10/11/2005, 15:00 — 16:00 — Amphitheatre Pa3, Mathematics Building Tom Graber, University of California, Berkeley
Quantum Cohomology of Orbifolds and their Crepant Resolutions
I will discuss joint work with Jim Bryan in which we show how to modify the definition of the small quantum cohomology of an orbifold to take into account divisorial twisted sectors. This quantum cohomology is conjecturally very closely related to the quantum cohomology of any crepant resolution of singularities. In particular, we can directly verify this conjecture in a strong form in the example where the orbifold is the n'th symmetric product of the complex plane, with crepant resolution given by the Hilbert scheme of points. REFERENCES:
J. Bryan, R. Pandharipande, The local Gromov-Witten theory of curves, math.AG/0411037. A. Okounkov, R. Pandharipande, The quantum cohomology of the Hilbert scheme of points in the plane, math.AG/0411210.
04/11/2005, 12:00 — 13:00 — Amphitheatre Pa3, Mathematics Building Emanuel Scheidegger, Institute for Theoretical Physics, TU Vienna
Topological String on K3 Fibrations
We explain that certain Gopakumar-Vafa
invariants (and, conjecturally, hence Gromow-Witten invariants) for
Calabi-Yau manifolds that admit a K3 fibration can be collected in
a generating function. This function is in general an automorphic
form determined by the fibration. In the class of K3 fibrations in
toric varieties in which the Picard lattice of the fiber has rank
one, we show how this automorphic form can be determined from the
topology of the fibration.
REFERENCES:
R. Gopakumar, C. Vafa, M-theory and topological
strings,II, hep-th/9812127.
S. Katz, A. Klemm, C. Vafa, M-Theory, Topological Strings and
Spinning Black Holes, hep-th/9910181.
J. Harvey, G. Moore, Algebras, BPS States, and Strings,
hep-th/9510182.
M. Marino, G. Moore, Counting higher genus curves in a
Calabi-Yau manifold, hep-th/9808131.
A. Klemm, M. Kreuzer, E. Riegler, E. Scheidegger, Topological
String Amplitudes, Complete Intersection Calabi-Yau Spaces and
Threshold Corrections, hep-th/0410018.
E. Scheidegger,Topological Strings on K3 fibrations, I, to
appear.
03/11/2005, 18:00 — 19:00 — Amphitheatre Pa3, Mathematics Building Victor Przyjalkowski, Mathematical Institute of Russian Academy of Science, Moscow
Generalized Givental's Theorem and classification of Fano threefolds
We present Golyshev's modularity conjecture that states that the counting equations for Fano threefolds with Picard group (which codes their Gromov-Witten invariants) are modular. To check it we find Gromov-Witten infariants of them. For this we generalise Givental's Theorem (in the Fano case) that gives us Gromov-Witten invariants for complete intersections in toric varieties with non-negative canonical bundle. REFERENCES:
V. Golyshev, The geometricity problem and modularity of some Riemann-Roch variations, Dokl. Akad. Nauk 386 (2002) 583-588.
V. Przyjalkowski, Quantum cohomology of smooth complete intersections in weighted projective spaces and singular toric varieties, math.AG/0507232.
