Minh-Tam Trinh：Triply-Graded Link Homology and the Hilb-vs-Quot Conjecture

Abstract: Any complex algebraic plane curve singularity defines a link up to isotopy, via its intersection with a surrounding 3-sphere. The Oblomkov–Rasmussen–Shende conjecture describes the a, q, t-graded Khovanov–Rozansky homology of the link in terms of the punctual Hilbert schemes of the singularity. The t = –1 limit is a theorem of Maulik. However, no prior results managed to incorporate all three gradings, except in very simple examples. Oscar Kivinen and I introduce a different kind of Quot scheme, and prove that an analogue of the ORS conjecture for these Quot schemes does hold, with all three gradings, for many singularities of the form y^n = x^d. This motivates a further conjecture about how these new Quot schemes are related to the Hilbert schemes by a motivic substitution of variables. I have proven it for singularities y^3 = x^d with d coprime to 3. This, combined with the previous result, establishes the ORS conjecture for the same class of singularities. If time permits, I will mention how ORS relates to Gromov–Witten theory.

Filippo Viviani: On the classification of fine compactified Jacobians of nodal curves

Abstract: We study the problem of characterizing fine compactified Jacobians of nodal curves that can arise as limits of Jacobians of smooth curves. The answer is given in terms of a new class of fine compactified Jacobians, that we call fine V-compactified Jacobians, and that is strictly larger than the class of fine classical compactified Jacobians, as constructed by Oda-Seshadri, Simpson, Caporaso and Esteves. We give several characterizations of fine V-compactified Jacobians. Furthermore, we show that most of the known properties of fine classical compactified Jacobians extend to  fine V-compactified Jacobians: the relation to the Neron models of Jacobians, the autoduality property, the Fourier-Mukai equivalences, the perverse filtration of their cohomology, the relation to Mumford models of Jacobians.  

Changping Fan：Stability manifolds of Kuznetsov components of prime fano threefolds

Abstract.I will present work done in collaboration with Zhiyu Liu and Songtao Kenneth Ma.  Let X be a cubic threefold, quartic double solid or Gushel--Mukai threefold, and Ku(X) be its Kuznetsov component. We show that a stability condition on Ku(X) is Serre-invariant if and only if its homological dimension is at most 2. As a corollary, we prove that all Serre-invariant stability conditions on Ku(X) form a contractible connected component of the stability manifold.

Alessandro Chiodo：Mumford's formula on the universal Picard stack

I will present work done in collaboration with David Holmes. We construct a derived pushforward of the rth root of the universal line bundle over the Picard stack of genus g prestable curves carrying a line bundle. We prove a number of basic properties, and give a formula in terms of standard tautological generators. After pullback, our formula recovers formulae of Mumford, of Pagani-Ricolfi-van Zelm and my Grothendieck-Riemann-Roch formula for r-spin curves. We apply these constructions to prove a conjecture expressing the double ramification cycle and several generalizations. 

Rahul Pandharipande：Log intersection theory of the moduli space of curves

Abstract: I will discuss definitions, motivations, and some results concerning the log intersection theory of the moduli space of curves.

Ce Ji: Toward a generalization of the Witten conjecture from spectral curve

Abstract: Over decades of development of the Witten conjecture, Many enumerative geometries are proven to be related to integrable hierarchies. Simultaneously, such theories can also be reconstructed from topological recursion, an algorithm producing multi-differential forms from the underlying spectral curve. In this talk, we propose a generalization of the Witten conjecture from spectral curve, which produce descendent potential functions related to certain reductions of (multi-component) KP hierarchy. Proof for genus zero spectral curve with one boundary will be sketched, which can be applied to deduce the rKdV integrability of deformed negative r-spin theory, conjectured by Chidambaram--Garcia-Falide--Giacchetto. 

Matthew Satriano: Beyond twisted maps: crepant resolutions of log terminal singularities and a motivic McKay correspondence

Abstract: Crepant resolutions have inspired connections between birational geometry, derived categories, representation theory, and motivic integration. In this talk, we prove that every variety with log-terminal singularities admits a crepant resolution by a smooth stack. We additionally prove a motivic McKay correspondence for stack-theoretic resolutions. Finally, we show how our work naturally leads to a generalization of twisted mapping spaces. No prior knowledge of stacks will be assumed. This is joint work with Jeremy Usatine.

