Speaker: Anar Ahmadov, Georgia Tech
Title of talk: Small Exotic 4-Manifolds
Abstract: In this talk we present new examples of symplectic 4-manifolds with same integral cohomology as S^2 x S^2. We also discuss the generalization of these examples to (2n-1)S^2 x S^2 case for n > 1. As an application of these symplectic building blocks, we construct
` 1) An exotic smooth (symplectic for b_2^+ = 1) structure on CP^2#3(-CP^2),
3CP^2#5(-CP^2), and 3CP^2#7(-CP^2).
2) An exotic symplectic CP^2#5(-CP^2).
3) An infinite family of distinct smooth 4-manifolds homeomorphic but not
diffemorphic to CP^2#3(-CP^2), 3CP^2#5(-CP^2) and 3CP^2#7(-CP^2).
Part of this is joint work with I. Baykur and D. Park.
Speaker: John Baldwin, Columbia University
Title of talk: Comultiplication and the Ozsvath-Szabo contact invariant
Abstract: Let S be a surface with boundary and suppose that g and h are diffeomorphisms of S which restrict to the identity on the boundary. We show that the Ozsvath-Szabo contact invariants associated to the open books (S,g), (S,h), and (S,hg) are natural with respect to a comultiplication on the corresponding Heegaard-Floer homology groups. In particular, it follows that if the contact invariants associated to the open books (S,g) and (S,h) are non-zero, then so is the contact invariant associated to the open book (S,hg). If time permits, we discuss an extension of this comultiplication to HF^+ and an obstruction to the compatibility of a contact structure with a planar open book.
Speaker: Scott Baldridge, Louisiana State
Title of talk: The symplectic geopgraphy problem for minimal s.c. 4-manifolds with signature leg 0
Abstract: In this talk we will show how to fill out the geography for all simply connected minimal symplectic 4-manifolds with odd intersection form where the signature is less than -1, except for 4 examples. We also show how to fill out all examples with signature \leq 0 for all but 96 examples. This is joint work with Paul Kirk, Inanc Baykur, D. Park, and A Akhmedov.
As you can tell from the description, the results are brand new but very much in line with the theme of the conference and in honor of Ron Stern's birthday (three of the authors are students of Ron and Ron).
Speaker: Ronald Fintushel, Michigan State University
Title of talk: Techniques for constructing 4-manifolds
Abstract: I'll discuss the techniques which are known to be useful in constructing smooth 4-manifolds.
Speaker: Paolo Ghiggini, UQAM, Montreal
Title of talk: Giroux's 2\pi-torsion kills the contact invariant
Abstract: We say that a contact 3-manifold (M, \xi) has positive 2\pi-torsion if there is a contact embedding of (T^2 \times [0,1], sin(2 \pi z)dx+cos(2 \pi z)dy=0) into (M, \xi). We will prove that the untwisted Ozsváth-Szabó invariant of a contact 3-manifolds with positive 2\pi-torsion is trivial.
Our proof uses a new relative contact invariant for contact manifolds with convex boundary defined by Hondad, Kazez and Matic in Juhász's sutured Floer homology. This is a joint work with Ko Honda and Jeremy Van Horn-Morris.
Speaker: Elisanda Grigsby, Columbia University
Title of talk: Knot Concordance and Heegaard Floer homology in Cyclic Branched Covers
Abstract: Let $K \subset (S^3=\partial B^4)$ be a classical knot. The smooth concordance order of $K$ is defined as the smallest $n \in \mathbb{Z}_+$ for which the connected sum of $n$ copies of $K$ bounds a smoothly-imbedded disk in $B^4$.
I will describe two new invariants which yield an obstruction to a knot having finite smooth concordance order. These invariants are defined by examining analogues of "classical'' Heegaard Floer homology invariants in the double-branched cover of $K$. Using a simple combinatorial description of these invariants in the case where $K$ is a two-bridge knot, we are able to conclude that all two-bridge knots of 12 or fewer crossings for which the concordance order was previously unknown have infinite concordance order.
This is joint work with Daniel Ruberman and Sa\v{s}o Strle.
Speaker: Matt Hedden, MIT
Title of talk: Symplectic four-manifolds with torsion first Chern class
Abstract: While new examples of exotic symplectic four-manifolds of various descriptions continue to be produced at a rapid rate, the set of known symplectic four-manifolds whose first Chern class is torsion is still fairly small and has received no new additions since 1992. I'll discuss evidence for and against the belief that the known examples might in fact be the only ones.
Speaker: Andras Juhasz, Princeton University
Title of talk: Floer homology and surface decompositions
Abstract: We define an invariant of balanced sutured manifolds called sutured Floer homology. In this talk we give a formula that shows how this invariant changes under surface decompositions. Using our formula we can simplify the proofs of a result of Ozsvath and Szabo that link Floer homology detects the Thurston norm, and a theorem of Ni that knot Floer homology detects fibred knots. We also show that for a wide class of knots if the top term of knot Floer homology has rank < 4 then the knot complement admits a depth one taut foliation.
Speaker: Joan Licata, Yale University
Title of talk: Local link alterations and the Thurston norm
Abstract: The Thurston norm measures the combinatorial complexity of surfaces representing second homology classes in a link complement. In this talk we discuss the use of the Heegaard Floer link invariant in computing the Thurston norm. We will begin with an infinite family of pretzel links and then present two theorems which address the question of how local changes to a link affect the Thurston norm of its complement.
