## Thomas E. Mark## Associate Professor, Mathematics |

Office: 327 Kerchof Hall | Phone: (434) 924-4948 | Email: tmark at virginia.edu | Math Department Home |

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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:**by appointment during the summer.

- I was at the workshop Low-Dimensional Topology after Floer at the Centre de Recherches Mathematiques in Montreal July 8-12, and the workshop Geometry and Topology of Smooth 4-manifolds at the Max Planck Institute for Mathematics in Bonn, June 3-7.
- I expect to be at the spring sectional meeting of the AMS in Baltimore, March 29-30, 2014.
- Some other upcoming conferences are:
- Workshop and Conference on the Topology and Invariants of Smooth 4-Manifolds at the University of Minnesota, July 31--August 10
- 28th British Topology Meeting, University of Aberdeen, September 9-11.
- Tech Topology Conference, Georgia Institute of Technology, December 2014.

I am an organizer for the UVa geometry seminar.

Classes I'm teaching Fall 2013:

- Math 3250, Ordinary Differential Equations.
- Math 7820, Differential Topology.

Class I taught, Spring 2013:

- Math 7810, Algebraic Topology I.

Classes I taught, Fall 2012:

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

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. To appear in
*Journal of Symplectic Geometry*. 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 b_{2}^{+}>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)

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