Thomas E. Mark Professor, Mathematics

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• Visitor Information
• News
• Courses
• Research
• Professional information and links

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, spring 2020: Wed. 12--1 and Th. 11--12.

• Geometry and topology group at UVa
• Summer undergraduate research in topology, number theory, and representation theory (summer 2020)
• Admissions information for the Virginia graduate program in math

News

• Some recent and upcoming conferences:
  • I'm co-organizing a conference on Holmorphic curves and low-dimensional topology in Dubrovnik, Croatia, June 22-26, 2020.
  • UVa will host a sectional meeting of the AMS, March 13-15, 2020. I'm co-organizing a session on knots and links in low-dimensional topology.
  • The 2018 Virginia Topology Conference was at the University of Virginia, December 12-14, 2018.
  • I was 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. A follow-up conference, Geometric Structures on 3- and 4-Manifolds, was held June 11-15, 2018.

Courses

I am an organizer for the UVa geometry seminar.

Classes I'm teaching Spring 2020:

• Math 7800, Algebraic Topology I
• Math 3310, Basic Real Analysis.
• A reading course on Heegaard Floer homology.

Research

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

Preprints and publications:

• "Cylindrical contact homology of 3-dimensional Brieskorn manifolds," with Sebastian Haney. Preprint.
• "Exotic Mazur manifolds and knot trace invariants," with Kyle Hayden and Lisa Piccirillo. Preprint.
• "Irreducible 3-manifolds that cannot be obtained by 0-surgery on a knot," with Matthew Hedden, Min Hoon Kim, and Kyungbae Park. Trans. AMS 372, no. 11 (2019), 7619--7638
• "On the Stein framing number of a knot," with Lisa Piccirillo and Faramarz Vafaee. To appear in Journal of Symplectic Geometry.
• "Obstructing pseudoconvex embeddings and contractible Stein fillings for Brieskorn spheres," with Bülent Tosun. Adv. Math. 335 (2018), 878--895
• "Naturality of Heegaard Floer invariants under positive rational contact surgery," with Bülent Tosun. J. Diff. Geom. 110 (2018), no. 2, 281--344
• "Floer homology and fractional Dehn twists," with Matthew Hedden. Advances in Mathematics 324 (2018) 1--39. (Arxiv version)
• "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. (Arxiv version)
• "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 b₂⁺ >1, there are infinitely many embedded surfaces topologically isotopic to the original but not smoothly isotopic to it. (Arxiv version)
• "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.