Thomas E. Mark Assistant Professor, Mathematics

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Visitor Information

I'm a low-dimensional topologist specializing in the topology of smooth and symplectic 4-manifolds. My recent work has been focused on developing and applying the technology of Heegaard Floer homology to fibered 3-manifolds and Lefschetz fibered 4-manifolds. I arrived here at Virginia in the fall of 2006 after a three-year postdoc at the University of California, Berkeley and three years as an assistant professor at Southeastern Louisiana University.

My office hours for the summer are intermittent (see first news item below). Between June 10 and July 8 I will be in the department every weekday and you will likely be able to find me in the early afternoon; to be more certain, send me an email.

News

• Eclipsing all other items in this list, my son Julian Maxwell Mark-Williams was born May 15, 2008. Click here for the photo tour.
• There is a workshop on low-dimensional topology at MSRI (Berkeley) in August, in honor of Rob Kirby's 70th birthday; you'll likely find me there.
• Subsequently there is a workshop on holomorphic curves at Stanford that I also hope to make.
• I attended the AMS 2008 spring sectional meeting in Baton Rouge last March, with particular interest in the special session on gauge theory in smooth and symplectic topology.
• During the spring of 2008 I visited Rice and Princeton; in the fall of 2007 I visited the Georgia Institute of Technology and the University of Miami.

Courses

Class I am teaching, Fall 2008:

• Math 875 Topology of Manifolds. A review of differential topology (transversality) will be followed by handlebody theory and the h-cobordism theorem. Subsequently we will focus on low-dimensional issues: constructions and examples in 3 and 4 dimensions, the work of Freedman in dimension 4, and the failure of the smooth h-cobordism theorem in that dimension.

Class I taught, Summer 2008:

• Math 770E Problems in Topology (preparation for general exam)

Class I taught, Spring 2008:

• Math 872 Differential Geometry (with emphasis on Riemannian geometry)

Research

My current work is focused on applying the technology of Heegaard Floer homology developed by Peter Ozsvath and Zoltan Szabo to the study of smooth 3- and 4-dimensional manifolds. In particular I (and my collaborator Slaven Jabuka) are studying Heegaard Floer invariants of Lefschetz fibered 4-manifolds with a view toward questions in the topology of symplectic 4-manifolds.

The following is a list of my preprints and publications:

• "Knotted surfaces in 4-manifolds." Submitted for publication. We extend a result of Fintushel and Stern to show that given a symplectic surface with simply-connected complement and self-intersection at least 2-2g in a symplectic 4-manifold with b₂⁺ >1, there are infinitely many embedded surfaces topologically isotopic to the original but not smoothly isotopic to it.
• "A note on Stein fillings of contact manifolds," with A. Akhmedov, J. Etnyre, and I. Smith. Submitted for publication. We give an infinite family of examples of contact three-manifolds that each admit infinitely many simply connected, homeomorphic but smoothly distinct Stein fillings.
• "Product formulae for Ozsvath-Szabo 4-manifold invariants," with S. Jabuka. Geometry and Topology 12 (2008) 1557--1651. We develop a general formalism for calculating Ozsvath-Szabo invariants for closed 4-manifolds obtained by gluing two manifolds with boundary, in terms of relative invariants of the pieces. The approach makes use of "perturbed" Heegaard Floer homology, that is, Floer homology with coefficients in modules over certain Novikov rings. As a motivating example and illustration of the formalism, we determine the behavior of Ozsvath-Szabo 4-manifold invariants under fiber sums of manifolds along surfaces with trivial normal bundle.
• "Heegaard Floer homology of a surface times a circle," with S. Jabuka. Advances in Mathematics 218 (2008) 728--761. We compute the Heegaard Floer homology of the 3-manifold named in the title, and show in particular that the integral Floer homology of the product of a surface of genus at least 3 with the circle contains nontrivial torsion.
• "Triple products and cohomological invariants for closed 3-manifolds." To appear in Michigan Mathematical Journal 56 (2008).
• "Heegaard Floer homology of mapping tori II," with S. Jabuka. In Geometry and Topology of Manifolds, Fields Institute Communications 47. H. Boden, I. Hambleton, A. J. Nicas, B. D. Park, eds. American Mathematical Society, Providence, RI, 2005.
• "Heegaard Floer homology of certain mapping tori," with S. Jabuka. Algebraic and Geometric Topology 4 (2004) 685--719.
• "Torsion, TQFT, and Seiberg-Witten Invariants of 3-manifolds." Geometry and Topology 6 (2002) 27--58.

Professional Information

I am a member of the American Mathematical Society and a reviewer for Mathematical reviews.