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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 HeegardFloer theory from gl(11) to arbitrary simple Lie (super) algebras. The corresponding category of Abranes 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 1013, 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 

9:3010:30 
J. Sawon  L. Wang  J. Fine  L. Ma (online lecture) 
11:0012:00 
J. Grimminger  C. Cifarelli  D. Baldwin  D. Platt 
13:1514:15 
V. Tosatti  
14:3015:30 
M. Li  M. Aganagic (online lecture) 
14:3015:00: J. Lente 15:0015:30: J. Li 

16:0017:00 
16:0016:30 A. Payne 16:3017:00 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 KählerEinstein metrics with cone singularities, and CalabiYau metrics
 Charles Cifarelli (Nantes), Steady gradient KählerRicci solitons and CalabiYau metrics on C^{n}
 Joel Fine (Université Libre de Bruxelles), Knot invariants from hyperbolic, SU(3) and G_2 geometries
 Julius Grimminger (Oxford), Stratified hyperKähler moduli spaces and physics
 Jonas Lente (Freiburg), Modular MathaiQuillen currents
 Jin Li (Freiburg), On the geometry of resolutions of G2manifolds with ICS
 Mingyang Li (UC Berkeley), Classification results for Hermitian nonKahler gravitational instantons
 Langte Ma (Shanghai Jiao Tong), Instantons on Joyce’s G_{2}manifolds
 Alec Payne (Duke), Closed G2Structures with Negatively Pinched Ricci Curvature
 Daniel Platt (Imperial College London), Approximations of harmonic 1forms on real loci of CalabiYau 3folds
 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 CalabiYau manifolds asymptotic to cones
This conference will be immediately preceded by our Seventh annual meeting held at the Simons Foundation in New York City.
Fabio Apruzzi: Lectures
March 13, 2023
TITLE: Generalized symmetries from string theory
ABSTRACT: String theory provides a systematic way of constructing quantum field theories (QFTs) via geometric engineering. In particular, this can involve noncompact CalabiYau spaces in various dimensions, as well as other special holonomy manifolds. I will describe the dictionary between the generalized symmetry data of the QFTs and some specific cohomology of the geometric engineering spaces by focusing on explicit examples.
Lakshya Bhardwaj: Lectures
March 13, 2023
TITLE: Overview of Generalized Symmetries
ABSTRACT: I will provide an overview of various types (higherform, highergroup and noninvertibles) of generalized symmetries and how they arise in gauge theories. This will set stage for later talks that will describe how generalized symmetries are encoded in geometric engineering.
Federico Bonetti: Lectures
March 14, 2023
TITLE: SymTFTs, Differential Cohomology, and Geometric Engineering
ABSTRACT: The symmetry data of a quantum field theory (QFT) in d spacetime dimensions is conveniently captured by an auxiliary topological field theory in d+1 spacetime dimensions, referred to as the Symmetry Topological Field Theory (SymTFT). After a brief introduction to the SymTFT, I will focus on the following question: how can we compute the SymTFT for a QFT engineered geometrically in string theory/Mtheory? The formalism of differential cohomology provides systematic tools to address this problem, as I will illustrate in some examples.
Saman Habibi Esfahani: Lectures
March 16, 2023
TITLE: Towards a Monopole Fueter Floer Homology
ABSTRACT: Monopoles appear as the dimensional reduction of instantons to 3manifolds. An interesting feature of the monopole equation is that it can be generalized to certain higherdimensional spaces. The most interesting examples appear on CalabiYau 3folds and G2manifolds. Monopoles, conjecturally, can be used to define invariants of 3manifolds, CalabiYau 3folds, and G2manifolds. These monopole invariants, conjecturally, are related to certain counts of calibrated submanifolds, similar to the Taubes’ theorem, which relates the SeibergWitten and Gromov invariants of symplectic 4manifolds.
Motivated by this conjecture, we propose a Floer theory for 3manifolds, generated by Fueter sections on hyperkähler bundles with fibers modeled on the moduli spaces of monopoles on R3. A major difficulty in defining these homology groups is related to the noncompactness problems. We prove partial results in this direction, examining the different sources of noncompactness, and proving some of them, in fact, do not occur.
Constantin Teleman: Lectures
March 13, 2023
TITLE: Introduction to topological symmetries and higher groups
ABSTRACT: I will review the setting of an algebra of symmetries acting on a QFT. Special emphasis will be placed on symmetries arising from finite homotopy types (aka higher finite groups) and the way homotopical calculations quantize to a ‘higher categorical group ring’ of a space.
Victoria Hoskins: Lectures
January 9, 2023
TITLE: Motivic mirror symmetry for Higgs bundles
ABSTRACT: Moduli spaces of Higgs bundles for Langlands dual groups are conjecturally related by a form of mirror symmetry. For SLn and PGLn, Hausel and Thaddeus conjectured a topological mirror symmetry given by an equality of (twisted orbifold) Hodge numbers, which was proven by GroechenigWyssZiegler and also MaulikShen. We lift this to an isomorphism of Voevodsky motives, and thus in particular an equality of (twisted orbifold) rational Chow groups. Our method is based on Maulik and Shen’s approach to the HauselThaddeus conjecture, as well as showing certain motives are abelian, in order to use conservativity of the Betti realisation on abelian motives. This is joint work with Simon Pepin Lehalleur.
Oscar GarciaPrada: Lectures
January 10, 2023
TITLE: Vinberg pairs and Higgs bundles
ABSTRACT: A finite order automorphism of a complex semisimple Lie group determines a cyclic grading of its Lie algebra. Vinberg’s theory is concerned with the geometric invariant theory associated to this grading. Important examples include the case of involutions and representations of cyclic quivers. After reviewing some basic facts about Vinberg’s theory, in this talk I will discuss about its relation to the geometry of moduli spaces of Higgs bundles over a compact Riemann surface.
Katrin Wendland: Lectures
January 11, 2023
TITLE: An application of folding ADE to BCFG
ABSTRACT: We consider families of CalabiYau threefolds which are obtained from the deformation spaces of ADE type surface singularities. For these noncompact CalabiYau threefolds, Diaconescu, Donagi and Pantev discovered in 2007 that the associated CalabiYau integrable systems agree with the ADE type Hitchin integrable systems. In joint work with Beck and Donagi we show that these integrable systems allow `folding´ by automorphisms of the underlying ADE root systems, and we investigate the corresponding orbifoldings of CalabiYau threefolds.