Past Geometry and Analysis Seminar

Let $\Gamma$ be a finite subgroup of $SL(2, \mathbb{C})$. We can attach several different moduli spaces to the action of $\Gamma$ on $\mathbb{C}^2$, and we show how Nakajima's quiver varieties provide constructions of them. The definition of such a quiver variety depends on a stability parameter, and we are especially interested in what happens when this parameter moves into a specific ray in its associated wall-and-chamber structure. Some of the resulting quiver varieties can be understood as moduli spaces of certain framed sheaves on an appropriate stacky compactification of the Kleinian singularity $\mathbb{C}^2/\Gamma$. As a special case, this includes the punctual Hilbert schemes of $\mathbb{C}^2/\Gamma$.

Much of this is joint work with A. Craw, Á. Gyenge, and B. Szendrői.

Examples of 2CY categories include the category of coherent sheaves on a K3 surface, the category of Higgs bundles, and the category of modules over preprojective algebras or fundamental group algebras of compact Riemann surfaces.  Let p:M->N be the morphism from the stack of semistable objects in a 2CY category to the coarse moduli space.  I'll explain, using cohomological DT theory, formality in 2CY categories, and structure theorems for good moduli stacks, how to prove a version of the BBDG decomposition theorem for the exceptional direct image of the constant sheaf along p, even though none of the usual conditions for the decomposition theorem apply: p isn't projective or representable, M isn't smooth, the constant mixed Hodge module complex Q_M isn't pure...  As an application, I'll explain how this allows us to extend nonabelian Hodge theory to Betti/Dolbeault stacks.

Donaldson-Thomas (DT) theory is an enumerative theory which produces a virtual count of stable coherent sheaves on a Calabi-Yau 3-fold. Motivic Donaldson-Thomas theory, originally introduced by Kontsevich-Soibelman, is a categorification of the DT theory. This categorification contains more refined information of the moduli space. In this talk, I will explain the role of d-critical locus structure in the definition of motivic DT invariant, following the definition by Bussi-Joyce-Meinhardt. I will also discuss results on this structure on the Hilbert schemes of zero dimensional subschemes on local toric Calabi-Yau threefolds. This is based on joint works with Sheldon Katz. The results have substantial overlap with recent work by Ricolfi-Savvas, but techniques used here are different. 

In 1989, Bryant and Salamon constructed the first Riemannian manifolds with holonomy group $\Spin(7)$. Since a crucial aspect in the study of manifolds with exceptional holonomy regards fibrations through calibrated submanifolds, it is natural to consider such objects on the Bryant-Salamon manifolds.

In this talk, I will describe the construction and the geometry of (possibly singular) Cayley fibrations on each Bryant-Salamon manifold. These will arise from a natural family of structure-preserving $\SU(2)$ actions. The fibres will provide new examples of Cayley submanifolds.

TBA

The homogeneous coordinate ring of a projective variety may be constructed by geometrically quantizing the multiples of a symplectic form, using the complex structure as a polarization. In this talk, I will explain how a holomorphic Poisson structure allows us to deform the complex polarization into a generalized complex structure, leading to a non-commutative deformation of the homogeneous coordinate ring. The main tool is a conjectural construction of a category of generalized complex branes, which makes use of the A-model of an associated symplectic groupoid. I will explain this in the example of toric Poisson varieties. This is joint work with Marco Gualtieri (arXiv:2108.01658).

The Hull--Strominger system is a system of non-linear PDEs on heterotic string theory involving a pair of Hermitian metrics $(g,h)$ on a six dimensional manifold $M$. One of these equations dictates the metric $g$ on $M$ to be conformally balanced. We will begin the talk by giving a description of the geometry of cohomogeneity one manifolds and SU(3)-structures. Then, we will look for solutions to the Hull--Strominger system in the cohomogeneity one setting. We show that a six-dimensional simply connected cohomogeneity one manifold under the almost effective action of a connected Lie group $G$ admits no $G$-invariant balanced non-Kähler SU(3)-structures. This is a joint work with F. Salvatore.

Cohomological Hall algebras and vertex algebras are two structures whose origins are (at least in part) from physics. I will explain what these objects are, how the latter was related to moduli stacks by Joyce, and a theorem relating these two structures. The main tool is torus localisation, a method for "turning geometry into combinatorics", or rather a new formulation of it which works in the singular setting.

