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Author Archives: Victoria Hain
Costante Bellettini: 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.
Yanki Lekili: Lectures
June 4, 2019
TITLE: Homological mirror symmetry for higher dimensional pants
ABSTRACT: Any Riemann surface can be glued together from pairsofpants. This provides a way of proving homological mirror symmetry for Riemann surfaces by first constructing a mirror to a single pairofpants and then categorically gluing several copies.
A theorem of Mikhalkin says that complex hypersurfaces in CP^{n} admit decompositions into higher dimensional pairsofpants. We prove that the wrapped Fukaya category of the complement of (n+2)generic hyperplanes in CP^{n} (ndimensional pants) is equivalent to the derived category of the singular affine variety x_{1}x_{2}..x_{n+1}=0. By taking covers, we also get some simple examples of gluing of pairsofpants and the corresponding mirror symmetry statements. Our proof is simple but combines ideas from lowdimensional topology (HeegaardFloer) and noncommutative resolutions of singularities.
This is joint work with A. Polishchuk.
Eric Zaslow: Lectures
June 6, 2019
TITLE: Graphs, Lagrangians and Open GromovWitten Conjectures
ABSTRACT: This talk picks up on the one of David Treumann, describing our joint work. A trivalent graph on a sphere defines a highergenus Legendrian surface in complex threespace. 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 AganagicVafa, we conjecture that the defining function for this chromatic Lagrangian is a generating function for open GromovWitten 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 GromovWitten invariants at all genus and relate them to cohomological quiver invariants.
David Treumann: Lectures
June 6, 2019
TITLE: Constructible sheaves and Lagrangian cones
ABSTRACT: The KashiwaraSchapira 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 R^{2n}. I’ll describe the categories you get for some examples in R^{2}, R^{4}, and R^{6}. 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.
Jake Solomon: Lectures
June 3, 2019
TITLE: Geodesics in the space of positive Lagrangian submanifolds
ABSTRACT:
It is a problem of fundamental importance in symplectic geometry to determine when a Lagrangian submanifold of a CalabiYau 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 nonpositive 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 nonlinear PDE. I will discuss some results on the existence of solutions to this PDE.
Yohsuke Imagi: Lectures
June 4, 2019
TITLE:On ThomasYau’s Uniqueness Theorem
ABSTRACT: ThomasYau’s seminal work on Fukaya categories and special
Lagrangians includes a uniqueness theorem for special Lagrangians of a fixed
isomorphismclass 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.
Mohammed Abouzaid: Lectures
June 4, 2019
TITLE: On nearby special Lagrangians
ABSTRACT:
Given a closed, embedded, special Lagrangian in a CalabiYau
manifold, we consider the question of classifying the C^{0}close (nearby) special
Lagrangians. The corresponding classification result in the C^{∞} topology is
classical, as such Lagrangians correspond to the graphs of harmonic 1forms. 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.
Rodrigo Barbosa: Lectures
April 10, 2019
TITLE: Deformations of G_{2}structures, String Dualities and Flat Higgs Bundles
ABSTRACT: We study Mtheory compactifications on (resolutions of) G_{2}orbifolds given by total spaces of ALEfibrations over a compact flat Riemannian 3manifold 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 BrieskornGrothendieck 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 G_{2}structures in terms of IIBbranes. The net result is two algebrogeometric descriptions of the moduli space of complexified G_{2}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
Dan Xie: Lectures
April 10, 2019
TITLE: New Five Dimensional SasakiEinstein Manifolds and the ADS/CFT Correspondence
ABSTRACT:
Pavel Putrov: Lectures

 4/10/2019: 6d (1,0) theories on 4manifolds
 6/4/2019 Cayley cycles and modular forms
April 10, 2019
TITLE: 6d (1,0) theories on 4manifolds
Lecture cancelled.
June 4, 2019
TITLE: Cayley cycles and modular forms
ABSTRACT: In the M/stringtheory setting, 5branes wrapping a Cayley cycle in a
local Spin(7)manifold define an effective 2d conformal field theory
with minimal supersymmetry. From mathematical point of view, this
predicts existence of an invariant of smooth spin 4manifolds valued in
the ring of (topological) modular forms. In my talk I will describe
general properties of this map and consider an example.