Home » Lectures
Category Archives: Lectures
01/09/2020: Counting embedded curves in 3-folds
January 9, 2020
TITLE: Counting embedded curves in 3-folds
ABSTRACT: There are several ways of counting holomorphic curves in 3-folds. Counting them as maps gives rise to the Gromov-Witten 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 Calabi-Yau 3-folds. 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 anti-symplectic involution). The latter is based on joint work with Penka Georgieva.
01/08/2020: Stable pair invariants for Calabi-Yau 4-folds
January 8, 2020
TITLE: Stable pair invariants for Calabi-Yau 4-folds
ABSTRACT: In this talk, I’ll survey integrality conjectures for curve-counting on CY 4-folds (following Klemm-Pandharipande) and a proposal (joint with Y. Cao and Y. Toda) for how to match them with sheaf-theoretic invariants.
01/08/2020: Lp-bounded curvatures in high dimensions
January 8, 2020
TITLE: Lp-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 Lp-control of their curvature. We will first recall the classical theory in critical dimension (i.e. p=n/2) and then move to the super-critical 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 Yang-Mills lagrangian as well as the weak closure of smooth Yang-Mills 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 1-1 curvatures.
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 Quasi-Hamiltonian 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 Shende-Takeda 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
01/09/2020: Real Gromov-Witten theory
January 9, 2020
TITLE: Real Gromov-Witten theory
ABSTRACT: For a symplectic manifold with an anti-symplectic involution one can consider J-holomorphic maps invariant under the involution. These maps give rise to real Gromov-Witten 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 Gromov-Witten theory and discuss some properties of the invariants.
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 Calabi-Yau 3-fold. 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.
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 G2-monopoles as well as categorified and twice-categorified Donaldson—Thomas invariants.
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.
MON 6 JAN
TUE 7 JAN
WED 8 JAN
THU 9 JAN
FRI 10 JAN
|Q&A / Discussion||Q&A / Discussion||Q&A / Discussion||Q&A / Discussion||Q&A / Discussion|
|Pantev 1||Pantev 2||Maulik||Haydys|
|Donaldson 1||Safronov||Q&A / Discussion||Shende|
|Q&A / Discussion||Q&A / Discussion||Q&A / Discussion||Q&A / Discussion|
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 Gromov-Witten theory
- Andriy Haydys (Freiburg), On the blow up set for the Seiberg-Witten equations with two spinors
- Siqi He (Simons Center, Stony Brook), The counting problem of the Kapustin-Witten equations
- Eleny Ionel (Stanford), Counting embedded curves in 3-folds
- Dominic Joyce (Oxford), Shifted symplectic Derived Algebraic Geometry for dummies
- Yang Li (Institute for Advanced Study), High codimension phenomena for Hermitian Yang-Mills connections
- Davesh Maulik (MIT), Stable pair invariants for Calabi-Yau 4-folds
- Tony Pantev (University of Pennsylvania), Constructing shifted symplectic structures
- Tristan Rivière (ETH, Zürich), Lp-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 Hermitian-Yang-Mills connections and reflexive sheaves
- Richard Thomas (Imperial College London), Borisov-Joyce in algebraic geometry
The workshop will have two main themes:
(A) The Donaldson-Segal programme for defining enumerative invariants of compact G2-manifolds by counting G2 instantons, with correction terms from associative 3-folds. Analysis of G2-instanton moduli spaces, singularities of G2-instantons. Analysis of Seiberg-Witten type equations on 3-manifolds used to define correction terms in Donaldson-Segal programme. Related gauge theory problems, including singularities of Hermitian-Yang Mills connections.
(B) The Pantev-Toën-Vaquié-Vezzosi theory of shifted symplectic derived algebraic geometry, giving geometric structures on Calabi-Yau moduli spaces, and its applications to generalizations of Donaldson-Thomas theory of Calabi-Yau 3- and 4-folds. 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. Algebro-geometric enumerative invariants (and Floer theories, etc), particularly of Calabi-Yau manifolds, related to the PTVV theory: Donaldson-Thomas, Gromov-Witten, and Vafa-Witten invariants, Donaldson-Thomas type invariants of Calabi-Yau 4-folds, the Gopakumar-Vafa conjecture.
Attendance at the workshop is by invitation only (except for London locals). Non-invitees 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.
September 11, 2019
TITLE: CY 3-folds over finite fields, Black hole attractors, and D-brane masses
ABSTRACT: The integer coefficients of the numerator of the Hasse-Weil Zeta function for one parameter Calabi-Yau 3-folds are expected to be Hecke eigenvalues of Siegel modular forms. For rigid CY 3-folds 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 L-function 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 Weil-Peterssen 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.
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 4-manifolds in terms of Gromov invariants.