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Category Archives: Lectures
June 7, 2019
TITLE 1: Analysis of singular sets in geometric analysis
TITLE 2: Analysis of singular sets in calibrated geometric analysis
ABSTRACT: I will give an overview of known regularity results for area minimizing currents and their (known or potential) applications to problems in calibrated geometry. In the first talk I will focus on the classical second order theory (Almgren, etc.), while in the second one I will emphasise tools that can be used to obtain regularity information by exploiting the first order information given by the calibrating condition. If time permits, I will address other variational problems (e.g. harmonic maps) that have instances in which a first order information can provide finer conclusions on the singular behaviour.
June 4, 2019
TITLE: Homological mirror symmetry for higher dimensional pants
ABSTRACT: Any Riemann surface can be glued together from pairs-of-pants. This provides a way of proving homological mirror symmetry for Riemann surfaces by first constructing a mirror to a single pair-of-pants and then categorically gluing several copies.
A theorem of Mikhalkin says that complex hypersurfaces in CPn admit decompositions into higher dimensional pairs-of-pants. We prove that the wrapped Fukaya category of the complement of (n+2)-generic hyperplanes in CPn (n-dimensional pants) is equivalent to the derived category of the singular affine variety x1x2..xn+1=0. By taking covers, we also get some simple examples of gluing of pairs-of-pants and the corresponding mirror symmetry statements. Our proof is simple but combines ideas from low-dimensional topology (Heegaard-Floer) and noncommutative resolutions of singularities.
This is joint work with A. Polishchuk.
June 6, 2019
TITLE: Graphs, Lagrangians and Open Gromov-Witten Conjectures
ABSTRACT: This talk picks up on the one of David Treumann, describing our joint work. A trivalent graph on a sphere defines a higher-genus Legendrian surface in complex three-space. This Legendrian serves as a boundary condition for Lagrangian Fukaya objects, equivalently as a singular support condition for constructible sheaves. The moduli space of objects in this category of sheaves itself embeds as a “chromatic Lagrangian” subspace of the space local systems on the surface. Generalizing work of Aganagic-Vafa, we conjecture that the defining function for this chromatic Lagrangian is a generating function for open Gromov-Witten invariants counting holomorphic disks with boundary on the Lagrangian Fukaya objects.
Time permitting, I will discuss joint work with Linhui Shen on how to exploit cluster structures to compute these conjectural open Gromov-Witten invariants at all genus and relate them to cohomological quiver invariants.
June 6, 2019
TITLE: Constructible sheaves and Lagrangian cones
ABSTRACT: The Kashiwara-Schapira theory of singular support for constructible sheaves, which I will assume is not familiar to anyone, can be used to attach a triangulated category to a Lagrangian cone in R2n. I’ll describe the categories you get for some examples in R2, R4, and R6. The examples are all special, or else conjectured to be special; I wish I knew how to put that to use. Joint work with Eric Zaslow.
Arrival date: Saturday, September 7.
Departure date: Wednesday afternoon, September 11, or Thursday, September 12.
