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Author Archives: Victoria Hain

Jake Solomon: Lectures

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.

Yohsuke Imagi: Lectures

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.

Mohammed Abouzaid: Lectures

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.

Rodrigo Barbosa: Lectures

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

Dan Xie: Lectures

April 10, 2019
TITLE: New Five Dimensional Sasaki-Einstein Manifolds and the ADS/CFT Correspondence


Pavel Putrov: Lectures

April 10, 2019
TITLE: 6d (1,0) theories on 4-manifolds

Lecture cancelled.

June 4, 2019
TITLE: Cayley cycles and modular forms

ABSTRACT: In the M/string-theory setting, 5-branes 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 4-manifolds valued in
the ring of (topological) modular forms. In my talk I will describe
general properties of this map and consider an example.

Alex Kinsella: Lectures

April 11, 2019
TITLE: Heterotic Duals of M-Theory on Joyce Orbifolds

Slides of Lecture

Gordon Kane: Lectures

April 11, 2019
TITLE: Hidden Sectors and Associated Physics — Supersymmetry Breaking, Dark Matter, de Sitter Vacuum

ABSTRACT: Compactifications on G2 manifolds lead to theories that describe the world well. As we understand G2 manifolds better we can test and improve this situation. Hidden sectors generically accompany G2 manifolds, and in turn lead to important physics – in particular supersymmetry breaking, De Sitter vacua, and candidates for the dark matter of the universe.
Slides of Lecture

Shamit Kachru: Lectures

April 8, 2019
TITLE: K3 Metrics from Little String Theory

ABSTRACT: I explain how one can find formulae for Ricci flat metrics on K3 surfaces by using data about the BPS spectrum of an auxiliary (compactified) little string theory. Well known approximations to such metrics, like the semi-flat approximation and the Gross-Wilson metric, arise as limits of this formalism.

Slides of lecture

Jonathan Heckman: Lectures

April 12, 2019
TITLE: F-Theory and Spin7 Manifolds

We discuss some recent progress in understanding 4D vacua obtained from F-theory on Spin(7) manifolds. Such backgrounds provide a potentially attractive way to avoid fine-tuning to cancel zero point energies. We present a general proposal for cosmological solutions in this framework, and also develop the physical formalism of local Spin(7) geometries containing a four-manifold of ADE singularities.

Based on work with C. Lawrie, L. Lin, and G. Zoccarato, hep-th/1811.01959

and C. Lawrie, L. Lin, J. Sakstein and G. Zoccarato, hep-th/1901.10489.