Dolbeault hypertoric manifolds are hyperkahler integrable systems generalizing the Ooguri-Vafa space. They approximate the Hitchin fibration near a totally degenerate nodal spectral curve. On the other hand, Betti hypertoric varieties are smooth affine varieties parametrizing microlocal sheaves on the same nodal spectral curve. I will review joint work with Zsuzsanna Dansco and Vivek Shende (arXiv:1910.00979) which constructs a diffeomorphism between the Dolbeault and Betti hypertorics, and proves that it intertwines the perverse and weight filtrations on their cohomologies. I will describe our main tool : deletion-contraction sequences arising from either smoothing a node of the spectral curve or separating its branches. I will also discuss some more recent developments and open questions.

A field extension $F/K$ in characteristic $p$ is purely inseparable if for each $x$ in $F$, some power $x^{p^n}$ belongs to $K$. Using methods from homotopy theory, we construct a Galois correspondence for finite purely inseparable field extensions $F/K$, generalising a classical result of Jacobson for extensions of exponent one (where $x^p$ belongs to $K$ for all $x$ in $F$). This is joint work with Waldron.

TBA

It is expected that complete noncompact Calabi-Yau manifolds are in some sense governed by their asymptotics at infinity. In the maximal volume growth case, the asymptotics at infinity are given by Calabi-Yau cones. We are interested in deformations of such metrics that fix the asymptotic cones at infinity. In the asymptotically conical case, Conlon-Hein proved uniqueness under such deformations. Their method is based on the corresponding linearized problem, namely the study of subquadratic harmonic functions. We generalize their work to the maximal volume growth case, allowing the tangent cones at infinity to have non-isolated singularities. Part of the talk is based on work in progress joint with Gabor Szekelyhidi.

A central theme of complex geometry is the relationship between differential-geometric PDEs and algebro-geometric notions of stability. Examples include Hermitian Yang-Mills connections and Kähler-Einstein metrics on the PDE side, and slope stability and K-stability on the algebro-geometric side. I will describe a general framework associating geometric PDEs on complex manifolds to notions of stability, and will sketch a proof showing that existence of solutions is equivalent to stability in a model case. The framework can be seen as an analogue in the setting of varieties of Bridgeland's stability conditions on triangulated categories.

In the mean curvature flow, translating solutions are an important model for singularity formation. In this talk, I will describe the asymptotic structure of 2D mean curvature flow translators embedded in R^3 which have finite total curvature, which turns out to be highly rigid. I will outline the proof of this asymptotic description, in particular focusing on some novel and unexpected features of this proof.

Fix a Calabi-Yau 3-fold X. Its DT invariants count stable bundles and sheaves on X. The generalised DT invariants of Joyce-Song count semistable bundles and sheaves on X. I will describe work with Soheyla Feyzbakhsh showing these generalised DT invariants in any rank r can be written in terms of rank 1 invariants. By the MNOP conjecture the latter are determined by the GW invariants of X.
Along the way we also show they are determined by rank 0 invariants counting sheaves supported on surfaces in X. These invariants are predicted by S-duality to be governed by (vector-valued, mock) modular forms.

Free boundary minimal surfaces are a notoriously elusive object in geometric analysis. From 2011, Fraser and Schoen's research program found a relationship between free boundary minimal surfaces in unit balls and metrics which maximise the first nontrivial Steklov eigenvalue. In this talk, I will explain how we can adapt homogenisation theory, a branch of applied mathematics, to a geometric setting in order to obtain surfaces with first Steklov eigenvalue as large as possible, and how it leads to the existence of free boundary minimal surfaces which were previously thought not to exist.

I will describe joint work with Abouzaid which constructs a stable homotopy theory refinement of Floer homology that has coefficients in the Morava K-theory spectra. The classifying spaces of finite groups satisfy Poincare duality for the Morava K-theories, which allows us to use this version of Floer homology to produce virtual fundamental chains for moduli spaces of Floer trajectories. As an application, we prove the Arnold conjecture for ordinary cohomology with coefficients in finite fields.

A celebrated result in geometry is the Kobayashi-Hitchin correspondence, which states that a holomorphic vector bundle on a compact Kähler manifold admits a Hermite-Einstein metric if and only if the bundle is slope polystable. Recently, Dervan and Sektnan have conjectured an analogue of this correspondence for fibrations whose fibres are compact Kähler manifolds admitting Kähler metrics of constant scalar curvature. Their conjecture is that such a fibration is polystable in a suitable sense, if and only if it admits an optimal symplectic connection. In this talk, I will provide an introduction to this theory, and describe my recent work on the conjecture. Namely, I show that existence of an optimal symplectic connection implies polystability with respect to a large class of fibration degenerations. The techniques used involve analysing geodesics in the space of relatively Kähler metrics of fibrewise constant scalar curvature, and convexity of the log-norm functional in this setting. This is work for my PhD thesis, supervised by Frances Kirwan and Ruadhaí Dervan.

