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Category Archives: Lectures
Valentino Tosatti: Lectures
September 12, 2023
TITLE: Holomorphic Lagrangian fibrations and special Kähler geometry
ABSTRACT: Consider a compact hyperkähler manifold (aka irreducible holomorphic symplectic) with a nontrivial fiber space structure onto a lower-dimensional space. Classical work of Matsushita shows that the base must be half-dimensional, and the smooth fibers are holomorphic Lagrangian tori. A basic conjecture is that the base of such holomorphic Lagrangian fibrations should be projective space. I will discuss a new proof, joint with Yang Li, of a theorem of Hwang which shows that this conjecture is true when the base is smooth. Our arguments exploit crucially the differential geometry of a “special Kähler metric” that exists on the base away from the discriminant locus.
Langte Ma: Lectures
September 13, 2023
TITLE: Instantons on Joyce’s G2-manifolds
ABSTRACT: As 7-manifolds with special holonomy, examples of compact G2-manifolds were first constructed by Joyce as resolutions of flat G2-orbifolds. Later, Walpuski constructed non-trivial G2-instantons over Joyce’s manifolds via gluing techniques. In this talk, I will first explain how to define a deformation invariant of G2-orbifolds by counting flat connections, then describe the moduli space of instantons over certain non-compact G2-manifolds that appeared in Joyce’s construction, with the aim to give a complete description of moduli spaces over some examples in Joyce’s list.
Mingyang Li: Lectures
September 10, 2023
TITLE: Classification results for Hermitian non-Kahler gravitational instantons
ABSTRACT: We will discuss some classification results for Hermitian non-Kähler gravitational instantons. There are three main results: (1) Non-existence of certain Hermitian non-Kähler ALE gravitational instantons. (2) Complete classification for Hermitian non-Kähler ALF/AF gravitational instantons. (3) Non-existence of Hermitian non-Kähler gravitational instantons under suitable curvature decay condition, when there is more collapsing at infinity (ALG, ALH, etc.). These are achieved by a thorough analysis of the collapsing geometry at infinity and compactifications. These results are based on a previous work and an upcoming work by the speaker.
Jin Li: Lectures
September 12, 2023
TITLE: On the geometry of resolutions of G-2-manifolds with ICS
ABSTRACT: Given a compact G_2 manifold with isolated conical singularities (ICS), the process of resolutions of these singularities gives us a one-parameter family of torsion-free G_2 structures, which can be viewed as a curve in some moduli space. This talk reports the progress in estimating the length of the curve under the L^2 Riemannian metric on the moduli space.
Jonas Lente: Lectures
September 12, 2023
TITLE: Modular Mathai-Quillen currents
ABSTRACT: The Mathai-Quillen current is a correction term that appears in the Poincaré-Hopf theorem for manifolds with boundary, similar to the eta invariant in the Atiyah-Patodi-Singer theorem.
In this talk, I will explain how to extract a modular Mathai-Quillen current from modular cobordism invariants, such as the Witten genus. On the way, I will present an analogue of the Poincaré-Hopf theorem for supermanifolds. The hope is that this modular current provides a modular extension of the nu invariant for G_2-manifolds. This is part of my ongoing PhD project supervised by S. Goette and K. Wendland.
Julius Grimminger: Lectures
September 10, 2023
TITLE: Stratified hyper-Kähler moduli spaces and physics
ABSTRACT: Singular hyper-Kähler varieties are stratified into smooth subsets called symplectic leaves. In recent years the 3d Coulomb branch construction of hyper-Kähler varieties has been a powerful tool to study this stratification, going under the name of quiver subtraction. This algorithm is derived from intuition coming from physics, and in particular brane systems. We will review the concept of this stratification, as well as some computational methods and physical interpretations.
Charles Cifarelli: Lectures
September 11, 2023
TITLE: Steady gradient Kähler-Ricci solitons and Calabi-Yau metrics on Cn
ABSTRACT: I will present a new construction of complete steady gradient Kähler-Ricci solitons on C^n, using the theory of hamiltonian 2 forms, introduced by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, as an Ansatz. The metrics come in families of two types with distinct geometric behavior, which we call Cao type and Taub-NUT type. In particular, the Cao type and Taub-NUT type families have a volume growth rate of r^n and r^{2n-1}, respectively. Moreover, each Taub-NUT type family contains a codimension 1 subfamily of complete Ricci-flat metrics. This is joint work with V. Apostolov.
