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Gianluca Zoccarato: Lectures

September 16, 2020 (jointly with Mirjam Cvetic)

TITLE: Unification of Higgs Bundles

ABSTRACT: We introduce a program to study compactifications of M-/F-theory on spaces with G2 and Spin(7) holonomy, which results in N=1 supersymmetry in four- and three-dimensions, respectively, by studying the world-volume gauge dynamics of Higgs bundles, which are associated with higher co-dimension singularities of special holonomy spaces.

Specifically, we will consider solutions of M-theory on local geometries with Spin(7) holonomy. The local geometry is an ADE fibration over a four manifold which gives a compactification of 7d supersymmetric Yang-Mills theory on Spin(7) manifold. The configurations of this system are controlled by a Higgs bundle on the four manifold. This Higgs bundle, which we will call the Spin(7) system, has the remarkable property of unifying other known Higgs bundles. We will show how solutions of other known Higgs bundles fit in the Spin(7) system and show how it is possible to use the Spin(7) system to create interpolating solutions between different Higgs bundles.

Slides of Lecture, Part I

Slides of Lecture, Part II

Justin Sawon: Lectures

Recorded June 2020
TITLE: Singular fibres of holomorphic Lagrangian fibrations

ABSTRACT: Fibrations on compact holomorphic symplectic manifolds are Lagrangian: their fibres must be Lagrangian with respect to the holomorphic symplectic structure. Moreover, the general fibre must be an abelian variety and singular fibres must occur in codimension one. In this talk I will survey results of Matsushita, Hwang-Oguiso, Christian Lehn, and myself that describe the structure of the singular fibres that can occur in these Lagrangian fibrations.
Slides of Lecture

Antoine Bourget: Lectures

Recorded June 2020
TITLE: Hasse diagrams for Symplectic Singularities via Magnetic Quivers

ABSTRACT:
In this lecture I review the construction of a finite Hasse diagram encoding the structure of singularities and symplectic leaves in conic symplectic singularities. This construction uses the concept of magnetic quiver, which is a combinatorial description of certain conic symplectic singularities, and the algorithm known as “quiver subtraction”. I then show how this can be used to gain insight about these spaces, and how this is connected to various effects in classical and quantum physics (Higgs mechanism, structure of the chiral ring, moduli space of strongly coupled SCFTs in 5 and 6 dimensions).
Slides of Lecture

Roger Bielawski: Lectures

Recorded June 2020
TITLE: Twistor spaces and hyperkahler metrics

ABSTRACT: I discuss several questions about twistor spaces of hyperkaehler manifolds:
multiple components of the Kodaira moduli space of sections; differential
geometry of spaces of sections which include sections with “wrong” normal
bundle; and (pseudo-)hyperkaehler metrics arising on Hilbert schemes of higher
degrees curves in a twistor space.
On this last topic, in addition to older results, I discuss the current project
with Lorenzo Foscolo, where we construct (presumably QALF) hyperkaehler manifolds
from W-invariant hypertoric varieties. We suggest that a subclass of these
corresponds precisely to the Coulomb branches of 3-dimensional N=4 SUSY gauge theories.
Slides of Lecture

Claire Voisin: Lectures

Recorded June 2020
TITLE: Hodge theory and the topology of hyper-Kähler manifolds: an introduction

ABSTRACT: After a short discussion of hyper-Kähler manifolds in the Riemannian and complex geometry contexts,
I discuss the general topological properties of hyper-Kähler
manifolds obtained from the local study of the period map. At the end of the talk, I start discussing
hyper-Kähler manifolds in the algebraic geometry setting, which will be the subject of the second lecture.

The audience for this lecture is encouraged to download the slides and to follow
along in the slides while watching the video.

Slides of lecture

June 3, 2020
TITLE: Polarized variations of Hodge structures of hyper-Kähler type

ABSTRACT: Like for abelian varieties and complex tori, the theory of projective hyper-Kähler manifolds is very different from the theory of general Kähler ones. The former is related via the period map and Torelli theorem to the study of certain quotients of bounded Hermitian symmetric domains by arithmetic groups. I will discuss various results concerning these moduli spaces and constructions of hyper-Kähler manifolds via algebraic geometry.

Giulia Saccà: Lectures

June 3, 2020
TITLE: Hyperkahler manifolds are higher dimensional K3 surfaces

ABSTRACT: This talk will survey some aspects of the theory of hyperkahler manifolds. The focus will be on how many of the properties enjoyed by this class of manifolds are the natural generalization of similar properties enjoyed by K3 surfaces.

Andrew Neitzke: Lectures

June 3, 2020
TITLE: An update on hyperkahler metrics on moduli of Higgs bundles

ABSTRACT: In joint work with Davide Gaiotto and Greg Moore, we gave a new conjectural construction of the hyperkahler metric on moduli spaces of Higgs bundles, in which the key new ingredient was counts of BPS states (Donaldson-Thomas-type invariants).

Through the work of various authors, including me, Dumas, Fredrickson, Mazzeo, Swoboda, Weiss, Witt, there is now some evidence that this conjectural picture is correct. On the one hand, some of the asymptotic predictions which follow from the conjecture have been proven; on the other hand, there is numerical evidence that the conjecture is correct even far away from the asymptotic limit. I will review these developments.

Slides of Lecture

Jaroslaw Wisniewski: Lectures

June 1, 2020
TITLE: Quaternion-Kähler Manifolds via algebraic torus action on projective contact manifolds

ABSTRACT: Let M be a positive quaternion-Kähler manifold of (real) dimension 4n. A conjecture by LeBrun and Salamon asserts that M is symmetric. The twistor space of M is known to be a projective Fano X manifold of (complex) dimension 2n+1 with reductive group of automorphisms. I will report on two projects which aim at proving LeBrun-Salamon conjecture.

In a joint project with Buczyński and Weber we proved the conjecture for n equal 3 and 4. In a project with Romano, Occhetta, and Sola Conde, we proved the conjecture for quaternion-Kähler manifolds admitting an action of a torus of rank bigger or equal to max(2,(n-3)/2). Both proofs are in complex settings, using action of a Cartan (algebraic) torus in a reductive group on a complex contact manifold. Eventually, via the torus action, the problem is reduced to questions in birational algebraic geometry which apparently are related to classical objects and transformations.

Slides of Lecture

Balázs Szendrői: Lectures

June 5, 2020
TITLE: Global aspects of Calabi-Yau moduli space

ABSTRACT: I will give a review of global aspects of the moduli space of Calabi-Yau manifolds (holonomy SU(n)) and the period map, concentrating on the three-dimensional case. I will in particular explain what’s known about the Torelli problem in this setting. 
Slides of lecture

Donaldson awarded the 2020 Wolf Prize in Mathematics

Collaboration Principal Investigator Sir Simon Donaldson, as well as Stanford’s Yakov Eliashberg, have been awarded the 2020 Wolf Prize in Mathematics

Awarded since 1978, the Wolf Prize recognizes “outstanding scientists and artists from around the world … for achievements in the interest of mankind and friendly relations among peoples.” Along with the Fields Medal and Abel Prize, it is considered the closest equivalent to a Nobel Prize in mathematics.

The prize presentation will take place at a special ceremony at the Knesset (Israel´s Parliament) in Jerusalem on June 11, 2020.

Read the news article.