We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in ℙ4.

I briefly give an overview on the key ML frameworks involved in this analysis (neural networks, auto-differentiation). This talk is mainly based on 2012.04656.

(joint with Indranil Biswas, Jacques Hurtubise, Sean Lawton, arXiv:2104.05589)

Let $G$ be either a compact Lie group or a reductive Lie group. Let $\pi$ be the fundamental group of a 2-manifold (possibly with boundary).
We can define a character variety by ${\rm Hom}(\pi, G)/G$, where $G$ acts by conjugation.

We explore the mappings between character varieties that are induced  by mappings between surfaces. It is shown that these mappings are generally Poisson.

In some cases, we explicitly calculate the Poisson bi-vector.

We give a brief introduction to Floer homotopy, from the Seiberg-Witten point of view.  We will then discuss Manolescu's version of finite-dimensional approximation for rational homology spheres.  We prove that a version of finite-dimensional approximation for the Seiberg-Witten equations associates equivariant spectra to a large class of three-manifolds.  In the process we will also associate, to a cobordism of three-manifolds, a map between spectra.  We give some applications to intersection forms of four-manifolds with boundary. This is joint work with Hirofumi Sasahira. 

Counting coherent sheaves on Calabi--Yau fourfolds is a subject in its infancy. An evidence of this is given by how little is known about perhaps the simplest case - counting ideal sheaves of length $n$. On the other hand, the parallel story for surfaces while with many open questions has seen many new results, especially in the direction of understanding virtual integrals over Quot-schemes. Motivated by the conjectures of Cao--Kool and Nekrasov, we study virtual integrals over Hilbert schemes of points of top Chern classes $c_n(L^{[n]})$ and their K-theoretic refinements. Unlike lower-dimensional sheaf-counting theories, one also needs to pay attention to orientations. In this, we rely on the conjectural wall-crossing framework of Joyce. The same methods can be used for Quot-schemes of surfaces and we obtain a generalization of the work of Arbesfeld--Johnson--Lim--Oprea--Pandharipande for a trivial curve class. As a result, there is a correspondence between invariants for surfaces and fourfolds in terms of a universal transformation.

The Atiyah-Floer conjecture asserts the instanton Floer homology of a closed three-manifold (constructed via gauge theory) is isomorphic to the Lagrangian Floer homology of a pair of Lagrangian submanifolds associated to a splitting of the three manifold (constructed via symplectic geometry). This conjecture has remained open for more than three decades. In this talk I will explain two compactness results for the SO(3) case of the conjecture in the neck-stretching process. One result is related to the construction of a natural bounding chain in the Lagrangian Floer theory and a conjecture of Fukaya.

I'll be presenting my PhD work, in which I define two new algebraic structures on the equivariant symplectic cohomology of a convex symplectic manifold. The first is a collection of shift operators which generalise the shift operators on equivariant quantum cohomology in algebraic geometry. That is, given a Hamiltonian action of the torus T, we assign to a cocharacter of T an endomorphism of (S1 × T)-equivariant Floer cohomology based on the equivariant Floer Seidel map. The second is a connection which is a multivariate version of Seidel’s q-connection on S1 -equivariant Floer cohomology and generalises the Dubrovin connection on equivariant quantum cohomology.

Grothendieck's Quot schemes — moduli spaces of quotient sheaves — are fundamental objects in algebraic geometry, but we know very little about them. This talk will focus on a relatively simple special case: the Quot scheme Quotˡ(E) of length l quotients of a vector bundle E of rank r on a smooth surface S. The scheme Quotˡ(E) is a cross of the Hilbert scheme of points of S (E=O) and the projectivisation of E (l=1); it carries a virtual fundamental class, and if l and r are at least 2, then Quotˡ(E) is singular. I will explain how the ADHM description of Quotˡ(E) provides a conjectural description of the singularities, and show how they can be resolved in the l=2 case. Furthermore, I will describe the relation between Quotˡ(E) and Quotˡ of a quotient of E, prove a functoriality result for the virtual fundamental class, and use it to compute certain tautological integrals over Quotˡ(E).