Xinyu Zhang: Physical approach to K-theoretic Donaldson invariants

Abstract: We consider the path integral of topologically twisted 5d N=1 SU(2) supersymmetric Yang-Mills theory on M x S1, where M is a smooth closed four-manifold. We derive the topological correlation functions, which produce K-theoretic Donaldson invariants of M.

Xinyu Zhang: Tetrahedron instantons

Abstract: In this talk, I will give an overview of tetrahedron instantons. I will describe the construction of tetrahedron instantons in string theory and in noncommutative field theory. The tetrahedron instanton partition function lies between the higher-rank Donaldson-Thomas invariants on Calabi-Yau threefolds and fourfolds, and can be computed exactly. It admits a free field representation, suggesting the existence of a novel kind of symmetry which acts on the cohomology of the moduli spaces of tetrahedron instantons. Tetrahedron instantons can also be used to study the duality between type IIA superstring theory and M-theory.

Pengfei Huang：Regular and irregular nonabelian Hodge correspondence for meromorphic G-connections

Abstract: Algebraic (integrable) connections on an algebraic  curve/variety are known to correspond to (Stokes) local systems, but a  similar statement for G-connections is not obvious. In this talk, we  will describe such a correspondence and place it within a general  framework, namely the nonabelian Hodge correspondence. The talk will  begin with a quick review of the classical theory and then proceed to  explore this nonabelian Hodge correspondence. It will end up with the  construction of moduli spaces for various filtered local systems. Based  on some recent joint work with G. Kydonakis, H. Sun and L. Zhao, and  with H. Sun.

Daniel S Halpern-Leistner: Theta-stratifications and gauged Gromov-Witten invariants I & II

Abstract: The moduli of vector bundles on a curve has a beautiful structure called the Harder-Narasimhan stratification, which has many applications to studying the geometry of this classical moduli problem. The theory of Θ-stratifications provides an intrinsic description of this stratification that generalizes to other moduli problems. Recently, with Andres Fernandez-Herrero, we use this theory, along with a newly developed method of “infinite dimensional geometric invariant theory,” to generalize the Harder-Narasimhan stratification to the moduli of maps from a smooth curve to a quotient stack V/G, where G is a reductive group and V is an affine G-variety.I will give an overview of these results. Then I will explain how to apply them in the case where V is a linear representation to give a formula for the K-theoretic gauged Gromov-Witten invariants of V in arbitrary genus. This includes some instances of quasi-maps invariants as a special case. If time allows, I will remark on the extension of these results to complete intersections.

Yingchun Zhang: A cluster algebra structure on the quantum cohomology ring of a quiver variety

In this work, we will introduce the cluster algebra and propose a cluster algebra structure on the quantum cohomology of a quiver variety. In particular, we will give a proof for the A-type cluster algebra on quantum cohomology of flag variety.This is a joint work with Weiqiang He.

Mina Aganagic：Homological link invariants from Floer theory

Abstract. A new relation between homological mirror symmetry and representation theory solves the knot categorification problem.  The symplectic side geometry side of mirror symmetry is a theory which generalizes Heegaard-Floer theory from gl(1|1) to arbitrary simple Lie (super) algebras. The corresponding category of A-branes has many special features, which render it solvable explicitly. In this talk, I will describe how the theory is solved, and how homological link invariants arise from it.

Meng-Chwan TAN: Vafa-Witten Theory: Invariants, Floer Homologies, Higgs Bundles, a Geometric Langlands Correspondence, and Categorification

We revisit Vafa-Witten theory in the more general setting whereby the underlying moduli space is not that of instantons, but of the full Vafa-Witten equations. We physically derive (i) a novel Vafa-Witten four-manifold invariant associated with this moduli space, (ii) their relation to Gromov-Witten invariants, (iii) a novel Vafa-Witten Floer homology assigned to three-manifold boundaries, (iv) a novel Vafa-Witten Atiyah-Floer correspondence, (v) a proof and generalization of a conjecture by Abouzaid-Manolescu in [1] about the hypercohomology of a perverse sheaf of vanishing cycles, (vi) a Langlands duality of these invariants, Floer homologies and hypercohomology, and (vii) a quantum geometric Langlands correspondence with purely imaginary parameter that specializes to the classical correspondence in the zero-coupling limit, where Higgs bundles feature in (ii), (iv), (vi) and (vii). We also explain how these invariants and homologies will be categorified in the process, and discuss their higher categorification. In essence, we will relate differential and enumerative geometry, topology and geometric representation theory in mathematics, via a maximally-supersymmetric topological quantum field theory with electric-magnetic duality in physics.