simplify by handle slides (Kirby Problem 1.82). We will consider an explicit family of 2-component links, and see via 4-manifold theory that these are probably counterexamples to the conjecture. If this is true, then work of Scharlemann and Thompson implies there is a 2-component counterexample containing a genus-1 knot. We will see explicitly how the extra move of Hopf-pair addition (introducing a canceling 1-2 handle pair) can simplify links, leading to more plausible versions of the conjecture that could provide insight into 4-dimensional problems such as the Smooth 4D Poincare Conjecture.

Title: Exceptional Dehn filling

Abstract: If M is a hyperbolic 3-manifold with cusps, it is rare for M to have two non-hyperbolic Dehn fillings M(r) and M(s) along a given cusp. If this happens, we call (M;r,s) an exceptional pair. We will discuss to what extent, and in what sense, one might expect to be able to classify all exceptional pairs. In particular we will suggest that this might be possible when the intersection number between r and s is at least 3. We will survey the progress that has been made in this direction and identify the problems that remain.

Title: On Khovanov homology, Heegaard Floer homology, and sutured manifolds.

Abstract: I will discuss an algebraic relationship, first discovered by Ozsvath and Szab, between (a reduced version of) the Khovanov homology of a knot and the Heegaard Floer homology of its branched double cover. Sutured manifold theory enters the picture in a beautiful way, leading to nice naturality results under certain "TQFT-type" operations. This is joint work with Stephan Wehrli.

Title: Sutures and contact homology

Abstract: We define a relative version of contact homology for contact manifolds with convex boundary, and prove basic properties of this relative contact homology. Similar considerations also hold for embedded contact homology. This is joint work with Vincent Colin, Paolo Ghiggini, and Michael Hutchings.

Title: The maximal number of exceptional Dehn surgeries.

Abstract: I will outline the proof of two old conjectures of Cameron Gordon. The first states that the maximal number of exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 10. The second states the maximal distance between exceptional Dehn surgeries on a 1-cusped hyperbolic 3-manifold is 8. The proof uses a combination of new geometric techniques and rigorous computer-assisted calculations. This is joint workwith Rob Meyerhoff.

Title: An algorithm to determine the Heegaard genus of a 3-manifold

Title: The Tunnel Leveling Addendum

Abstract: The Tunnel Leveling Theorem of Goda, Scharlemann, and Thompson says that when a tunnel number 1 knot is in (minimal) bridge position, any tunnel arc can be slid to lie in a level 2-sphere. The Tunnel Leveling Addendum says, roughly speaking, that when a tunnel arc is leveled, the other two knots of the theta-curve formed by the knot and tunnel arc are also in minimal bridge position, after trivial repositioning (although there are certain complications for "eyeglass" tunnels). This yields considerable information about the bridge numbers of tunnel number 1 knots. We will discuss the proof, which is an application of the theory of tunnel number 1 knots developed by Sangbum Cho and the speaker. We will also discuss recent work on the tunnels of (1,1)-knots, that is, knots admitting a genus-1 1-bridge position, and on knots with more than one equivalence class of tunnels. All of this work is joint with Cho.

Title: Fat train tracks, waves and Heegaard splittings

Abstract: Two full dimensional simplexes in the curve complex of a surface of genus greater or equal than two determine a Heegaard splitting for some manifold. Given an integer n we give sufficient conditions for when two simplexes of distance greater or equal to n will determine a Heegaard splitting of distance greater or equal to n. Since the disk complex of a handlebody embedded in the curve complex is unbounded the fact that the simplexes are far apart does not imply that the handlebodies are far apart. The conditions are combinatorial in nature and can be "effectively" satisfied.

Title: Distance and Heegaard genera of surface sums

Abstract: Let $M$ be a compact, orientable 3-manifold, and $F$ be a properly embedded surface in $M$. Then $M$ is called the surface sum or self-surface sum of $M-\eta(F)$ where $\eta(F)$ is a open regular neighborhood of $F$ according to the separation of $F$. Some results on Heegaard genera of the surface and self-surface sums will be presented.

Title: Classifying minimal triangulations of some 3-manifolds

Abstract: A useful measure of complexity of a 3-manifold is the minimum number of tetrahedra required to describe it. There are various census of small complexity 3-manifolds and often the minimum triangulations are unique and can be used to find geometric structures. Some bounds are known but until recently no infinite classes of examples for which the complexity and minimum triangulations had been determined. In joint work with Stephan Tillmann and Bus Jaco and recently Kei Nakamaura, we have determined minimum triangulations for some classes of lens spaces, small Seifert fibred spaces and are writing up the case of once punctured torus bundles over a circle. General bounds in terms of norms of Z_2 homology classes are given.

Title: The Kakimizu complex is simply connected

Abstract: In 1988, Scharlemann and Thompson proved that the Kakimizu complex is connected. The main construction in their proof can be translated into the setting of infinite abelian covers. This leads to an argument proving that the Kakimizu complex is simply connected.