A. Givental, A mirror theorem for toric complete intersections, Topological field theory, primitive forms and related topics (Kyoto, 1996), 141-175, alg-geom/9701016
02/11/2005, 17:00 — 18:00 — Amphitheatre Pa3, Mathematics Building Ralph Cohen, Stanford University
String Topology and Holomorphic Curves in the Cotangent Bundle
In this lecture I will give an overview of ``String topology''. This is a purely topological theory first introduced by Chas and Sullivan, that has developed into a vast topological field theory of structure on the homology of loop spaces of manifolds, and in spaces of paths with boundary values in D-branes. I will describe a Morse theoretic viewpoint of string topology. This involves representing moduli space of Riemann surfaces by a category of ribbon graphs. Using this together with analytic work of Salaman and Weber, we relate string topology operations on the loop space LM, with the Gromov Witten theory of the cotangent bundle T*M. REFERENCES:
M. Chas and D. Sullivan, String Topology, math.GT/9911159. To appear in Ann. of Math. R.L. Cohen, Morse theory, graphs, and string topology, math.GT/0411272. To appear in Proc. SMS/NATO Adv. study inst. on Morse theoretic methods in nonlinear analysis and symplectic topology, Kluwer press, 2005. R.L. Cohen and V. Godin, A polarized view of string topology, Topology, Geometry, and Quantum Field Theory, London Math. Soc. Lecture Notes, vol. 308 (2004) 127-154. R.L. Cohen and J.D.S. Jones, A homotopy theoretic realization of string topology, Math. Annalen, 324 (2002) 773-798. R.L. Cohen and A. Voronov, Notes on String Topology, math.GT/0503625. To appear in CRM Lecture Notes from summer school on string topology and Hochschild homology, Almeria Spain, 2005. D. Sullivan, Open and closed string field theory interpreted in classical algebraic topology, Topology, geometry and quantum field theory, London Math. Soc. Lecture Notes, 308, Cambridge Univ. Press, Cambridge, 2004, math.QA/0302332, p. 344-357.
31/10/2005, 11:00 — 12:00 — Amphitheatre Pa3, Mathematics Building Leonardo Rastelli, Princeton University
Topological Strings in
26/10/2005, 16:30 — 17:30 — Amphitheatre Pa2, Mathematics Building Sergei Gukov, Harvard University, Cambridge
Knot Homology and Topological Strings - II
26/10/2005, 15:00 — 16:00 — Amphitheatre Pa2, Mathematics Building Sergei Gukov, Harvard University, Cambridge
Knot Homology and Topological Strings - I
We start with a brief introduction into knot
homology theories and categorification of polynomial knot
invariants. Of particular interest are homology theories of
Ozsvath-Szabo-Rasmussen and Khovanov-Rozansky which provide a
homological lift of the Alexander polynomial and the quantum sl(N)
invariant, respectively. Motivated by the ideas from physics, we
then present a framework for unifying the sl(N) Khovanov-Rozansky
homology (for all N) with the knot Floer homology. We argue that
this unification should be accomplished by a triply graded homology
theory which categorifies the HOMFLY polynomial. We also describe
the geometric meaning of the new knot invariants in terms of the
enumerative geometry of Riemann surfaces with boundaries in a
certain Calabi-Yau three-fold.
REFERENCES:
S. Gukov, A. Schwarz and C. Vafa, Khovanov-Rozansky
homology and topological strings, hep-th/0412243.
N.M. Dunfield, S. Gukov and J. Rasmussen, The Superpolynomial
for Knot Homologies, math.gt/0505662.
25/10/2005, 17:00 — 18:00 — Amphitheatre Pa3, Mathematics Building Yuan-Pin Lee, University of Utah
Invariance of Tautological Equations
Finding relations in the tautological rings of
the moduli spaces of stable curves has been an important question.
In this talk, I will propose a conjectural framework of deriving
all/some tautological equations. These conjectures are inspired by
a study of Givental's axiomatic Gromov-Witten theory, for which
some background will be provided.
REFERENCES:
25/10/2005, 14:00 — 15:00 — Amphitheatre Pa1, Mathematics Building Vyacheslav S. Rychkov, Scuola Normale Superiore, Pisa
Symplectic Structure and Quantization on Moduli Spaces of Regular (Super)Gravity Solutions
Motivated by the AdS/CFT and black hole microstate counting, we discuss quantization of infinitely-dimensional families of regular supergravity solutions with Anti-de-Sitter or flat asymptotics. We demonstrate the general methods on two recent examples of such families - the "bubbling AdS" family of Lin-Lunin-Maldacena and the "D1-D5 with angular momentum" family of Lunin-Maldacena-Maoz. REFERENCES:
L.Maoz, V.S.Rychkov, Geometry Quantization from Supergravity: the Case of 'Bubbling Ads', hep-th/0508059. L.Grant, L.Maoz, J.Marsano, K.Papadodimas, V.S.Rychkov, Minisuperspace Quantization of 'Bubbling Ads' and Free Fermion Droplets, hep-th/0505079. H.Lin, O.Lunin, J.Maldacena, Bubbling Ads Space and 1/2 BPS Geometries, hep-th/0409174. O.Lunin, J.Maldacena, L.Maoz, Gravity Solutions for the D1-D5 System with Angular Momentum, hep-th/0212210.
