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Eleny Ionel: Lectures
01/09/2020: Counting embedded curves in 3folds
January 9, 2020
TITLE: Counting embedded curves in 3folds
ABSTRACT: There are several ways of counting holomorphic curves in 3folds. Counting them as maps gives rise to the GromovWitten invariant, but in general, these are not integer counts due to the presence of multiple covers with symmetries. In joint work with Tom Parker we constructed an integer count of embedded curves in symplectic CalabiYau 3folds. It conjecturally satisfies a finiteness property. In this talk I outline some of the ideas behind this construction, as well as some of the new ingredients that enter in extending it to the Real setting (in the presence of an antisymplectic involution). The latter is based on joint work with Penka Georgieva.
Davesh Maulik: Lectures
01/08/2020: Stable pair invariants for CalabiYau 4folds
January 8, 2020
TITLE: Stable pair invariants for CalabiYau 4folds
ABSTRACT: In this talk, I’ll survey integrality conjectures for curvecounting on CY 4folds (following KlemmPandharipande) and a proposal (joint with Y. Cao and Y. Toda) for how to match them with sheaftheoretic invariants.
Tristan Rivière: Lectures
01/08/2020: L^{p}bounded curvatures in high dimensions
January 8, 2020
TITLE: L^{p}bounded curvatures in high dimensions
ABSTRACT: In this talk we shall be presenting some analysis aspects of gauge theory in high dimension. In the first part we will study the completion of the space of arbitrary smooth bundles and connections under L^{p}control of their curvature. We will first recall the classical theory in critical dimension (i.e. p=n/2) and then move to the supercritical dimension (i.e. p<n/2). In a second part we will explain how the previous analysis can be used to study the variations of YangMills lagrangian as well as the weak closure of smooth YangMills Fields in arbitrary dimension. If time permits, in the last part of the talk, we will study the space of weak unitary connections over complex manifolds with 11 curvatures.
Tony Pantev: Lectures
01/06/2020 and 01/07/2020: Constructing shifted symplectic structures
January 6, 2020 and January 7, 2020
TITLE: Constructing shifted symplectic structures
ABSTRACT: I will explain an important extension of Hamiltonian and QuasiHamiltonian reduction which uses derived geometry in an essential way. This extension has a lot of built in flexibility and provides a universal construction of many known and new symplectic structures in algebraic geometry. The generalized reduction construction relies on the notion of a relative shifted symplectic structure along the stalks of a constructible sheaf of derived stacks on a stratified space. I will introduce relative shifted symplectic forms and will describe a general pushforward construction, together with explicit techniques for computing such forms. As applications I will discuss a relative lift of recent results of ShendeTakeda on moduli of objects in topological Fukaya categories, and a universal construction of symplectic structures on derived moduli of Stokes data on smooth varieties. This is a joint work with Dima Arinkin and Bertrand Toën.
Slides of Lecture 1
Slides of Lecture 2
Penka Georgieva: Lectures
01/09/2020: Real GromovWitten theory
January 9, 2020
TITLE: Real GromovWitten theory
ABSTRACT: For a symplectic manifold with an antisymplectic involution one can consider Jholomorphic maps invariant under the involution. These maps give rise to real GromovWitten invariants and are related to real enumerative geometry in the same spirit as their more classical counterparts; in physics they are related to orientifold theories. In this talk I will give an overview of the developments in real GromovWitten theory and discuss some properties of the invariants.
Vivek Shende: Lectures
01/09/2020: Skeins on Branes
January 9, 2020
TITLE: Skeins on Branes
ABSTRACT: 30 years ago, Witten explained how the Jones polynomial and its relatives – at the time, the latest word in knot invariants – arise naturally from a certain quantum field theory. Ten years later, Ooguri and Vafa used string theory to argue that the same invariants should count holomorphic curves in a certain CalabiYau 3fold. In this talk, I will explain how to understand their proposal in mathematical terms, and sketch a proof that indeed, the coefficients of the HOMFLY polynomial count holomorphic curves, and, conversely, that the skein relations of knot theory are the key ingredient in defining invariant counts of higher genus holomorphic curves with boundary. This is joint work with Tobias Ekholm.
Pavel Safronov: Lectures
January 7, 2020
TITLE: Enumerative invariants from supersymmetric twists
ABSTRACT: I will recall the notion of supersymmetric twisting to obtain a topological field theory from a supersymmetric one. I will work through some examples in dimensions 7 and 8 to explain the appearance of the problem of counting Spin(7)instantons and G2monopoles as well as categorified and twicecategorified Donaldson—Thomas invariants.
Geometry and Analysis of Moduli Spaces, Imperial College, 610 January 2020
Arrival Date: Sunday January 5, 2020. Departure Date: Saturday January 11, 2020.
All talks will be at Huxley Bldg 140. [ click for map ]
Registration and coffee/tea are in the Huxley 5th floor common room.
Scientific Organizers: Dominic Joyce joyce@maths.ox.ac.uk, Aleksander Doan doan@math.columbia.edu.
