Workshop and Conference on Holomorphic Curves and Low Dimensional Topology
July 30 to August 11, 2012
Stanford University

Organizers: S. Akbulut (MSU) , A. Akhmedov (UMN), D. Auroux (Berkeley), Y. Eliashberg (Stanford), K. Honda (USC), C. Karakurt (UT Austin), P. Ozsváth (Princeton).

The main focus of this workshop will be on holomorphic curve techniques in low-dimensional topology and symplectic geometry. The workshop is a part of the FRG: Collaborative Research: Topology and Invariants of Smooth 4-Manifolds. It is funded by NSF Focused Research Grant DMS-1065955. Additional funds come from the Stanford University Mathematics Research Center. There will be one week of mini-courses, primarily for graduate students, followed by a conference during the second week.

Mini-course speakers:

Abstract of mini courses :

4-manifolds via their handlebodies (by Selman Akbulut)

Abstract:

Handlebody descriptions of 3 and 4-manifolds will be discussed, and as time permits their various applications to 4-manifold problems will be given, such as carving, branch coverings, knot complements, complex surfaces, Stein manifolds, Lefschetz fibrations, BLF's, corks and plugs. From these techniques various exotic manifolds will be constructed, such as going from handlebody of logarithmic transforms to Dolgachev surfaces, and from handlebody of surface bundles over surfaces to Akhmedov-Park exotic manifolds.

Symplectic topology of Weinstein manifolds (by Yasha Eliashberg)

Abstract:

Weinstein manifolds are symplectic counterparts of affine complex manifolds. We discuss in the lectures techniques (due to Bourgeois, Ekholm and the lecturer) for computation of symplectic invariants of Weinstein manifolds via their handlebody decompositions.

The geometry and physics of Knot Homologies (by Sergei Gukov)

Abstract:

HF=ECH via open book decompositions (by Ko Honda)

Abstract:

The goal of this minicourse is to give a brief introduction to Heegaard Floer homology (due to Ozsvath-Szabo) and embedded contact homology (due to Hutchings), and to prove the equivalence between the two homology theories. The equivalence is joint work with Vincent Colin and Paolo Ghiggini.

Computing with bordered Heegaard Floer homology (by Robert Lipshitz)

Abstract:

Heegaard Floer homology is an extension of the Seiberg-Witten invariants of closed 4-manifolds to invariants of closed 3-manifolds and 4-manifolds with boundary. Bordered Heegaard Floer homology is a further extension, to closed surfaces and 3-manifolds with boundary. We will start by discussing the formal structure of bordered Heegaard Floer homology and discuss how it is defined in a toy model. We will then sketch the definitions in general and spend the rest of the time discussing various ways it can be used for computations. (This is joint work with Peter Ozsvath and Dylan Thurston.)

Pseudoholomorphic quilts and low dimensional topology (by Katrin Wehrheim)

Abstract:

I will give an introduction to Lagrangian correspondences and holomorphic quilts, their analysis, and some sample applications in both symplectic topology and the construction of 3- and 4-manifold invariants.

Short talks :

  • SCHEDULE, TITLES AND ABSTRACTS OF SHORT TALKS

    Lecture notes from the mini-courses and slides from the conference talks:

  • S. Akbulut - 07-30-12 - 4-manifolds via their handlebodies
  • K. Wehrheim - 08-01-12 - Pseudoholomorphic quilts and low dimensional topology I
  • K. Wehrheim - 08-02-12 - Pseudoholomorphic quilts and low dimensional topology II
  • K. Wehrheim - 08-03-12 - Pseudoholomorphic quilts and low dimensional topology III
  • S. Akbulut - 08-09-12 - Corks, plugs, Glucks, and ropes
  • M. Abreu - 08-10-12 - Displacing Lagrangian toric fibers by extended probes
  • A. Akhmedov - 08-11-12 - Exotic smooth structures on 4-manifolds

    Schedule of the workshop:

    All the workshop lectures will take place in room 380C at the Stanford University Mathematics Department.

