Equivariant Gromov-Witten theory and symplectic vortices CIRM, Luminy July 6- 10 Monday 9.00 - 10.00 B. Kim 10.15 - 11.15 D. Salamon 11:30 - 12.30 K. Hori (16.00 - 17.00) 17.15 - 18.15 H.H. Tseng 18.30 - 19.30 F. Ziltener Tuesday 8.50 - 9.50 H. Iritani 10.15 - 11.15 C. Teleman 11.25 - 12.25 I. Mundet (16.00 - 17.00) 17.00 - 18.00 E. Gonzalez 18.15 - 19.15 R. Pandharipande Wednesday 8.50 - 9.50 D. Diaconescu 10.15 - 11.15 U. Frauenfelder 11.25 - 12.25 J. Brown Afternoon free Thursday 8.50 - 9.50 Y.-P. Lee 10.15 - 11.15 A. Ott 11.25 - 12.25 I. Ciocan-Fontanine (16.00 - 17.00) 17.00 - 18.00 L. Mihalcea 18.15 - 19.30 M. Bader Friday 9:00 - 10:00 Jan Wehrheim 10:10 - 11:10 Aleksey Zinger 11:20- 12:20 Katrin Wehrheim 2:00 - 2:45 Ziltener II 2:45 - 3:30 Mundet II Abstracts ------------------------------------------ Speaker: M. Bader Title: Cohomotopy invariants and gauge theoretical Gromov-Witten theory Speaker: J. Brown Title: Gromov--Witten invariants of toric fibrations Abstract: We'll state a mirror theorem for toric bundles, and remark on some of the techniques that are involved. Speaker: U. Frauenfelder Title: On Rabinowitz Floer homology Abstract: Rabinowitz action functional is a Lagrange multiplier action functional whose critical points are Reeb orbits on an energy hypersurface in a symplectic manifold. Gradient flow lines of Rabinowitz action functional satisfy some vortex type equation. I discuss compactness issues for the gradient flow equation and applications of Rabinowitz Floer homology to obstructions of exact contact embeddings and existence of leafwise intersection points. Speaker: H. Iritani Title: Quantum cohomology D-module of toric stacks Abstract: Borisov-Chen-Smith introduced toric Deligne-Mumford stacks and described the orbifold cohomology ring in terms of a stacky fan. In this talk, we ``quantize" the Borisov-Chen-Smith description and give the (equivariant) orbifold quantum cohomology D-module in terms of the stacky fan. We also mention to the relationship to mirror symmetry and application to wall-crossings of quantum cohomology. This is based on the joint work with Coates, Corti and Tseng. Speaker: B. Kim Title: The abelian/nonabelian correspondence Abstract: I will explain the proposal of the abelian/non-abelian correspondence starting from an introduction to the equivariant Gromov-Witten theory. The proposal is a joint work with Ciocan-Fontanine and Sabbah, began by joint works with Bertram and Ciocan-Fontanine. Speaker: L. Mihalcea, Duke Univ. Title: Quantum K-theory of Grassmannians Speaker: I. Mundet Title: Hamiltonian Gromov-Witten invariants for circle actions on compact symplectic manifolds Abstract: We will describe a set of invariants of compact symplectic manifolds endowed with a Hamiltonian action of the circle. These invariants satisfy some properties which are analogous to those of Gromov-Witten invariants. We will emphasize the main differences with Gromov-Witten theory, both in the properties of the invariants and in the technical difficulties which one encounters when defining them. As an application we will define a new associative ring structure on the equivariant cohomology of the symplectic manifold, which may be seen as a deformation of the orbifold cohomology of its symplectic quotients. This is joint work with G. Tian. Speaker: Andreas Ott, ETH Zurich Title: Gauged Gromov-Witten invariants via perturbation of the vortex equations Abstract: We will discuss the problem of defining gauged Gromov-Witten invariants for compact strongly semipositive symplectic manifolds equipped with a Hamiltonian action of a compact Lie group. Our method is based on a perturbation of the vortex equations, and extends the pseudocycle approach to the definition of Gromov-Witten invariants to the equivariant case. Speaker: Rahul Pandharipande Title: The moduli space of stable quotients. Abstract: I will talk about a new moduli space of stable quotients of the trivial sheaf on stable curves. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular, irreducible, and modular compactification of the moduli of maps from genus 1 curves to projective space is obtained. The moduli space of stable quotients carries a canonical 2-term obstruction theory and thus a virtual class. I will discuss the resulting theory in several toric and CY cases. Questions about the behavior of stable quotients for arbitrary targets will be raised. Joint work with A. Marian and D. Oprea. Speaker: D. Salamon, ETH Zurich Title: A brief introduction to the symplectic vortex equations Abstract: The symplectic vortex equations can be viewed as a natural adaptation of the nonlinear Cauchy-Riemann equations to the equivariant setting of symplectic manifolds equipped with Hamiltonian group actions. The goal of this lecture is to explain the basic setup for the symplectic vortex equations and the resulting invariants, discuss how they are related to other equations that play a central role in geometry (eg the self-dual Yang-Mills equations or the Seiberg-Witten equations), and, if time permits, outline one or two applications. Speaker: A.J. Tolland (Stonybrook) Title: Gromov-Witten Theory for [pt/GL(1)] Abstract: I will explain recent work with E. Frenkel and C. Teleman, in which we constructed K-theoretic Gromov-Witten invariants for the quotient stack [pt/GL(1)]. These invariants are like Gromov-Witten invariants in that they result from integrating evaluation classes over a moduli stack of marked algebraic curves carrying maps to [pt/GL(1)] (i.e., carrying principal GL(1)-bundles), but they differ in that the moduli stack is an Artin stack and very far from compact. We'll show that the invariants are well-defined by proving a vanishing theorem for the local cohomology of our stack. Speaker: H.-H. Tseng Title: On mirror theorem for complete intersections. Abstract: One aspect of mirror symmetry states that the genus 0 Gromov-Witten invariants of a Calabi-Yau manifold X can be computed from Hodge-theoretic data associated to the mirror of X, which is another Calabi-Yau. A large class of X for which this is proven, by many people's work, are complete intersections in toric varieties. In this talk we will give a survey on a more recent two-step approach to this, and discuss how to generalize this approach to orbifolds. Speaker: F. Ziltener, Toronto Title: Symplectic Vortices and a Quantum Kirwan Map Abstract: A Hamiltonian action of a Lie group on a symplectic manifold $(M,\omega)$ gives rise to a gauge theoretic deformation of the Cauchy-Riemann equations, called the symplectic vortex equations. Counting solutions of these equations over the plane $R^2$ leads to a quantum version of the Kirwan map. In joint work with Christopher Woodward, we interpret this map as a weak morphism of cohomological field theories. " Participant List: * Markus BADER, Universität Zürich Institut für Mathematik Universität Zürich Winterthurerstrasse 190 8057 Zürich, SWITZERLAND * Joao BAPTISTA, ITFA - University of Amsterdam ITFA, Valckenierstraat 65 1018 XE Amsterdam, NETHERLANDS (THE) * ANDREA BRINI, SISSA/ISAS - Trieste Via Beirut 2-4 34014 Trieste, ITALY * Jeff BROWN, UC Berkeley (PASSPORT ISSUE; ARRIVING LATE) UC Berkeley Mathematics Dept. 970 Evans Hall 94720 Berkeley, U.S.A. * Olguta BUSE, IUPUI 405 North blackford street LD270C 46202 Indianapolis, U.S.A. * Pierre-Emmanuel CHAPUT, Universite Nantes 29, rue d'Anjou 49230 Montfaucon-Montigne, FRANCE * Linda CHEN, Swarthmore College Department of Mathematics Swarthmore College 19081 Swarthmore, U.S.A. * Daewoong CHEONG, KIAS(Korea Institute for Advan 207-43 Cheongnyangni 2-dong Dongdaemun-g 130-722 Seoul, SOUTH KOREA * Ionut CIOCAN-FONTANINE, University of Minnesota School of Mathematics University of Minnesota 206 Church Street SE 55455 Minneapolis, U.S.A. * Alessio CORTI, Imperial College London Department of Mathematics