Research*
  My current research is focused on contact topology. My collaborators are Ko Honda (USC) and Gordana Matic (University of Georgia). I've also written several papers on laminations of 3-manifolds and branched coverings of surfaces with David Gabai (Princeton University).
 

(with Ko Honda and Gordana Matic) Contact structures, sutured Floer homology and TQFT, (preprint 2008).
    We describe the natural gluing map on sutured Floer homology which is induced by the inclusion of one sutured manifold (M',\Gamma') into a larger sutured manifold (M,\Gamma), together with a contact structure on M-M'. As an application of this gluing map, we produce a (1+1)-dimensional TQFT by dimensional reduction and study its properties.

  (with Ko Honda and Gordana Matic) The contact invariant in sutured Floer homology, (to appear in Invent. Math.)
    We describe an invariant of a contact 3-manifold with convex boundary as an element of Juhasz's sutured Floer homology. It specializes to Ozsvath-Szabo's contact invariant in Heegaard Floer homology, via the paper below on Heegaard Floer homology.
  (with Ko Honda and Gordana Matic) On the contact class in Heegaard Floer homology, (to appear in JDG).
     We present an alternate description of the Ozsv\'ath-Szab\'o contact class in Heegaard Floer homology. Using the new description of the contact class, they prove that if a contact structure $(M,\xi)$ has an adapted open book decomposition whose page $S$ is a once-punctured torus, then the monodromy is right-veering if and only if the contact structure is tight.
  (with Ko Honda and Gordana Matic) Right-veering diffeomorphisms of a compact surface with boundary II, Geometry & Topology 12 (2008).
    In the first portion of the paper, the difference between Veer and Dehn for the punctured torus is analyzed. In the second portion the authors show that certain open books with monodromies having large enough boundary twisting are isotopic to small perturbations of taut foliations, and hence are weakly symplectically fillable.
  (with Ko Honda and Gordana Matic) Right-veering diffeomorphisms of a compact surface with boundary, Invent. Math. (2007).
    The monoid of right-veering diffeomorphisms Veer \subset Aut(S,\bdry S) is introduced. Several equivalent definitions of right-veering are given and the concept is related to the Thurston's classification of surface automorphisms. The main result of the paper is a criterion for a contact structure to be tight in the open book framework of Giroux the theorem that a contact 3-manifold $(M,\xi)$ is tight if and only if all of its adapted open book decompositions have right-veering monodromy.
  (with Ko Honda and Gordana Matic) Pinwheels and bypasses, Algebr. Geom. Top. (2005).
    Bypasses were defined by Ko Honda as the most basic, non-product changes in a contact structure. They correspond to elementary changes in the dividing set of a convex surfaces under isotopy of surfaces when the isotopy goes through a non-convex level. They study the tightness of a contact structure obtained by attaching a family of bypasses to a tight product neighbourhood of a convex surface. The paper introduces a very usefull tool of "bypass rotation".
  A cut-and-paste approach to contact topology, Boletin de la Sociedad Matematica Mexicana (2004).
    In this expository paper, contact structures on 3-manifolds are analyzed by decomposing the manifold along convex surfaces. Background results of Giroux, Eliashberg, Colin, and Honda are discussed with an emphasis on examples. Convex decompositions are then used to give a new proof of the Gabai-Eliashberg-Thurston Theorem on the existence of universally tight contact structures and also to study the contact topology of a space in the presence or absence of tori. Classification of tight contact structures on fibred manifolds and related open questions are also discussed. This paper is based on a series of talks given at the Tokyo Institute of Technology from Jun 3-7, 2002.
  (with Ko Honda and Gordana Matic) On the Gabai-Eliashberg-Thurston theorem, Comment. Math. Helv. (2004).
    We reprove, using purely three-dimensional cut-and paste methods, the theorem of Gabai-Eliashberg-Thurston which states that a closed, oriented, irreducible 3-manifold with nonzero second homology carries a universally tight contact structure.
  (with Ko Honda and Gordana Matic) Tight contact structures on fibered hyperbolic 3-manifolds, J. Differential Geom. (2003).
    The paper gives a classification of tight contact structures in the extremal case on surface bundles which fiber over the circle with pseudo-Anosov monodromy.
  (with Ko Honda and Gordana Matic) Convex decomposition theory, Int. Math. Res. Not. (2002).
    A continuation of Tight contact structures and taut foliations. We prove the existence of universally tight contact structures on 3-manifolds which are `large', in a completely 3-dimensional, cut and paste topology manner. We also prove that a toroidal 3-manifold carries infinitely many isomorphism classes of universally tight contact structures.
  (with Ko Honda and Gordana Matic) Tight contact structures and taut foliations, Geom. Topol. (2000).
    In this paper we unite sutured manifolds and their decompositions with their contact siblings, the `convex structures' and their decompositions.
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This research is supported by grants from the National Science Foundation:
DMS-0711341 Contact topology and automorphisms of surfaces (co-PI, Matic)
DMS-0406158 Cut-and-paste contact topology
DMS-0073029 Genuine laminations of 3-manifolds