SUN 8 SEP
MON 9 SEP
TUES 10 SEP
WED 11 SEP
|B. Lawson||A. Hanany||C. Hull||C. Vafa|
|M. Larfors||R. Conlon||G. Ball||A. Klemm|
|D. Morrison (SCGP Weekly Talk)|
|Leave for NYC (from hotel)|
|S.-T. Yau||N. Fornasin|
|A. Waldron||B. Willett||A. Moroianu|
- Gavin Ball (Duke University), Quadratic closed G2-structures
- Ronan Conlon (Florida International University), Classification results for expanding and shrinking gradient Kähler-Ricci solitons
- Nelvis Fornasin (University of Freiburg), The eta invariant under cone-edge degeneration
- Amihay Hanany (Imperial College London), Hasse diagrams for symplectic singularities
- Chris Hull (Imperial College London), Special Holonomy Metrics, Degenerate Limits and Intersecting Branes
- Albrecht Klemm (University of Bonn), CY 3-folds over finite fields, Black hole attractors, and D-brane masses
- Magdalena Larfors (Uppsala University), Heterotic string theory and G2 structure manifolds
- Blaine Lawson (Stony Brook University), Pseudo-Convexity for the Special Lagrangian Potential Equation
- Andrei Moroianu (CNRS), Toric nearly Kähler 6-manifolds
- David R. Morrison (UC Santa Barbara), Nonaelian gauge symmetry and charged chiral matter in G_2 compactifications
- Cumrun Vafa (Harvard University), G2 Structure and Physical Interpretation of Taubes Construction of SW Invariants
- Alex Waldron (Michigan State University), G2-instantons on the 7-sphere
- Brian Willett (UC Santa Barbara), Global aspects of the 3d-3d correspondence
- Shing-Tung Yau (Harvard University), String duality and G2 manifolds
This conference will be immediately followed by our Third annual meeting held at the Simons Foundation in New York City.
June 3, 2019
TITLE: Geodesics in the space of positive Lagrangian submanifolds
It is a problem of fundamental importance in symplectic geometry to determine when a Lagrangian submanifold of a Calabi-Yau manifold can be moved by Hamiltonian flow to a special Lagrangian. I will describe an approach to this problem based on the geometry of the space of positive Lagrangians. This space admits a Riemannian metric of non-positive curvature and a convex functional with critical points at special Lagrangians. Existence of geodesics in the space of positive Lagrangians implies uniqueness of special Lagrangians in a Hamiltonian isotopy class as well as rigidity of Lagrangian intersections. The geodesic equation is a degenerate elliptic fully non-linear PDE. I will discuss some results on the existence of solutions to this PDE.
June 4, 2019
TITLE:On Thomas-Yau’s Uniqueness Theorem
ABSTRACT: Thomas-Yau’s seminal work on Fukaya categories and special
Lagrangians includes a uniqueness theorem for special Lagrangians of a fixed
isomorphism-class in the derived Fukaya category; which will be explained with
an outline of the proof, including a later contribution made by Dominic Joyce,
Joana Oliveira dos Santos and the speaker.
June 4, 2019
TITLE: On nearby special Lagrangians
Given a closed, embedded, special Lagrangian in a Calabi-Yau
manifold, we consider the question of classifying the C0-close (nearby) special
Lagrangians. The corresponding classification result in the C∞ topology is
classical, as such Lagrangians correspond to the graphs of harmonic 1-forms. I
shall explain that, if the fundamental group of the Lagrangian is nilpotent, then
all embedded nearby Lagrangians which are unobstructed in the sense of Floer
theory are given by this construction, and will explain some basic examples of
unobstructed Lagrangians which are not graphical in some cases where the
fundamental group is not nilpotent. The proof relies on methods of geometric
analysis and Floer theory, building upon the ideas of Thomas and Yau. This is
joint work with Yohsuke Imagi.
April 10, 2019
TITLE: Deformations of G2-structures, String Dualities and Flat Higgs Bundles
ABSTRACT: We study M-theory compactifications on (resolutions of) G2-orbifolds given by total spaces of ALE-fibrations over a compact flat Riemannian 3-manifold Q. The flatness condition allows an explicit description of the moduli space of supersymmetric vacua: it is parametrized by flat sections of a bundle of Brieskorn-Grothendieck resolutions over Q. Moreover, when instanton corrections are neglected, we have an explicit description of the moduli space for the IIA dual compactification in terms of flat Higgs bundles on Q. We explain how it suggests a new interpretation of SYZ mirror symmetry, while also providing a description of G2-structures in terms of IIB-branes. The net result is two algebro-geometric descriptions of the moduli space of complexified G2-structures: one as a character variety, and another as a Hilbert scheme of points on a threefold. We show the moduli spaces match in an important example. This is joint work with Tony Pantev.
Slides of Lecture