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Thomas Walpuski: Lectures

March 17, 2023
TITLE: Progress and open problems around associative submanifolds

ABSTRACT: This is short (and biased) overview of some progress around associative submanifolds obtained during the collaboration and a discussion of a few (out of many) open problems.

Slides of Lecture

September 9 and 14, 2021
TITLE: The Gopakumar-Vafa finiteness conjecture

ABSTRACT: In 1998, using arguments from M–theory, Gopakumar and Vafa argued that there are integer BPS invariants of symplectic Calabi–Yau 3–folds. Unfortunately, they did not give a direct mathematical definition of their BPS invariants, but they predicted that they are related to the Gromov–Witten invariants by a transformation of the generating series. The Gopakumar–Vafa conjecture asserts that if one defines the BPS invariants indirectly through this procedure, then they satisfy an integrality and a (genus) finiteness condition.

The integrality conjecture has been resolved by Ionel and Parker. A key innovation of their proof is the introduction of the cluster formalism: an ingenious device to side-step questions regarding multiple covers and super-rigidity. Their argument could not resolve the finiteness conjecture, however. The reason for this is that it relies on Gromov’s compactness theorem for pseudo-holomorphic maps which requires an a priori genus bound. It turns out, however, that rather powerful tools from geometric measure theory imply a compactness theorem for pseudo-holomorphic cycles. This can be used to upgrade Ionel and Parker’s cluster formalism and prove both the integrality and finiteness conjecture.

This talk is based on joint work with Eleny Ionel and Aleksander Doan.

Slides of Lecture (9/9)


Slides of Lecture (9/14)

October 30, 2020
TITLE: Problems with counting G2-instantons and associatives, and generalized Seiberg–Witten equations

ABSTRACT: There is a naïve hope that counting G2–instantons (or associatives) might lead to invariants for G2–manifolds. However, numerous geometric degenerations threaten the invariance of such counts (pointed out by Donaldson–Segal, Joyce, Nordström, Doan–W.). Initially unrelated to this, generalized Seiberg–Witten equations are also plagued by severe non-compactness issues. A formal analysis suggests that the degeneration phenomena of the ADHM Seiberg–Witten equations and (some of) those encountered by G2–instantons and associatives have the same origin. Based on this observation, another (somewhat less?) naïve hope is that counting richer geometric objects involving G2–instantons, associatives, as well as solutions to the ADHM Seiberg–Witten equations might yields invariants for G2–manifolds after all. The purpose of this talk is to explain this in more detail and to report on some progress regarding the analysis of generalized Seiberg–Witten equations (joint with Aleksander Doan and Boyu Zhang).
Slides of Lecture

September 16, 2020
TITLE: More examples of G_2–instantons on twisted connected sums

ABSTRACT: Quite some time ago, Sá Earp and I proved a gluing theorem to
construct G_2–instantons on twisted connected sums. Embarrassingly,
there are only two(!) examples of G_2–instantons in the literature. I
constructed the first example on a TCS discovered by Crowley–Nordström
using the “rigid case” of the gluing theorem. The second example is due
to Menet–Sá Earp–Nordström and uses the “non-rigid case”.

It turns out that the input required for the “rigid case” of the gluing
theorem is guaranteed to exist provided certain mostly
arithmetic/lattice-theoretic conditions are met. For TCSs arising from
pairs of Fano 3-folds of Picard rank two there is only one solution to
the arithmetic problem. If one allows for Fano 3-folds with higher
Picard rank, then there are numerous solutions. So far, by a brute-force
computer search, I have found 298 new examples of G_2–instantons; in
particular, the first examples of irreducible, unobstructed
G_2–instantons on PU(r)–bundles for r ≠ 2.

I believe that similar ideas can also help with the construction of
G_2–instantons via the “non-rigid case” of the gluing theorem. Sadly, I
haven’t quite been able to make this work yet. If time permits, I will
explain what I have tried and where I have failed.

