09/13/2018: A Ringel-Hall type construction of vertex algebras

06/05/2018: On Mirror Symmetry, Fukaya categories, and Bridgeland stability, with a view towards Lagrangian Mean Curvature Flow

01/09/2018: Lie brackets on the homology of moduli spaces, and wall-crossing formulae

09/14/2017: Conjectures on counting associative 3-folds in G₂ manifolds

06/06/2017: Constructing compact 8-manifolds with holonomy Spin(7)

06/05/2017: Constructing compact 7-manifolds with holonomy G_{2}

01/11/2017: Counting problems for G_{2} manifolds

09/07/2016: Derived differential geometry and moduli spaces in differential geometry

### September 13, 2018

TITLE: A Ringel-Hall type construction of vertex algebras

ABSTRACT: `Vertex algebras’ are complicated algebraic structures coming from physics, which arise in 2D conformal field theory and string theory, and also play an important role in mathematics, in areas such as monstrous moonshine and geometric Langlands. I will explain a new geometric construction of vertex algebras, which seems to be unknown. I discovered it by accident, while working on wall-crossing formulae for Donaldson-Thomas type invariants of Calabi-Yau 4-folds. The construction applies in many situations in algebraic geometry, differential geometry and representation theory, and produces vast numbers of new examples. It is also easy to generalize the construction in several ways to produce different types of vertex algebra, quantum vertex algebras and representations of vertex algebras. The construction seems to be closely related to, and is maybe the “correct” explanation for, a large body of work started by Grojnowski, Nakajima and others, which produces representations of interesting infinite-dimensional Lie algebras on the homology of moduli schemes, such as Hilbert schemes. Suppose A is a nice abelian category (such as coherent sheaves coh (X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as Dbcoh (X) or Dbmod − CQ ) over C . Let M be the moduli stack of objects in A or T, as an Artin stack or higher stack. Consider the homology H*(M) over some ring R. Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H* (Mpl) of a projective linear version Mpl of the moduli stack M. For example, if we take T = Dbmod − CQ, the vertex algebra H*(M) is the lattice vertex algebra attached to the dimension vector lattice Z Q₀ of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra. There is also a differential-geometric version: if X is a compact manifold equipped with an elliptic complex E (such as the de Rham complex or the Dirac operator), and M is the moduli stack (as a topological stack) of either all unitary connections on complex vector bundles on X , or all unitary connections on X satisfying a curvature condition depending on E (e.g., instantons on 4-manifolds, Hermitian-Einstein connections on Kahler manifolds, G₂-instantons or Spin(7)-instantons), then we can define a vertex algebra structure on H*(M). This should be part of the big picture into which other work in this collaboration on G₂-instantons and the Donaldson-Segal program fits. There must be a physical explanation for these vertex algebras, but so far, string theorists have not been able to give me one.

### June 5, 2018

TITLE: On Mirror Symmetry, Fukaya categories, and Bridgeland stability, with a view towards Lagrangian Mean Curvature Flow

ABSTRACT: There will be no new research ideas in this talk, I will just cover well known background material. So experts (and indeed, the rest of the audience) are advised to go see the dinosaurs in the Natural History Museum instead.

My intended target audience is people who work on geometric flows, particularly (Lagrangian) Mean Curvature Flow, but may not be very familiar with areas of symplectic geometry around Homological Mirror Symmetry, Fukaya categories, etc.

I will briefly review the “big picture” of Homological Mirror Symmetry for Calabi-Yau manifolds M, M*, the derived Fukaya category D^{b}F(M) of Lagrangians L in a Calabi-Yau manifold M, and how it is expected that there exists a Bridgeland stability condition on D^{b}F(M) whose semistable objects are represented by special Lagrangian submanifolds L; also the role of obstructions to Lagrangian Floer cohomology, and how they determine whether a Lagrangian appears as an object in D^{b}F(M).