24/10/2005, 15:00 — 16:00 — Amphitheatre Pa3, Mathematics Building Davesh Maulik, Princeton University
Gromov Witten Theory of Surfaces
In this talk, we will explain how to give an explicit
closed-formula of all-genus Gromov-Witten theory for certain
noncompact surface geometries.
18/10/2005, 17:00 — 18:00 — Amphitheatre Pa3, Mathematics Building João Baptista, Cambridge University
A Topological Gauged Sigma Model
In this talk I will consider non-linear gauged
sigma-models with Kahler domain and target. For a special choice of
potential these models admit Bogomolny (or self-duality) equations,
the so-called symplectic vortex equations. I will then describe a
topological field theory that studies the moduli space of solutions
of these equations. The correlation functions of the theory are
closely related to the recently introduced Hamiltonian
Gromov-Witten invariants.
REFERENCES:
J.M.Baptista, "A Topological Gauged Sigma
Model", hep-th/0502152.
K.Cieliebak, A.R.Gaio, I.Mundet i Riera, D.Salamon, "The
Symplectic Vortex Equations and Invariants of Hamiltonian Group
Actions", math.SG/0111176.
18/10/2005, 14:00 — 15:00 — Amphitheatre Pa3, Mathematics Building Andy Neitzke, Institute for Advanced Study, Princeton
BPS Microstates and Open Topological Strings
I will discuss some recent attempts to
understand the meaning of nonperturbative topological strings---one
approach via counting of BPS microstates, and in particular its
extension to open topological strings, and another via topological
M-theory.
REFERENCES:
M.Aganagic, A.Neitzke, C.Vafa, "BPS Microstates and
the Open Topological String Wave Function",
hep-th/0504054.
R.Dijkgraaf, S.Gukov, A.Neitzke, C.Vafa, "Topological M-theory
as Unification of Form Theories of Gravity",
hep-th/0411073.
12/10/2005, 17:00 — 18:00 — Amphitheatre Pa2, Mathematics Building Radu Popescu, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos
Twisted K-theory, C*-algebras and groupoids.
We take a look at the nontorsion case, which correspond to infinite dimensional projective bundles. We cast this in the realm of C*-algebras and groupoids, using Morita equivalences and Brauer groups, with reference to the general analysis of the interplay between groupoids and algebras as a central theme of the noncommutative geometry.
M. Atiyah, G. Segal, Twisted K-theory, math.KT/0407054.
A. Connes, Noncommutative Geometry, Academic Press, 1994.
P. Donavan and M. Karoubi, Graded Brauer groups and K-theory with local coefficients, Publ. Math IHES 38 (1970) 5-25.
N.P. Landsman, Mathematical topics between classical and quantum mechanics. Springer Monographs in Mathematics. Springer-Verlag, New York, 1998
J. Rosenberg, Continuous-trace algebras from the bundle theoretic point of view. J. Austral. Math. Soc. Ser. A 47 (1989), no. 3, 368-381.
I. Raeburn, D.P. Williams, Morita equivalence and continuous-trace C*-algebras, Mathematical Surveys and Monographs, 60. American Mathematical Society, Providence, RI, 1998. xiv+327 pp.
A. Kumjian, P. S. Muhly, J. N. Renault, D. P. Williams, The Brauer Group of a Locally Compact Groupoid, funct-an/9706004.
11/10/2005, 17:00 — 18:00 — Amphitheatre Pa3, Mathematics Building Andrei Okounkov, Princeton University
GW/DT Correspondence for Descendents
I will explain the present status of the conjectural correspondence between descendents in Gromov-Witten and Donaldson-Thomas theories, in particular, the appearance of the Frankenstein series .