    For direction to the Stanford Mathematics Department, please click here

    Monday 7/30 Tuesday 7/31 Wednesday 8/1 Thursday 8/2 Friday 8/3
    9:00-9:30
    Registration and Coffee
    9:00-9:30
    Coffee
    9:00-9:30
    Coffee
    9:00-9:15
    Coffee
    9:00-9:30
    Coffee
    9:30-10:30
    Eliashberg I
    9:30-10:30
    Honda II
    9:30-10:30
    Gukov I
    9:15-10:15
    Gukov II
    9:30-10:30
    Lipshitz III
    11:00-12:00
    Honda I
    11:00-12:00
    Lipshitz I
    11:00-12:00
    Honda III
    10:30-11:30
    Wehrheim II
    11:00-12:00
    Wehrheim III
    Lunch Lunch Lunch 11:45-12:45
    Akbulut III
    Lunch
    2:00-3:00
    Akbulut I
    2:00-3:00
    Akbulut II
    2:00-3:00
    Lipshitz II
    2:45-5:45
    Short
    talks
    2:00-3:00
    Gukov III
    3:30-4:30
    Eliashberg II
    3:30-4:30
    Wehrheim I
    3:30-4:30
    Eliashberg III
    5:00
    Reception at Stanford
    Math Department
    courtyard


    Schedule of the conference:

    All the lectures will take place in the Jordan Hall (420), room 420-040 at the Stanford University. Jordan Hall is next to the Math Corner (380). For direction and campus map , please click here . The registration and the coffee breaks will be in the Mathematics Department courtyard.

    Monday 8/6 Tuesday 8/7 Wednesday 8/8 Thursday 8/9 Friday 8/10 Saturday 8/11
    9:00-9:30
    Registration and Coffee
    9:00-9:30
    Coffee
    8:30-9:00
    Coffee
    9:00-9:30
    Coffee
    9:00-9:30
    Coffee
    8:30-9:00
    Coffee
    9:30-10:30
    Taubes
    9:30-10:30
    Stipsicz
    9:00-10:00
    Ekholm
    9:30-10:30
    Akbulut
    9:30-10:30
    Bourgeois
    9:00-10:00
    Akhmedov
    11:00-12:00
    Gukov
    11:00-12:00
    Hedden
    10:15-11:15
    Ng
    11:00-12:00
    Wendl
    11:00-12:00
    Abreu
    10:15-11:15
    Borman
    Lunch Lunch 11:30-12:30
    Woodward
    Lunch Lunch 11:30-12:30
    Cornea
    1:30-2:30
    Perutz
    1:30-2:30
    Manolescu
    free
    afternoon
    1:30-2:30
    Hutchings
    1:30-2:30
    McLean
    2:45-3:45
    Lekili
    2:45-3:45
    Ghiggini
    2:45-3:45
    Yasui
    2:45-3:45
    Entov
    4:15-5:15
    Thurston
    4:15-5:15
    Hom
    4:15-5:15
    Ozbagci
    4:15-5:15
    Plamenevskaya
    6:30
    Conference banquet
    at Ming's restaurant

    Titles and abstracts of the talks :

    Displacing Lagrangian toric fibers by extended probes (by Miguel Abreu)

    Abstract:

    I will address a natural Lagrangian intersection problem in the context of toric symplectic manifolds (and orbifolds): displaceability of Lagrangian torus orbits. I will start with joint work with Leonardo Macarini on a symplectic reduction interpretation of known non-displaceability results, and will then describe more recent joint work with Matthew Strom Borman and Dusa McDuff, introducing extended probes as a generalization of McDuff's displaceability method of probes. I will finish with applications in several examples, including some simple ones with continuous sets of Lagrangian torus orbits whose displaceability status is unknown.

    Corks, plugs, Glucks, and ropes (by Selman Akbulut)

    Abstract:

    Corks and plugs are small fundamental pieces in smooth 4-manifolds which determine their smooth structure (twisting along them changes smooth structure). Roughly cork-twisting is related altering Floer homology, while plug-twisting is related to altering Spin^c structures. Also plug-twisting generalizes the Gluck-twisting operation. I will review some of the recent joint work with K.Yasui about them. There are also big alterations of 4-manifolds, by taking big chunks out and re-glueing them back, which can result change the smooth structures. A. Akhmedov and B. D. Park used this kind of operation to construct small exotic manifolds without boundary. Breaking these chunks into small corks is an open problem (although theoretically it is possible). One way to perform such operation on a handlebody is by what we call the "roping'' technique (imagine hanging shirts on hangers in a dress closet and then keep rearranging them, one way to recall where they were before is to connect them by ropes to one central hook on the ceiling then do rearranging).

    Exotic 4-manifolds with small Euler characteristics (by Anar Akhmedov)

    Abstract:

    It is known that many simply connected, smooth topological 4-manifolds admit infinitely many exotic smooth structures. However, the smaller the Euler characteristic, the harder it is to construct exotic smooth structure. In this talk, we will construct exotic smooth structures on various small 4-manifolds. We will also discuss interesting applications to the geography of spin and non-spin symplectic 4-manifolds. This is mostly a joint work with B. Doug Park.