Olivier Biquard: Lectures
September 12, 2023
TITLE: Limits of Kähler-Einstein metrics with cone singularities, and Calabi-Yau metrics
ABSTRACT: There exist various constructions of open Calabi-Yau metrics (Kähler Ricci flat metrics on quasiprojective varieties). There are general questions about obtaining them as limits of Kähler-Einstein metrics with cone singularities on compactifications. I will discuss several cases, in particular the case of the Tian-Yau metric on the complement of an anticanonical divisor in a Fano manifold. Joint work with Henri Guenancia.
Mina Aganagic: Lectures
September 11, 2023
TITLE: Homological link invariants from Floer theory
ABSTRACT: A new relation between homological mirror symmetry and representation theory solves the knot categorification problem. The symplectic geometry side of mirror symmetry is a theory which generalizes Heegard-Floer theory from gl(1|1) to arbitrary simple Lie (super) algebras. The corresponding category of A-branes has many special features, which render it solvable explicitly. In this talk, I will describe how the theory is solved, and how homological link invariants arise from it.
Progress and Open Problems 2023: September 10-13, 2023, SCGP, Stony Brook
Arrival date: Saturday, September 9.
Departure date: Wednesday afternoon, September 13, or Thursday, September 14.
All times are EDT. Organizers are Mark Haskins (Duke University) and Simon Salamon (King’s College London).
Schedule:
|
SUN 10 SEP |
MON 11 SEP |
TUES 12 SEP |
WED 13 SEP |
|
|
9:30-10:30 |
J. Sawon | L. Wang | J. Fine | L. Ma (online lecture) |
|
11:00-12:00 |
J. Grimminger | C. Cifarelli | D. Baldwin | D. Platt |
|
13:15-14:15 |
V. Tosatti | |||
|
14:30-15:30 |
M. Li | M. Aganagic (online lecture) |
14:30-15:00: J. Lente 15:00-15:30: J. Li |
|
|
16:00-17:00 |
16:00-16:30 A. Payne 16:30-17:00 E. Sabag |
J. Zhang | O. Biquard (Joint with Stony Brook Geometry) |
Speakers:
The links will take you to abstracts, slides of lectures, and/or video recordings of the lectures (when available).
- Mina Aganagic (UC Berkeley), Homological link invariants from Floer theory
- Daniel Baldwin (King’s College London), Coulomb and Higgs phases of G_2 manifolds
- Olivier Biquard (Sorbonne), Limits of Kähler-Einstein metrics with cone singularities, and Calabi-Yau metrics
- Charles Cifarelli (Nantes), Steady gradient Kähler-Ricci solitons and Calabi-Yau metrics on Cn
- Joel Fine (Université Libre de Bruxelles), Knot invariants from hyperbolic, SU(3) and G_2 geometries
- Julius Grimminger (Oxford), Stratified hyper-Kähler moduli spaces and physics
- Jonas Lente (Freiburg), Modular Mathai-Quillen currents
- Jin Li (Freiburg), On the geometry of resolutions of G-2-manifolds with ICS
- Mingyang Li (UC Berkeley), Classification results for Hermitian non-Kahler gravitational instantons
- Langte Ma (Shanghai Jiao Tong), Instantons on Joyce’s G2-manifolds
- Alec Payne (Duke), Closed G2-Structures with Negatively Pinched Ricci Curvature
- Daniel Platt (Imperial College London), Approximations of harmonic 1-forms on real loci of Calabi-Yau 3-folds
- Evyatar Sabag (Oxford), G2 Manifolds from 4d N=1 Theories
- Justin Sawon (UNC Chapel Hill), Lagrangian fibrations in four and six dimensions
- Valentino Tosatti (NYU), Holomorphic Lagrangian fibrations and special Kähler geometry
- Lu Wang (Yale), A mean curvature flow approach to density of minimal cones
- Junsheng Zhang (UC Berkeley), On complete Calabi-Yau manifolds asymptotic to cones
This conference will be immediately preceded by our Seventh annual meeting held at the Simons Foundation in New York City.
