On Calabi–Yau threefolds there are two types of integral invariants, quantum K-invariants and Gopakumar–Vafa invariants. In this talk, I will explain a joint project (with You-Cheng Chou) which aims to show that the quantum K-invariants and Gopakumar invariants are equivalent. At genus zero, this is a conjecture by Jockers–Mayr and Garoufalidis–Scheidegger (for the quintic), and a proof of the JMGS conjecture will be presented.

Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We constructed an algebraic virtual cycle. A key step is a localisation of Edidin-Graham’s square root Euler class for $SO(2n,\mathbb{C})$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We also develop a theory of complex Kuranishi structures on projective schemes which are sufficiently rigid to be equivalent to weak perfect obstruction theories, but sufficiently flexible to admit global complex Kuranishi charts. We apply the theory to the moduli spaces to prove the two virtual cycles coincide in homology after inverting 2 in the coefficients. In particular, when Borisov-Joyce’s real virtual dimension is odd, their virtual cycle is torsion. This is a joint work with Richard Thomas.

Givental and Lee introduced quantum K-theory, a K-theoretic generalization of Gromov–Witten theory. It studies holomorphic Euler characteristics of coherent sheaves on moduli spaces of stable maps to given target spaces. In this talk, I will introduce the quantum K-theory for orbifold target spaces which generalizes the work of Tonita-Tseng. In genus zero, I will define a quantum K-ring which specializes to the full orbifold K-ring introduced by Jarvis-Kaufmann-Kimura. As an application, I will give a detailed description of the quantum K-ring of weighted projective spaces, which generalizes a result by Goldin-Harada-Holm-Kimura. This talk is based on joint work with Yang Zhou.

In this talk, I will explain quantum spectrum and asymptotic expansions in FJRW theory of invertible singularities of general type. Inspired by Galkin-Golyshev-Iritani’s Gamma conjectures for quantum cohomology of Fano manifolds, we propose Gamma conjectures for FJRW theory of general type. Here the Gamma structures are essential to understand the connection between algebraic structures of the singularities (such as Orlov’s semiorthogonal decompositions of matrix factorizations) and the analytic structures, such as asymptotic expansions in FJRW theory. The talk is based on the work joint with Ming Zhang.

The DT/PT correspondence is a formula which relates Donaldson-Thomas invariants counting closed subschemes in Calabi-Yau 3-folds, and Pandharipnade-Thomas invariants counting stable pairs on them. It gives an economical way of formulating GW/DT correspondence by Maulik-Nekrasov-Okounkov-Pandharipande. The DT/PT correspondence was proved by wall-crossing in the derived category, where on the wall we have a special type of Ext-quivers called DT/PT quivers. In this talk, I will give a categorical wall-crossing formula for categories of matrix factorizations associated with DT/PT quivers. The resulting formula is a semiorthogonal decomposition which involves quasi-BPS categories, and is a categorical analogue of numerical DT/PT correspondence. This is a joint work with Tudor Padurariu.

Caldararu-Costello-Tu defined Categorical Enumerative Invariants (CEI) as a set of invariants associated to a cyclic A-infinity category (with some extra conditions/data), that resemble the Gromov-Witten invariants in symplectic geometry. In this talk I will explain how one can define these invariants for Calabi-Yau A-infinity categories - a homotopy invariant version of cyclic - and then show the CEI are Morita invariant. This has applications to Mirror Symmetry and Algebraic Geometry.

We define integer valued invariants of an orbifold Calabi-Yau threefold $X$ with transverse ADE orbifold points. These invariants contain equivalent information to the Gromov-Witten invariants of $X$ and are related by a Gopakumar-Vafa like formula which may be regarded as a universal multiple cover / degenerate contribution formula for orbifold Gromov-Witten invariants. We also give sheaf theoretic definitions of our invariants. As examples, we give formulas for our invariants in the case of a (local) orbifold K3 surface. These new formulas generalize the classical Yau-Zaslow and Katz-Klemm-Vafa formulas. This is joint work with S. Pietromonaco.

Sir Simon Donaldson conjectured that there should exist a virtual fundamental class on the moduli space of surfaces of general type inspired by the geometry of complex structures on the general type surfaces. In this talk I will present a method to construct the virtual fundamental class in the sense of Behrend-Fantechi on the moduli stack of lci (locally complete intersection) covers over the moduli stack of general type surfaces with only semi-log-canonical singularities. A tautological invariant is defined by taking the integration of the power of the first Chern class of the CM line bundle over the virtual fundamental class. This can be taken as a generalization of the tautological invariants on the moduli space of stable curves to the moduli space of stable surfaces. The method presented can also be applied to the moduli space of stable map spaces from semi-log-canonical surfaces to projective varieties.