    The size of a Weinstein neighborhood of a Lagrangian (by Matthew Borman)

    Abstract:

    The width of a Lagrangian is the largest capacity of ball that can be symplecticly embedded into the ambient manifold such that the ball intersects the Lagrangian exactly along the real part of the ball. This notion provides a quantitative answer to the question "What is the maximal size of a Weinstein neighborhood for the Lagrangian?" In this talk I will present a wrapped Floer cohomology construction that can upper bound a Lagrangian's width in terms of its displacement energy. This is joint work in progress with Mark McLean.

    S^1-equivariant symplectic homology and linearized contact homology (by Frederic Bourgeois)

    Abstract:

    We define an S^1-equivariant version of symplectic homology via various equivalent approaches. We show that, over rational coefficients, S^1-equivariant symplectic homology is isomorphic to linearized contact homology. This is joint work with Alexandru Oancea.

    Lagrangian Cobordism and Categorification (by Octav Cornea)

    Abstract:

    I will discuss recent work joint with Paul Biran (ETH) on the existence and properties of a functor relating a certain Lagrangian cobordism category and the derived Fukaya category.

    Exact Lagrangian immersions with a single double point (by Tobias Ekholm)

    Abstract:

    We show that if a closed orientable 2k-manifold K, k > 2, with Euler characteristic not equal to −2 admits an exact Lagrangian immersion into complex 2k-space with one transverse double point and no other self intersections, then K is diffeomorphic to the sphere. The proof combines Floer homological arguments with a detailed study of moduli spaces of holomorphic disks with Lagrangian boundary conditions determined by K. We will also discuss related results in the odd dimensional case. The talk reports on joint work with Ivan Smith.

    Symplectic rigidity and the Poisson bracket (by Michael Entov)

    Abstract:

    The Poisson bracket of two smooth functions on a symplectic manifold depends on their first derivatives. Nevertheless, as it has been discovered in "hard" symplectic topology in recent years, the Poisson bracket is rather sensitive to the uniform (that is, C^0) norm of functions. I will discuss a new manifestation of this phenomenon and its application to Hamiltonian dynamics, based on a joint work with L.Buhovsky and L.Polterovich.

    Knot filtrations in embedded contact homology (by Paolo Ghiggini)

    Abstract:

    Embedded contact homology is a Floer theoretic invariant of three-manifolds which is isomorphic to Heegaard Floer homology by the work of Kutluhan, Lee and Taubes and of Colin, Ghiggini and Honda. An important feature of Heegaard Floer homology is the existence of a knot filtration associated to any knot in a three-manifold. I will describe two equivalent ways to construct a similar filtration in embedded contact homology and discuss some conjectures and corollaries. This is a joint work with V. Colin, K. Honda and M. Hutchings.

    Super-A-polynomial (by Sergei Gukov)

    Abstract:

    The generalized volume conjecture states that "color dependence" of the colored Jones polynomial is governed by an algebraic variety, the zero locus of the A-polynomial (for knots) or, more generally, by character variety (for links or higher-rank quantum group invariants). This relation, based on SL(2,C) Chern-Simons theory, explains known facts and predicts many new ones. In particular, since the colored Jones polynomial can be categorified to a doubly-graded homology theory, one may wonder whether the generalized (or quantum) volume conjecture admits a natural categorification. In this talk, I will argue that the answer to this question is "yes" and introduce a two-parameter deformation of the A-polynomial that describes the "color behavior" of the HOMFLY homology, much like the ordinary A-polynomial does it for the colored Jones polynomial.This deformation, named the super-A-polynomial, is strong enough to distinguish mutants, and its most interesting properties include relation to knot contact homology and knot Floer homology.

    Recent progress on smooth concordance (by Matthew Hedden)

    Abstract:

    Call two knots concordant if they arise as the boundary of a smooth and properly embedded cylinder in the three-sphere times an interval. Concordance is an equivalence relation and, modulo concordance, knots form an abelian group. One can define an analogous group in a purely topological category, and the kernel of a natural homomorphism between the groups is generated by so-called topologically slice knots: those knots which bound topologically flat disks in the four-ball. These knots are particularly interesting, as they highlight the distinction between the smooth and topological categories in dimension four. In this talk I'll review what is known about topologically slice knots, and provide the first examples of such knots which represent (two-) torsion elements in the concordance group. This is joint work with Se-Goo Kim and Charles Livingston.