There is a conjectural framework known as the “Thomas-Yau Conjecture” for how Lagrangian MCF fits into all this, quite similar to Ricci flow in the Poincare Conjecture and geometrization of 3-manifolds. The Thomas-Yau conjecture has nothing at all to do with Yau, and consists of what Richard Thomas really meant to say, although he actually said something different. I’ll describe a personal view of the Thomas-Yau Conjecture.

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### January 9, 2018

TITLE: Lie brackets on the homology of moduli spaces, and wall-crossing formulae

ABSTRACT: Let be a field, and be the “projective linear” moduli stack of objects in a suitable -linear abelian category (such as the coherent sheaves on a smooth projective ) or triangulated category (such as the derived category ). I will explain how to define a Lie bracket [ , ] on the homology (with a nonstandard grading), making into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra.

There is also a differential-geometric version of this: if is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, manifold, manifold) then we can define Lie brackets both on the homology of the moduli spaces of all or connections on for all , and on the homology of the moduli spaces of instanton or connections on for all .

All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition.

Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with ), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for CY 3-folds, Fano 3-folds, and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear” moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs.

I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra (, [ , ]).

### September 14, 2017

TITLE: Conjectures on counting associative 3-folds in G₂ manifolds

ABSTRACT: Riemannian 7-manifolds with holonomy G₂ are a special class of Ricci-flat Riemannian manifolds, which are of interest to physicists working in M-theory. Associative 3-folds are calibrated 3-submanifolds in 7-manifolds with holonomy G₂, so they are a special kind of minimal submanifold.

There is a well-known analogy between G₂ manifolds X in dimension 7 and Calabi-Yau 3-folds Y in dimension 6. Under this analogy one should compare associative 3-folds in X with J-holomorphic curves in Y. Much of symplectic geometry — Gromov-Witten theory, Lagrangian Floer theory, and so on — is concerned with “counting” J-holomorphic curves, to get an answer which is independent of the (almost) complex structure J up to deformation. So we can ask: might there be interesting geometry of G₂ manifolds concerned with “counting” associative 3-folds, which gives an answer unchanged under deformations of the G₂ structure?

This talk, based on arXiv:1610.09836, presents a conjectural answer to this question. It is connected to conjectures of Donaldson and Segal on defining invariants by “counting” G₂ instantons on X with “compensation terms” counting pairs of a G₂ instanton and an associative 3-fold on X. At the end we will briefly discuss a proposed modification to the Donaldson-Segal conjecture, to correct for wall-crossing behaviour of associative 3-folds we discover during our investigation.

### June 6, 2017

TITLE: Constructing compact 8-manifolds with holonomy Spin(7)

### June 5, 2017

TITLE: Constructing compact 7-manifolds with holonomy G_{2}

### January 11, 2017

TITLE: Counting problems for G_{2} manifolds

ABSTRACT: This talk(s) reviews my recent preprint arXiv:1610.0983.

Many important areas of geometry involve “counting” some kind of geometric object to define an “invariant” which is then shown to be unchanged under deformations of the base geometry. Examples include Donaldson invariants counting instantons on a compact oriented Riemannian 4-manifold (X,g) with b^2_+(X)>1 (which are unchanged under deformations of g), Gromov-Witten invariants of complex algebraic / symplectic manifolds (unchanged under deformations of the (almost) complex structure J), and so on.

Floer cohomology theories and Fukaya categories are similar: they involve “counting” objects such as J-holomorphic curves with boundary, and while the numbers are not deformation-invariant, one uses the numbers and some homological algebra to construct things which are deformation-invariant.

I will discuss whether there may be interesting “invariant” theories for compact -manifolds which are unchanged under deformations of the -structure (perhaps deformations preserving closure of 3- or 4-form), as considered by Donaldson and Segal in their 2009 paper “Gauge theory in higher dimensions II”.

The three obvious classes of objects in a -manifold one could try to “count” are associative 3-folds, coassociative 4-folds, and -instantons. I haven’t much to say about coassociatives.

