Thomas E. Mark Associate Professor, Mathematics

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• Visitor Information
• News
• Courses
• Research
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Visitor Information

I'm a low-dimensional topologist specializing in 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.

• Office Hours, Fall 2012: Tuesday and Thursday, 9:30 to 11:00.

News

• I was at the Tokyo Workshop on Low-Dimensional Topology at the Tokyo Institute of Technology, March 19-23, and the Georgia Topology Conference, May 9-13.
Some other upcoming conferences are:
• The CAST Summer School and Conference in Budapest, July 9-20.
• A workshop in symplectic field theory in Munich, July 23-27.
• A Workshop and Conference on Holomorphic Curves and Low Dimensional Topology at Stanford, July 30-August 11.

Courses

I am an organizer for the UVa geometry seminar.

Classes I'm teaching Fall 2012:

• Math 5770, General Topology.
• Math 2310, Calculus III.

Class I taught, Spring 2012:

• Math 8720, Riemannian geometry. Tu-Th 9:30-10:45, Kerchof 317.

Classes I taught, Fall 2011:

• Math 7830, Fiber Bundles, Tu-Th 9:30-10:45, Kerchof 327
• Math 2310, Calculus III, Tu-Th 11:00-12:15, Cabell 431.

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-manifolds, as well as techniques from symplectic topology and geometry.

The following is a list of my preprints and publications:

• "Convex plumbing and Lefschetz fibrations," with David Gay. Submitted for publication. We show that many operations arising from monodromy substitution can be performed naturally in the symplectic category.
• "Knotted surfaces in 4-manifolds." To appear in Forum Mathematicum. 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.
• "Monodromy substitution and rational blowdowns," with H. Endo and J. Van Horn-Morris. Journal of Topology 4 (2011) 227--253. We exhibit several new families of relations in mapping class groups of planar surfaces that give rise to various rational blowdowns under the operation of monodromy substitution. (Arxiv version)
• "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. (Arxiv version)
• "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. (Arxiv version)
• "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. (Arxiv version)
• "Triple products and cohomological invariants for closed 3-manifolds." Michigan Mathematical Journal 56 (2008) 265--281. (Arxiv version)
• "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. (Arxiv version)
• "Torsion, TQFT, and Seiberg-Witten Invariants of 3-manifolds." Geometry and Topology 6 (2002) 27--58. (Arxiv version)

Professional Information

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