    Applications of the knot Floer complex to concordance (by Jennifer Hom)

    Abstract:

    We will discuss a concordance invariant, epsilon, associated to the knot Floer complex, and use this invariant to better understand knot concordance. In particular, by considering the knot Floer complex up to the weaker relation of epsilon-equivalence, rather than filtered chain homotopy equivalence, we can obtain a wealth of concordance information. We will discuss applications of this approach, such as a new filtration on the smooth concordance group and bounds on concordance genus.

    Embedded contact homology as a (symplectic) field theory (by Michael Hutchings)

    Abstract:

    We use Seiberg-Witten theory to complete embedded contact homology to a functor defined on the category of contact 3-manifolds and strong symplectic cobordisms between them. We give applications to functoriality of the ECH contact invariant, and to ECH capacities of certain closed symplectic four-manifolds.

    Symplectic topology of rational blowdowns (by Yanki Lekili)

    Abstract:

    We study some finite quotients of the A_n Milnor fibre which coincide with the Stein surfaces that appear in Fintushel and Stern's rational blowdown construction. We show that these Stein surfaces have no exact Lagrangian submanifolds by using the already available and deep understanding of the Fukaya category of the A_n Milnor fibre coming from homological mirror symmetry. On the contrary, we find Floer theoretically essential monotone Lagrangian tori, finitely covered by the monotone tori that we studied in the A_n Milnor fibre. We conclude that these Stein surfaces have non-vanishing symplectic cohomology. This is joint work with M. Maydanskiy.

    Monopole Floer homology and covering spaces (by Ciprian Manolescu)

    Abstract:

    I will discuss a Smith-type inequality for regular covering spaces in monopole Floer homology. A corollary is that if an oriented 3-manifold Y admits a p^n-sheeted regular cover that is a Z/p-L-space (for p prime), then Y itself is a Z/p-L-space. This is joint work with Tye Lidman.

    On the symplectic invariance of log Kodaira dimension (by Mark Mclean)

    Abstract:

    Every smooth affine variety has a natural symplectic structure coming from some embedding in complex Euclidean space. This symplectic form is a biholomorphic invariant. An important algebraic invariant of smooth affine varieties is log Kodaira dimension. One can ask, to what extent is this a symplectic invariant? We show some partial symplectic invariance results for smooth affine varieties of dimension less than or equal to 3.

    Knot contact homology and the augmentation polynomial (by Lenhard Ng)

    Abstract:

    The augmentation polynomial is a three-variable knot invariant derived from knot contact homology. I will introduce this polynomial and some of its properties, including a proposed relation to the HOMFLY polynomial, and discuss its recent appearances in the physics literature.

    Singularity links with exotic Stein fillings (by Burak Ozbagci )

    Abstract:

    In a recent paper of Akhmedov-Etnyre-Smith-Mark, it was shown that certain contact Seifert fibered 3-manifolds, each with a unique singular fiber, have infinitely many exotic simply-connected Stein fillings. Here we generalize this result to some contact Seifert fibered 3-manifolds with many singular fibers and observe that these 3-manifolds are links of some isolated complex surface singularities. In addition, we prove that the contact structures involved in the construction are the canonical contact structures on these singularity links. As a consequence we provide examples of isolated complex surface singularities whose links with their canonical contact structures have infinitely many exotic simply-connected Stein fillings---verifying a prediction of Andras Nemethi. For some of these singularity links, and for each positive integer n, we also construct an infinite family of exotic Stein fillings whose fundamental group is Z x Z_n. (This is a joint work with Anar Akhmedov).

    The arithmetic geometry of homological mirror symmetry (by Timothy Perutz)

    Abstract:

    Joint work with Yanki Lekili (Cambridge). Homological mirror symmetry (HMS) envisions an algebro-geometric interpretation of the Fukaya category of a polarized Calabi-Yau manifold X. The categories in question are defined over algebraically closed fields containing the complex numbers. The choice of an ample divisor in X makes available (in principle) a version of the Fukaya category defined over the much smaller ring of rational power series with integer leading term (in some cases Z[[q]] suffices). The mirror should then be a scheme defined over this same ring - an arithmetic variety. We discuss conjectural arithmetic refinements to HMS, which we expect to be present in all dimensions. In the case of the 2-torus, we prove that the Fukaya category relative to a point is derived-equivalent, over Z[[q]], to perfect complexes of coherent sheaves on the Tate curve, and that the wrapped Fukaya category of the complement of this point is derived-equivalent over Z to coherent sheaves on the Weierstrass curve y^2 + xy = x^3 .

