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 on the topology of smooth and symplectic 4-manifolds, using Lefschetz fibrations and tools from Heegaard Floer homology and gauge theory. 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.

Office hours, fall 2009: WF, 9:00--10:00.

News

• I will be speaking at the "International Conference on Symplectic Contact Low Dimensional Topology" at Seoul National University, November 9-13, 2009.
• I will be on sabbatical at MSRI in Berkeley for the spring semester of the coming academic year.
• Eclipsing the other items in this list, my son Julian Maxwell Mark-Williams was born May 15, 2008. Click here for the photo tour.

Courses

Class I am teaching, Fall 2009:

• Math 2310, Calculus III.
  • Section 100, MWF 10:00-10:50, New Cabell Hall 323.
  • Section 200, MWF 12:00-12:50, New Cabell Hall 323.

Class I taught, Summer 2009:

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

Classes I taught, Spring 2009:

• Math 533, Advanced Multivariable Calculus
• Math 351, Linear Algebra

Research

My current work is focused on the study of smooth 3- and 4-dimensional manifolds, and contact and symplectic structures on them, using Lefschetz fibrations and related structures. Central tools are Heegaard Floer homology and the associated invariants of 4-dimensional 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. Mathematical Research Letters 15, no. 6 (2008) 1127--1132. 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.
• "On the 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." Michigan Mathematical Journal 56 (2008) 265--281.
• "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.