Slides of Lecture

September 9, 2018
TITLE: Super-rigidity and Castelnuovo’s bound.

ABSTRACT: Castelnuovo’s bound is a very classical result in algebraic geometry. It asserts a sharp bound on the genus of a curve of degree d in n-dimensional projective space. It is an interesting question to ask whether analogues of Castelnuovo’s bound hold in almost complex geometry. There is a direct analogue in dimension four. In dimension at least eight genus bounds can be established for generic almost complex structures. These results leave open the case of dimension six.

Bryan and Panharipande introduced the notion of super-rigidity of an almost complex structure. They also speculated that this condition might hold for a generic almost complex structure (compatible with a fixed symplectic structure). It had been believed for a long time that super-rigidity will play an important role in the proof of the Gopakumar–Vafa conjecture. However, it turned that Ionel and Parker’s recent proof of this conjecture did not make use of it. Nevertheless, super-rigidity has important consequences. I will present one of these consequences, namely, a genus bound for index zero pseudo-holomorphic curves. This is joint work with Aleksander Doan and, heavily, relies on work by De Lellis, Spadaro, and Spolaor and ideas of Taubes.

There has been a lot of progress towards establishing Bryan and Pandharipande’s super-rigidity conjecture in the work of Wendl. In fact, based on his ideas, Aleksander Doan and I have developed an abstract framework for equivariant transversality/Brill–Noether type questions. Wendl’s work shows that the super-rigidity conjecture holds provided generic real Cauchy-Riemann operators satisfy an easy to state analytic condition. I will explain what this condition means and discuss a few cases in which this condition (or versions of it hold).

This is joint work with Aleksander Doan.

January 9, 2018
TITLE: On counting associative submanifolds and Seiberg–Witten monopoles

ABSTRACT: From Nordström and Joyce we learn that counting associatives cannot lead to an invariant of G_2-manifolds. Classical surgery formulae for the Seiberg-Witten invariant SW(Y) of a 3-manifold Y show that the count of associative submanifolds weighted by SW is invariant under the transitions described Nordström and Joyce. Unfortunately, the Seiberg-Witten invariant SW(Y) is only defined if b_1(Y) > 1. In this talk I will explain how one might replace SW by a version of Kronheimer-Mrowka’s monopole homology and construct a Floer homology group which has a chance of being both defined and invariant. This is joint work with Aleksander Doan.

January 8, 2018
TITLE: Introduction to gauge theory on Riemannian manifolds with special holonomy

ABSTRACT: Riemannian manifolds with special holonomy typically allow for instanton-type equations implying the the Yang-Mills equation. I will recall the most important examples of these equations; discuss foundational results and examples; and finally mention a number of questions.

September 13, 2017
TITLE: The (1,k)-ADHM Seiberg­-Witten equation and k­-fold covers of associatives

ABSTRACT: The (1,k) ADHM Seiberg–Witten equations are a class of generalized Seiberg–Witten equations associated with the hyperkähler quotient appearing in the Atiyah, Drinfeld, Hitchin, and Manin’s construction of the framed moduli space of ASD instantons on R4. Formally, degenerating solutions of this equation are related to Fueter sections of bundles of symmetric products of k copies of R4. In this talk I will explain this relation in more detail and discuss why we believe these equations to be relevant to issues of multiply covered associatives. This is joint work in progress with Aleksander Doan.

September 7, 2016
TITLE: From G2 gauge theory to the ADHM Seiberg–Witten equation

ABSTRACT: The compactness question for G_2–instantons naturally leads one to consider the local problem around an associative submanifold. I will explain how, in a neighborhood of an associative submanifold, the G_2–instanton equation can be understood as a Seiberg–Witten type equation. A result of Haydys then motivates introducing a second Seiberg–Witten type equation associated with the ADHM construction. I will end the talk by discussing a compactness theorem for the simplest of the ADHM Seiberg–Witten equations. This is joint work with Andriy Haydys.