## 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 Fall 2015.

- I am on leave during fall semester 2015.
- I will be at the Workshop on Flows, Foliations, and Contact Structures at the Institute for Advanced Study in Princeton, December 7-11, 2015.
- I'll also be at the Joint Mathematics Meetings in Seattle, January 6-9, 2016.
- I am an organizer for Perspectives in topology and geometry of 4-manifolds, a conference in honor of Ronald Fintushel and Ronald Stern's respective 70th birthdays, June 6-10, 2016, in Dubrovnik, Croatia.

I am an organizer for the UVa geometry seminar.

Classes I'm teaching Spring 2016:

- Math 7010, Seminar on Research in Mathematics.
- Math 7800, Algebraic Topology I.

My current work is focused on the study of smooth 3- and 4-dimensional manifolds, and contact and symplectic structures on them. Central tools are Lefschetz fibrations, Heegaard Floer homology and the associated invariants of 4-manifolds, gauge theory, and techniques from symplectic topology and geometry.

The following is a list of my **preprints and publications**:

- "Naturality of Heegaard Floer invariants under positive rational contact surgery," with Bülent Tosun. Preprint.
- "Floer homology and fractional Dehn twists," with Matthew Hedden. Preprint.
- "Convex plumbing and Lefschetz fibrations," with David Gay.
*Journal of Symplectic Geometry***11**, no. 3 (2013) 363--375. We show that many operations arising from monodromy substitution can be performed naturally in the symplectic category. - "Knotted surfaces in 4-manifolds."
*Forum Mathematicum***25**, no. 3 (2013) 597--637. 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.