The torus T of projective space also acts on the Hilbert
scheme of subschemes of projective space, and the T-graph of the
Hilbert scheme has vertices the fixed points of this action, and edges
the closures of one-dimensional orbits. In general this graph depends
on the underlying field. I will discuss joint work with Rob
Silversmith, in which we construct of a subgraph, which we call the
spine, of the T-graph of Hilb^N(A^2) that is independent of the choice
of field. The key technique is an understanding of the tropical ideal,
in the sense of tropical scheme theory, of the ideal of the universal
family of an edge in the spine.

We prove a homological mirror symmetry result for a one-parameter family of genus 2 curves (https://arxiv.org/abs/1908.04227), and then mention current joint work with H. Azam, H. Lee, and C.-C. M. Liu on generalizing this to the 6-parameter family of all genus 2 curves.

First we describe the B-model genus 2 curve in a 4-torus and the geometric construction of the generalized SYZ mirror. Then we set up the Fukaya-Seidel category on the mirror. Finally we will see the main algebraic HMS result on homogenous coordinate rings, which is at the level of cohomology. The method involves first considering mirror symmetry for the 4-torus, then restricting to the hypersurface genus 2 curve and extending to a mirror Landau-Ginzburg model with fiber the mirror 4-torus. 

Unlike for closed manifolds, the existence of positive scalar curvature (psc) metrics on connected manifolds with
nonempty boundary is unobstructed. We study and compare the spaces of psc metrics on such manifolds with various
conditions along the boundary: H ≥ 0, H = 0, H > 0, II = 0, doubling, product structure. Here H stands for the
mean curvature of the boundary and II for its second fundamental form. "Doubling" means that the doubled metric
on the doubled manifold (along the boundary) is smooth and "product structure" means that near the boundary the
metric has product form. We show that many, but not all of the obvious inclusions are weak homotopy equivalences.
In particular, we will see that if the manifold carries a psc metric with H ≥ 0, then it also carries one which is
doubling but not necessarily one which has product structure. This is joint work with Bernhard Hanke.

The idea of assigning weights to local coordinate functions is used in many areas of mathematics, such as singularity theory, microlocal analysis, sub-Riemannian geometry, or the theory of hypo-elliptic operators, under various terminologies. In this talk, I will describe some differential-geometric aspects of weightings along submanifolds. This includes a coordinate-free definition, and the construction of weighted normal bundles and weighted blow-ups. As an application, I will describe a canonical local model for isotropic embeddings in symplectic manifolds. (Based on joint work with Yiannis Loizides.)

In the talk I shall discuss an approach to the localisation technique, for spaces satisfying the curvature-dimension condition, by means of L1-optimal transport. Moreover, I shall present recent work on a generalisation of the technique to multiple constraints setting. Applications of the theory lie in functional and geometric inequalities, e.g. in the Lévy-Gromov isoperimetric inequality.

Geometric complexity theory is an approach towards solving computational complexity lower bounds questions using algebraic geometry and representation theory. This talk contains an introduction to geometric complexity theory and a presentation of some recent results. Along the way connections to the study of secant varieties and to classical combinatorial and representation theoretic conjectures will be pointed out.

Let G be a reductive group, E an elliptic curve, and Bun_G the moduli stack of principal G-bundles on E. In this talk, I will attempt to explain why Bun_G is a very interesting object from the perspectives of both singularity theory on the one hand, and shifted symplectic geometry and representation theory on the other. In the first part of the talk, I will explain how to construct slices of Bun_G through points corresponding to unstable bundles, and how these are linked to certain singular algebraic surfaces and their deformations in the case of a "subregular" bundle. In the second (probably much shorter) part, I will discuss the shifted symplectic geometry of Bun_G and its slices. If time permits, I will sketch how (conjectural) quantisations of these structures should be related to some well known algebras of an "elliptic" flavour, such as Sklyanin and Feigin-Odesskii algebras, and elliptic quantum groups.

In this talk, I will work on a smooth projective threefold X which satisfies the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the projective space P^3 or the quintic threefold. I will show certain moduli spaces of 2-dimensional torsion sheaves on X are smooth bundles over Hilbert schemes of ideal sheaves of curves and points in X. When X is Calabi-Yau this gives a simple wall crossing formula expressing curve counts (and so ultimately Gromov-Witten invariants) in terms of counts of D4-D2-D0 branes. This is joint work with Richard Thomas. 