Imperial College London 180 Queen's Gate SW7 2AZ London, UNITED KINGDOM (THE) * Duiliu DIACONESCU, 126 Frelinghuysen Road NHETC Rutgers University 08854 Piscataway, U.S.A. * Urs FRAUENFELDER, Seoul National University Seoul National University Department of Mathematics 151-747 Seoul, SOUTH KOREA * Edward FRENKEL, University of California (CANCELLED) Math Dept, 970 Evans Hall University of California CA 94720 Berkeley, U.S.A. * Damiano FULGHESU, Scuola Normale Superiore Pisa Piazza dei Cavalieri 7 5600 Pisa, ITALY * Amin GHOLAMPOUR, California Institute of Techno 1200 E California Blvd Mail code: 253-37 91125 Pasadena, U.S.A. * Eduardo GONZALEZ, UMASS Boston Department of Mathematics UMASS Boston 100 William T. Morrissey Boulevard 02125 Boston, U.S.A. * Tom GRABER, Caltech Mathematics 253-37 Caltech 91125 Pasadena, CA, U.S.A. * Tara HOLM, Cornell University Department of Mathematics Cornell University Ithaca, NY 14853 14853 Ithaca, U.S.A. * Kentaro HORI, IPMU, University of Tokyo 5-1-5 Kashiwanoha 277-8568 Kashiwa-shi, JAPAN * Hiroshi IRITANI, Imperial College London and Ky Imperial College London South Kensington Campus SW7 2AZ London, UNITED KINGDOM (THE) * Julien KELLER, Univ. Aix-Marseille 1 39, rue F. Joliot Curie, 13453 Marseille, FRANCE * Bumsig KIM, KIAS 87 Hoegiro, Dongdaemun-gu 130-722 Seoul, SOUTH KOREA * Yuan-Pin LEE, University of Utah Department of Mathematics, University of Utah 84112-0090 Salt Lake City, Utah, U.S.A. * Etienne MANN, Universite Montpellier place eugene bataillon 34095 Montpellier, FRANCE * Cristina MANOLACHE, SISSA SISSA, via Beirut 2-4 34014 Trieste, ITALY * Leonardo Constantin MIHALCEA, Duke University Duke University, Dept. of Mathematics, Box 90320 27708-0320 Durham, U.S.A. * Ignasi MUNDET I RIERA, Universitat de Barcelona Dep. Algebra i Geometria Gran Via de les Corts Catalanes 585 08007 Barcelona, SPAIN * Christian OKONEK, universitaet zuerich winterthurerstrasse 190 8057 zuerich, SWITZERLAND * Andreas OTT, ETH Zuerich/Rutgers Department of Mathematics 110 Frelinghuysen Road 08854 Piscataway, NJ, U.S.A. * Rahul PANDHARIPANDE, Princeton University Dept. of Mathematics Princeton University Princeton, NJ 08540 USA 08540 Princeton, U.S.A. * Nicolas PERRIN, HCM, Bonn Hausdorff Center for Mathematics Universität Bonn Villa Maria Endenicher Allee 62 53115 Bonn, GERMANY * Nuno ROMAO, CTQM, University of Aarhus Institut for Matematiske Fag Ny Munkegade 118, bygn 1530 8000 Aarhus, DENMARK * Claude SABBAH, CNRS Ecole polytechnique CMLS Ecole polytechnique 91128 Palaiseau, FRANCE * Dietmar SALAMON, ETH Zurich Departement Mathematik ETH Zentrum Ramistrasse 101 8092 Zurich, SWITZERLAND * Venugopalan SUSHMITA, Rutgers University Department of Mathematics, 110 Frelinghuysen Rd, Piscataway NJ-08854 08854 Piscataway, U.S.A. * Kaisa TAIPALE, University of Minnesota 206 Church St. S.E. 55455 Minneapolis, MN, U.S.A. * Andrei TELEMAN, CMI, Université de Provence 39, Rue F. Joliot-Curie 13453 Marseille, FRANCE * Constantin TELEMAN, UC Berkeley UC Department of Mathematics 970 Evans Hall #3840 CA Berkeley, U.S.A. * Andrew TOLLAND, UC--Berkeley (Cancelled, ill) Math. Dept., Evans Hall University of California 94703 Berkeley, U.S.A. * Hsian-Hua TSENG, University of Wisconsin-Madiso Department of Mathematics, University of Wisconsin-Madison, Van Vleck Hall, 480 Lincoln Dr. 53706 Madison, U.S.A. * Eva Gabriela TUDOR, University of Zurich Institut of Mathematics University of Zurich Winterthurerstrasse 190 CH-8057 Zurich 8051 Zurich, SWITZERLAND * Jan WEHRHEIM, IRMA Strasbourg Jan Wehrheim, IRMA 7 rue René Descartes 67000 Strasbourg, FRANCE * Katrin WEHRHEIM, MIT 277 Mass Ave 02139 Cambridge MA, U.S.A. * Christopher WOODWARD, Rutgers 110 Frelinghuysen Rd 08854 Piscataway NJ, U.S.A. * Fabian ZILTENER, Department of Mathematics, University of Toronto M5S 2E4 Toronto, CANADA * Aleksey ZINGER, Department of Mathematics SUNY Stony Brook 11794-3651 Stony Brook, NY, U.S.A.