Harder-Narasimhan (HN) theory gives a structure theorem for principal G bundles on a smooth projective curve. A bundle is either semistable, or it admits a canonical filtration whose associated graded bundle is semistable in a graded sense. After reviewing recent advances in extending HN theory to arbitrary algebraic stacks, I will discuss work in progress with Andres Fernandez Herrero to apply this general machinery to the stack of “gauged” maps from a curve C to a G-scheme X, where G is a reductive group and X is projective over an affine scheme. Our main application is to use HN theory for gauged maps to compute generating functions for K-theoretic enumerative invariants known as gauged Gromov-Witten invariants. This problem is interesting more broadly because it can be formulated as an example of an infinite dimensional analog of the usual set up of geometric invariant theory, which has applications to other moduli problems.

I this talk, I will give semiorthogonal decompositions of derived categories of several classical moduli spaces, e.g. symmetric products of curves, Brill-Noether loci, (relative) Quot schemes, Hilbert schemes of points. In particular, they contain a proof of Jiang’s conjecture for semiorthogonal decompositions of Quot schemes of locally free quotients. They are by-products of my research on categorifications of wall-crossing in Donaldson-Thomas theory, and the proofs involve techniques of derived algebraic geometry, categorical Hall algebras, matrix factorizations and Koszul duality.

We study non-commutative projective varieties in the sense of Artin-Zhang, which are given by non-commutative homogeneous coordinate rings, which are finite over their centre. We construct moduli spaces of stable modules for these, and construct a symmetric obstruction theory in the CY3-case. This gives deformation invariants of Donaldson-Thomas type. The simplest example is the Fermat quintic in quantum projective space, where the coordinates commute up to carefully chosen 5th roots of unity. We explore the moduli theory of finite length modules, which mixes features of the Hilbert scheme of commutative 3-folds, and the representation theory of quivers with potential. This is joint work with Yu-Hsiang Liu, with contributions by Atsushi Kanazawa.

I will report on joint work with Daniel Bragg. There are many ways to express the universality properties of moduli spaces. For example, Vakil established years ago that any singularity type defined over the integers appears in natural moduli spaces, proving what he called “Murphy’s Law”. We have discovered another, similar, phenomenon: many natural moduli spaces contain all finite gerbes. In particular, any finite gerbe over any field appears as the residual gerbe of some point of the stack of curves.

The moduli stack of complexes of vector bundles over a Gorenstein Calabi-Yau curve, considered up to chain isomorphisms, admits a 0-shifted Poisson structure in the sense of Calaque-Pantev-Toen-Vaquie-Vezzosi. We will give several classical examples of Poisson varieties appearing in representation theory and integrable system, that are naturally Poisson substacks of the above mentioned stack. Using derive algebraic geometry, we are able to prove some new results on these Poisson varieties. This is a joint work with Alexander Polishchuk.

The construction of stability conditions on the bounded derived category of coherent sheaves on smooth projective varieties is notoriously a difficult problem. In this talk, I will review some results and techniques related to this problem. I will specifically concentrate on the case of Hilbert schemes of points on K3 surfaces and on generic abelian varieties of any dimension. This is joint work in progress with C. Li, E. Macrì and P. Stellari.

We consider tautological bundles and their exterior and symmetric powers over the Quot scheme of zero dimensional quotients over the projective line. We prove several results regarding the vanishing of their higher cohomology, and we describe the spaces of global sections via tautological constructions. This is based on joint work with Alina Marian, Shubham Sinha and Steven Sam.

I will discuss joint work with Kenny Ascher, Dori Bejleri, Harold Blum, Giovanni Inchiostro, Yuchen Liu, and Xiaowei Wang on construction of moduli stacks and moduli spaces of log Calabi Yau pairs that can be realized as slc log Fano pairs with complements. Unlike moduli of canonically polarized varieties (respectively, Fano varieties) in which the moduli stack of KSB stable (respectively, K semistable) objects is bounded for fixed volume, dimension, the objects here form unbounded families. Despite this unbounded behavior, in the case of plane curve pairs (P2, C), we construct a projective good moduli space parameterizing S-equivalence classes of these slc Fanos with complements.

Grothendieck’s Existence Theorem asserts that a coherent sheaf on a scheme proper over a complete local noetherian ring is the same as a compatible system of coherent sheaves on the thickenings of its central fiber. This is a fundamental result with important applications to moduli theory. We will discuss generalizations of this result to algebraic stacks beginning with a review of the characteristic 0 situation where a satisfactory answer is known: any quotient stack $[{\rm Spec} A/G]$ whose invariant ring $A^G$ is a complete local k-algebra is coherently complete along its unique closed point. We will report on partial progress in joint work with Hall and Lim on extending this result to positive characteristic.