Provisional Timetable
MON 6 JAN 
TUE 7 JAN 
WED 8 JAN 
THU 9 JAN 
FRI 10 JAN 

9:00 9:30 
REGISTRATION  
9:3010:30 
Joyce  Donaldson 2  Doan  Georgieva  Li 
10:3011:00 
Q&A / Discussion  Q&A / Discussion  Q&A / Discussion  Q&A / Discussion  Q&A / Discussion 
11:0011:30 
Coffee  Coffee  Coffee  Coffee  Coffee 
11:3012:30 
Thomas  Sun  Riviere  Ionel  He 
12:3014:00 
Lunch  Lunch  Lunch  Lunch  Lunch 
14:0015:00 
Pantev 1  Pantev 2  Maulik  Haydys  
15:0015:30 
Coffee  Coffee  Coffee  Coffee  
15:3016:30 
Donaldson 1  Safronov  Q&A / Discussion  Shende  
16:3017:00 
Q&A / Discussion  Q&A / Discussion  Q&A / Discussion  Q&A / Discussion  
19:0021:00 
Social Dinner 
Speakers (click on title to see abstract and lecture slides):
 Aleksander Doan (Columbia and Cambridge), On complexification and categorification
 Sir Simon Donaldson (Simons Center, Stony Brook and Imperial College London), Gauge theory and special holonomy I,II
 Penka Georgieva (Jussieu), Real GromovWitten theory
 Andriy Haydys (Freiburg), On the blow up set for the SeibergWitten equations with two spinors
 Siqi He (Simons Center, Stony Brook), The counting problem of the KapustinWitten equations
 Eleny Ionel (Stanford), Counting embedded curves in 3folds
 Dominic Joyce (Oxford), Shifted symplectic Derived Algebraic Geometry for dummies
 Yang Li (Institute for Advanced Study), High codimension phenomena for Hermitian YangMills connections
 Davesh Maulik (MIT), Stable pair invariants for CalabiYau 4folds
 Tony Pantev (University of Pennsylvania), Constructing shifted symplectic structures
 Tristan Rivière (ETH, Zürich), L^{p}bounded curvatures in high dimensions
 Pavel Safronov (Universität Zürich), Enumerative invariants from supersymmetric twists
 Vivek Shende (UC Berkeley), Skeins on Branes
 Song Sun (UC Berkeley), Singular HermitianYangMills connections and reflexive sheaves
 Richard Thomas (Imperial College London), BorisovJoyce in algebraic geometry
The workshop will have two main themes:
(A) The DonaldsonSegal programme for defining enumerative invariants of compact G2manifolds by counting G2 instantons, with correction terms from associative 3folds. Analysis of G2instanton moduli spaces, singularities of G2instantons. Analysis of SeibergWitten type equations on 3manifolds used to define correction terms in DonaldsonSegal programme. Related gauge theory problems, including singularities of HermitianYang Mills connections.
(B) The PantevToënVaquiéVezzosi theory of shifted symplectic derived algebraic geometry, giving geometric structures on CalabiYau moduli spaces, and its applications to generalizations of DonaldsonThomas theory of CalabiYau 3 and 4folds. One particular aim is to present the theory in a way accessible to String Theorists, to encourage communication between mathematicians and physicists on this subject, and to facilitate interpretation of the implications of the theory in String Theory terms. Algebrogeometric enumerative invariants (and Floer theories, etc), particularly of CalabiYau manifolds, related to the PTVV theory: DonaldsonThomas, GromovWitten, and VafaWitten invariants, DonaldsonThomas type invariants of CalabiYau 4folds, the GopakumarVafa conjecture.
Attendance at the workshop is by invitation only (except for London locals). Noninvitees wishing to attend may contact the organizers, but are likely to be disappointed, as we are already inviting as many as the lecture theatre will comfortably hold, so we can only include others if we have refusals.
Albrecht Klemm: Lectures
September 11, 2019
TITLE: CY 3folds over finite fields, Black hole attractors, and Dbrane masses
ABSTRACT: The integer coefficients of the numerator of the HasseWeil Zeta function for one parameter CalabiYau 3folds are expected to be Hecke eigenvalues of Siegel modular forms. For rigid CY 3folds as well as at conifold — and rank two attractor points this numerator contains factors of lower degree, which can be shown to be the Hecke eigenvalues of weight two or four of modular cusp forms of . We show that the Hecke Lfunction at integer arguments or more generally the periods of these modular forms give the brane masses as well as the value and the curvature of the WeilPeterssen metric at the points. The coefficients of the connection matrix from the integer symplectic basis to a Frobenius basis at the conifold are given by the quasi periods of these modular forms.
Cumrun Vafa: Lectures
September 11, 2019
TITLE: G2 Structure and Physical Interpretation of Taubes Construction of SW Invariants
ABSTRACT: In this talk I review a joint recent work with Sergio Cecotti, where we use the structure to shed light on Taubes reformulation of invariant for symplectic 4manifolds in terms of Gromov invariants.