For associative 3-folds, I argue that one cannot define deformation-invariant Gromov-Witten style invariants counting associatives, as there are singular behaviours of associatives which would change the numbers under deformation of the -structure. However, I suggest that there may still be interesting deformation-invariant information encoded in “numbers” of associatives, and I outline how one might define a supercommutative “ quantum cohomology algebra”, similar to quantum cohomology in symplectic geometry, but with features of a Floer theory.

On the way to this conclusion, I introduce some new ideas about associative 3-folds, including a way to orient moduli spaces of associatives, and make some conjectures on their singular behaviour.

For instantons, Donaldson and Segal proposed to define invariants, similar to Donaldson invariants of 4-manifolds, which “count” -instantons on the -manifold (X,). It is known that -instantons can “bubble” on an associative 3-fold under deformation of , and this would change the numbers of -instantons. So to make their invariants unchanged under deformation, they proposed to add “compensation terms” C(N,(P,A)) which count pairs of an associative N in X and a -instanton (P,A) on X with some weight, which would have the property that C(N,(P,A)) jumps by 1 when a -instanton bubbles on N leaving (P,A) after removal of singularities, so that the compensated sum remains unchanged. The precise definition of the weight C(N,(P,A)) has remained mysterious, though Walpuski and Haydys are working towards a definition.

I argue that for -instantons with group SU(2), it is not possible to define “compensation terms” C(N,(P,A)) with the required properties (for any conceivable definition, not just for current definitions), so that I do not believe that the Donaldson-Segal programme as currently formulated will succeed. The argument why not is actually a fairly simple thought-experiment using material from Donaldson and Segal’s paper, but it is based on the ideas about orienting associative moduli spaces mentioned earlier.

I won’t be able to say all this in an hour. So what I’ll probably actually do is give an 1-hour overview in three parts: 1. Introduction; 2. Counting associatives; 3. Counting -instantons; and then we can continue afterwards according to demand from those audience members who have not already run away screaming, if this is a nonempty set.

### September 7, 2016

TITLE: Derived differential geometry and moduli spaces in differential geometry

ABSTRACT: Derived Differential Geometry (DDG) is the study of “derived manifolds” and “derived orbifolds”, where “derived” is in the sense of the Derived Algebraic Geometry of Jacob Lurie and Toen-Vezzosi. They include ordinary manifolds and orbifolds, but also many spaces which are singular at the classical level. There are several approaches to DDG, due to Spivak, Borisov-Noel and myself, all more-or-less equivalent. The “Kuranishi spaces” studied in symplectic geometry by Fukaya-Oh-Ohta-Ono are a prototype notion of derived orbifold.

I claim that moduli spaces M of solutions of a nonlinear elliptic p.d.e. on a compact manifold should naturally have the structure of a derived manifold (if solutions have no symmetries) or a derived orbifold (if solutions have finite symmetry groups, e.g. Deligne-Mumford stable J-holomorphic curves). This includes many very interesting problems — instantons on 4-manifolds and other gauge-theoretic moduli problems, J-holomorphic curves in symplectic geometry and so on. In particular, it includes moduli spaces used to define enumerative invariants (Donaldson invariants, Gromov-Witten invariants, etc.). This is because compact, oriented derived manifolds and derived orbifolds have virtual cycles in homology, and these virtual cycles may be used to define the invariants.

I also claim that many natural (partial) compactifications of such moduli spaces M, e.g. by including J-holomorphic curves with nodes, should naturally have the structure of derived manifolds or derived orbifolds with corners, where the boundary represents the extra singular solutions.

I will outline a method to prove the existence of natural derived manifold and derived orbifold structures on differential geometric moduli spaces by a method of “universal families”, based on Grothendieck’s representable functors in algebraic geometry. That is, given a moduli problem, we define a notion of family F of solutions over a base derived manifold or orbifold S. A “universal family” is a family with a universal property w.r.t. all other families. If a universal family exists (I claim it should, under reasonable conditions) it is unique up to equivalence, and the base M of the family is the moduli space, with a derived manifold/orbifold structure.