Given a nested pair X and Y of complex projective varieties, there is a single positive integer e which measures the singularity type of X inside Y. This is called the Hilbert-Samuel multiplicity of Y along X, and it appears in the formulations of several standard intersection-theoretic constructions including Segre classes, Euler obstructions, and various other multiplicities. The standard method for computing e requires knowledge of the equations which define X and Y, followed by a (super-exponential) Grobner basis computation. In this talk we will connect the HS multiplicity to complex links, which are fundamental invariants of (complex analytic) Whitney stratified spaces. Thanks to this connection, the enormous computational burden of extracting e from polynomial equations reduces to a simple exercise in clustering point clouds. In fact, one doesn't even need the polynomials which define X and Y: it suffices to work with dense point samples. This is joint work with Martin Helmer.

In the talk I will survey the fast growing field of metric measure spaces satisfying a lower bound on Ricci Curvature, in a synthetic sense via optimal transport. Particular emphasis will be given to discuss how such (possibly non-smooth) spaces naturally (and usefully) extend the class of smooth Riemannian manifolds with Ricci curvature bounded below.

In this talk, I will discuss some results on the structure of the cohomology of the moduli space of stable SL_n Higgs bundles on a curve. 

One consequence is a new proof of the Hausel-Thaddeus conjecture proven previously by Groechenig-Wyss-Ziegler via p-adic integration.

We will also discuss connections to the P=W conjecture if time permits. Based on joint work with Junliang Shen.

We describe how Smith theory applies in the setting of Hamiltonian Floer homology filtered by the action functional, and provide applications to questions regarding Hamiltonian diffeomorphisms, including the Hofer-Zehnder conjecture on the existence of infinitely many periodic points and a question of McDuff-Salamon on Hamiltonian diffeomorphisms of finite order.

The Spin(7) and SU(4) structures on a Calabi-Yau 4-fold give rise to certain first order PDEs defining special Yang-Mills connections: the Spin(7) instanton equations and the Hermitian Yang-Mills (HYM) equations respectively. The latter are stronger than the former. In 1998 C. Lewis proved that -over a compact base space- the existence of an HYM connection implies the converse. In this talk we demonstrate that the equivalence of the two gauge-theoretic problems fails to hold in generality. We do this by studying the invariant solutions on a highly symmetric noncompact Calabi-Yau 4-fold: the Stenzel manifold. We give a complete description of the moduli space of irreducible invariant Spin(7) instantons with structure group SO(3) on this space and find that the HYM connections are properly embedded in it. This moduli space reveals an explicit example of a sequence of Spin(7) instantons bubbling off near a Cayley submanifold. The missing limit is an HYM connection, revealing a potential relationship between the two equation systems.

This is a report on joint work with Martijn Kool. 

Recently, Marian-Oprea-Pandharipande established a generalization of Lehn’s conjecture for Segre numbers associated to Hilbert schemes of points on surfaces. Extending work of Johnson, they provided a conjectural correspondence between Segre and Verlinde numbers. For surfaces with holomorphic 2-form, we propose conjectural generalizations of their results to moduli spaces of stable sheaves of higher rank. 

Using Mochizuki’s formula, we derive a universal function which expresses virtual Segre and Verlinde numbers of surfaces with holomorphic 2-form in terms of Seiberg- Witten invariants and intersection numbers on products of Hilbert schemes of points. We use this to  verify our conjectures in examples. 

In this talk, I will explain a new wall-crossing phenomenon on P^3 that induces non-Q-factorial singularities and thus cannot be understood as an operation in the MMP of the moduli space, unlike the case for many surfaces.  If time permits, I will explain how the wall-crossing could help to understand the geometry of the associated Hilbert scheme and PT moduli space.

I will describe an ongoing research project which aims to encode the DT invariants of a CY3 triangulated category in a geometric structure on its space of stability conditions. More specifically we expect to find a complex hyperkahler structure on the total space of the tangent bundle. These ideas are closely related to the work of Gaiotto, Moore and Neitzke from a decade ago. The main analytic input is a class of Riemann-Hilbert problems involving maps from the complex plane to an algebraic torus with prescribed discontinuities along a collection of rays.

Calibrated geometry, more specifically Calabi-Yau geometry, occupies a modern, rather sophisticated, cross-roads between Riemannian, symplectic and complex geometry. We will show how, stripping this theory down to its fundamental holomorphic backbone and applying ideas from classical complex analysis, one can generate a family of purely holomorphic invariants on any complex manifold. We will then show how to compute them, and describe various situations in which these invariants encode, in an intrinsic fashion, properties not only of the given manifold but also of moduli spaces.