Speaker: Thomas Mark, University of Virginia
Title of talk: Perturbed Heegaard Floer invariants and applications
Abstract: We describe a version of Heegaard Floer homology with coefficients in certain Novikov rings. The construction depends on the choice of a 2-dimensional real cohomology class; when this "perturbation" is nonzero the reducible part of the Floer homology vanishes, in close analogy with Seiberg-Witten theory. We use this construction to describe a version of Ozsvath-Szabo invariants for 4-manifolds with b^+ > 0, and to define well-behaved relative invariants for 4-manifolds with boundary. We will also describe some calculations and applications of these relative invariants.
Speaker: Tomasz Mrowka, MIT
Title of talk: Knot homologies via singular instantons
Abstract: I will discuss some work in progress with Peter Kronheimer on a variant of Floer's instanton homology dealing with connections that are singular along codimension two submanifolds. As observed by Kronheimer earlier there are some interesting relations with Khovanov homology.
Speaker: Swatee Naik, University of Nevada Reno
Title of talk: The Knot Concordance Group
Abstract: We will discuss smooth, topological and algebraic concordance
of knots and work related to finite order classes.
Speaker: Yi Ni, Princeton University
Title of talk: Knot Floer homology detects fibred knots
Abstract: Knot Floer homology is a knot invariant introduced by Ozsv\'ath and Szab\'o, and by Rasmussen. The Euler characteristic of knot Floer homology gives rise to the Alexander polynomial of a knot, so many properties of Alexander polynomial can be generalized to knot Floer homology. For example, if a knot is fibred, then its knot Floer homology is "monic". Ozsv\'ath and Szab\'o conjectured that the converse of the previous fact is also true, namely, if the knot Floer homology is monic, then the knot is fibred. In this talk, we will discuss a proof of this conjecture, based on the works of Paolo Ghiggini and of the speaker. A corollary is that if a knot in S3 admits a lens space surgery, then the knot is fibred.
Speaker: Burak Ozbagci, Koc University, Turkey
Title of talk: Invariants of contact structures from open books
Abstract: We define three invariants of contact structures in terms of open books supporting the contact structures. These invariants are the support genus (which is the minimal genus of a page of a supporting open book for the contact structure), the binding number (which is the minimal number of binding components of a supporting open book for the contact structure with minimal genus pages) and the norm (which is minus the maximal Euler characteristic of a page of a supporting open book (joint work with John Etnyre)
Speaker: Danny Ruberman, Brandeis University
Title of talk: Periodic Dirac operators and positive scalar curvature on 4-manifolds
Abstract:Which smooth manifolds admit a Riemannian metric whose scalar curvature is positive? This question has been attacked using minimal surface theory (Schoen-Yau) and the Dirac operator (Lichnerowicz, Gromov-Lawson, and many others). Using Taubes' theory of periodic-end operators, we will discuss the Dirac operator on a non-compact 4-manifold that is an infinite cyclic cover of a compact spin manifold X. We show that such an operator is Fredholm for a generic metric, and use this to give a new analytical interpretation of the Rohlin invariant of X. This new interpretation gives rise to a new obstruction to the existence of metrics of positive scalar curvature. This is joint work with Nikolai Saveliev (U. of Miami)
Speaker: Sucharit Sarkar, Princeton University
Title of talk: An algorithm to compute some Heegaard Floer homologies
Abstract: We shall review the definition and some of the properties of Heegaard Floer homology, and find a chain complex which computes one version of it. We shall also try to investigate a few properties of this chain complex.
Speaker: Zoltan Szabo, Princeton University
Title of talk: A cube of resolutions for knot Floer homology
Abstract: This is a joint work with Peter Ozsvath. We develop a skein exact sequence for knot Floer homology that involves singular knots. This leads to an algebraic description of knot Floer homology in terms of the braid projection of the knot.
Speaker: Dylan Thurston, Barnard College, Columbia
Title of talk: "Combinatorial Link Floer Homology and Transverse Knots"
Speaker: Michael Usher, Princeton University
Title of talk: Periodic Dirac operators and positive scalar curvature on 4-manifolds
Abstract: Which smooth manifolds admit a Riemannian metric whose scalar curvature is positive? This question has been attacked using minimal surface theory (Schoen-Yau) and the Dirac operator (Lichnerowicz, Gromov-Lawson, and many others). Using Taubes' theory of periodic-end operators, we will discuss the Dirac operator on a non-compact 4-manifold that is an infinite cyclic cover of a compact spin manifold X. We show that such an operator is Fredholm for a generic metric, and use this to give a new analytical interpretation of the Rohlin invariant of X. This new interpretation gives rise to a new obstruction to the existence of metrics of positive scalar curvature. This is joint work with Nikolai Saveliev (U. of Miami)
Speaker: Jiajun Wang, Columbia University
Title of talk: Nice Heegaard diagrams and Floer homology
Abstract: The Heegaard Floer homology provides invariants to various structures in low dimensional topology. I will talk about the combinatorial description of the hat version Heegaard Floer theory and knot Floer homology, via nice Heegaard diagrams. The corresponding hat invariant for four-manifolds with boundary is strong enough to detect exotic smooth structures. These are joint work with Sarkar, and joint work with Lipshitz and Manolescu.
Speaker: Hao Wu, University of Massachusetts
Title of talk: Perturbations of the Khovanov-Rozansky Cohomology
Abstract: In this talk, I will review the construction of the (perturbed) Khovanov-Rozansky cohomology of links, briefly discuss the proof of its invariance, and then give some applications.