    Flexibility properties of plastikstufes (by Olga Plamenevskaya)

    Abstract:

    In contact manifolds of dimension greater than 3, a plastikstufe (aka overtwisted family) is a possible generalization of an overtwisted disk. It is known that a contact manifold containing a plastikstufe cannot be fillable. We show that a plastikstufe also has some flexibility properties. In particular, under certain conditions, non-isotopic contact structures become isotopic after connect-summing with a sphere that contains a plastikstufe. (This is work in progress, joint with E. Murphy, K. Niederkruger, and A. Stipsicz.)

    Knots in Lattice homology (by Andras Stipsicz)

    Abstract:

    We introduce a filtration on the lattice homology of a negative definite plumbing tree associated to a further vertex and show how to determine lattice homologies of surgeries on this last vertex. We discuss the relation with Heegaard Floer homology.

    SL(2;C) connections with L^2 bounds on curvature (by Clifford Taubes)

    Abstract:

    Non-Abelian gauge theories are used to study the structure of 3 and 4-dimensional spaces; and all such applications require a theorem of Karen Uhlenbeck about connections with integral bounds on their curvatures. Uhlenbeck's theorem only applies to a gauge theory with compact Lie group. I will describe an extension of Uhlenbeck's theorem that applies to gauge theories on 3-manifolds with the non-compact group SL(2,C). This extension will likely have analogs for the higher rank non-compact groups and it may have an analog for certain generalizations of the Seiberg-Witten equations. There is also a possibility that this extension will have an analog that can be used to define SL(2,C) Floer homology and SL(2,C) Donaldson invariants for 4-manifolds.

    Tight but nonfillable contact manifolds in all dimensions (by Chris Wendl)

    Abstract:

    Contact topology in dimension three is shaped by the fundamental dichotomy between "tight" and "overtwisted" contact structures, and while it is not known whether any such dichotomy exists in higher dimensions, there are certainly contact structures in all dimensions that have all the trappings of overtwistedness (e.g. nonfillability, vanishing contact homology), or tightness (e.g. no contractible orbits, lack of flexibility). In dimension 3, the invariant known as "Giroux torsion" has played a central role in classifying tight contact structures, and in this talk I will explain how one can generalize it to find the first examples in all dimensions of contact structures that must be considered tight but do not admit any symplectic fillings. A crucial ingredient for this is the existence (also in all dimensions) of symplectic manifolds with disconnected convex boundary, which requires a surprising digression into algebraic number theory. This is joint work with Patrick Massot and Klaus Niederkrueger.

    Non-displaceable Lagrangians and toric minimal model programs (by Christopher Woodward)

    Abstract:

    (w/ E. Gonzalez) We give a computation of the quantum cohomology of symplectic toric orbifolds which exhibits, for a toric minimal model program (tmmp), a splitting of the quantum cohomology so that each transition in the tmmp corresponds to a collection of Hamiltonian non-displaceable Lagrangian tori.

    Corks and exotic 4-manifolds (by Kouichi Yasui)

    Abstract:

    We discuss how to construct exotic 4-manifolds using corks. In particular, from any 4-dimensional compact oriented handlebody X without 3- and 4-handles and with b_2>0, we construct arbitrary many compact Stein 4-manifolds which are all homeomorphic but mutually non-diffeomorphic, so that their topological invariants coincide with those of X. Time permitting we also construct exotic Stein fillings using log transform. This talk is based on joint work with Selman Akbulut.

    List of Participants:

    Registration:

    Thanks to the generous support of the National Science Foundation and the Stanford University Mathematics Research Center, funds are available for partial support of participant expenses. Students, recent Ph.D.'s, women, and members of underrepresented minorities are particularly encouraged to apply. Applicants are requested to register, send a CV, including a brief description of research interests, and have one reference letter sent to akhmedov@math.umn.edu. The reference letter is optional for people more than two years after receiving a Ph.D. ***Please note that we are no longer accepting application*** .

    Housing:

    We have reserved rooms at the Stanford Housing for most of the participants. Please visit the following links to check the details of the reservation.

    Reservations for both single and shared housing, together with the travel dates . The website also includes information on housing check in, maps, and directions to your housing.

    Please let us know if you wish to stay at the hotel. We have a special conference rate with the following hotel: The Cardinal Hotel (650)-323-5101.

    The rates are as follows:

    Standard rooms with private bath $125.00 + tax
    Shared bath style rooms $75.00 + tax

    Booking is subject to availability

    Please follow this link for more information.

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    Stanford Resources

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