Interest in these topics, if initially lacking, will arise spontaneously during this informal presentation.

Given a log Calabi-Yau surface Y with maximal boundary D, I'll explain how to construct a mirror Landau-Ginzburg model, and sketch a proof of homological mirror symmetry for these pairs when (Y,D) is distinguished within its deformation class (this is mirror to an exact manifold). I'll explain how to relate this to the total space of the SYZ fibration predicted by Gross--Hacking--Keel, and, time permitting, explain ties with earlier work of Auroux--Katzarkov--Orlov and Abouzaid. Joint work with Paul Hacking.

I will discuss some ongoing work on three-dimensional supersymmetric gauge theories and their relationship to (equivariant) quantum K-theory. I will emphasise the interplay between the physical and mathematical motivations and approaches, and attempt to build a dictionary between the two.  As an interesting example, I will discuss the quantum K-theory of flag manifolds. The QK ring will be related to the vacuum structure of a gauge theory with Chern-Simons interactions, and the (genus-0) K-theoretic invariants will be computed in terms of explicit residue formulas that can be derived from the relevant supersymmetric path integrals.

I will describe the results of two projects on the construction of Calabi-Yau threefolds and certain real Lagrangian submanifolds. The first concerns the construction of a novel dataset of Calabi-Yau threefolds via an application of the Gross-Siebert algorithm to a reducible union of toric varieties obtained by degenerating anti-canonical hypersurfaces in a class of (around 1.5 million) Gorenstein toric Fano fourfolds. Many of these constructions correspond to smoothing such a hypersurface; in contrast to the famous construction of Batyrev-Borisov which exploits crepant resolutions of such hypersurfaces. A central ingredient here is the construction of a certain 'integral affine structure with singularities' on the boundary of a class of polytopes from which one can form a topological model, due to Gross, of the corresponding Calabi-Yau threefold X. In general, such topological models carry a canonical (anti-symplectic) involution i and in the second project, which is joint work with H. Argüz, we describe the fixed point locus of this involution. In particular, we prove that the map i*-1 on graded pieces of a Leray filtration of H^3(X,Z2) can be identified with the map D -> D^2, where D is an element of H^2(X',Z2) and X' is mirror-dual to X. We use this to compute the Z2 cohomology group of the fixed locus, answering a question of Castaño-Bernard--Matessi.

An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which 'count' $\tau$-(semi)stable objects $E$ with fixed topological invariants $[E]=\alpha$ in some geometric problem, by means of a virtual class $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ of the moduli spaces ${\mathcal M}_\alpha^{\rm st}(\tau)\subseteq{\mathcal M}_\alpha^{\rm ss}(\tau)$ of $\tau$-(semi)stable objects in some homology theory. Examples include Mochizuki's invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds.

We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces ${\mathcal M},{\mathcal M}^{\rm pl}$, where my big vertex algebras project http://people.maths.ox.ac.uk/~joyce/hall.pdf gives $H_*({\mathcal M})$ the structure of a graded vertex algebra, and $H_*({\mathcal M}^{\rm pl})$ a graded Lie algebra, closely related to $H_*({\mathcal M})$. The virtual classes $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ take values in $H_*({\mathcal M}^{\rm pl})$. In most such theories, defining $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ when ${\mathcal M}_\alpha^{\rm st}(\tau)\ne{\mathcal M}_\alpha^{\rm ss}(\tau)$ (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define $[{\mathcal M}_\alpha^{\rm ss}(\tau)]_{\rm virt}$ in homology over $\mathbb Q$, and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition $\tau$, written using the Lie bracket on $H_*({\mathcal M}^{\rm pl})$. We prove our conjectures for moduli spaces of representations of quivers without oriented cycles.

This is joint work with Jacob Gross and Yuuji Tanaka.

There are many known ways to compute the homology of the moduli space of algebraic vector bundles on a curve. For higher-dimensional varieties however, this problem is very difficult. It turns out that the moduli stack of objects in the derived category of a variety X, however, is topologically simpler than the moduli stack of vector bundles on X. We compute the rational homology of the moduli stack of complexes in the derived category of a smooth complex projective variety. For a certain class of varieties X including curves, surfaces, flag varieties, and certain 3- and 4-folds we get that the rational cohomology is freely generated by Künneth components of Chern characters of the universal complex––this allows us to identify Joyce's vertex algebra construction with a super-lattice vertex algebra on the rational cohomology of X in these cases.