Abstract: I will give some background on the Gamma Conjecture, which says that mirror symmetry does *not* respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the symplectic mirror, multiplied by a certain transcendental characteristic class called the Gamma class. In the second part of the talk I will explain a new geometric approach to the Gamma Conjecture, which is based on the SYZ viewpoint on mirror symmetry. We find that the appearance of zeta(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration. This is based on joint work with Abouzaid, Ganatra, and Iritani.

Abstract: (joint work with Maxim Smirnov) Let X be a Fano variety. Dubrovin’s conjecture predicts, among other things, an equivalence between the semi-simplicity of QH(X) the big quantum cohomology of X and the existence of a full exceptional collection in D(X) the bounded derived category of X. Recently, Kuznetsov and Smirnov proposed a refinement of this conjecture if qH(X) the small quantum cohomology is not semi-simple. I will present this conjecture, discuss it in the case of coadjoint varieties and make a connection with simple surface singularities.   

Abstract: Just over half a century ago, Bacry and Lévy-Leblond asked the question: “Which are the kinematical symmetries of space and time”?  For the physically interesting case of four dimensions, they gave an answer subject to some conditions which were removed twenty years later by Bacry and Nuyts, resulting in a small list of kinematical Lie algebras.  Two years ago in collaboration with Stefan Prohazka we took this pioneering work to its logical conclusion and classified kinematical Klein geometries in any dimension.  These geometries arrange themselves in several families: lorentzian, galilean, carrollian and aristotelian.  In this talk I will re-interpret these geometries as G-structures on the spacetime manifold and will present an initial classification of their intrinsic torsion. This talk is based on the paper https://arxiv.org/abs/2009.01948.

Abstract: The Grassmannian has a cluster structure (due to Scott) that can be used in several ways to define Landau--Ginzburg models. Two specific examples are given by Marsh--Ritesch's superpotential (and Rietsch--Williams' work relating it to the cluster structure) and Gross--Hacking--Keel--Kontsevich's superpotential. I will explain how the two are related by a p^*-map from cluster theory. This is based on joint work with Cheung, Magee and Nájera Chávez.

Abstract: I will outline, in a down-to-earth way, how invariants are extracted from moduli spaces in modern enumerative algebraic geometry.The key tool is something called a “virtual cycle”, which can be understood as some kind of “localised Euler class”. We will see this works for 3 complex dimensional Calabi-Yau manifolds, but not for 4-folds. Then I will explain a fix using “localised square root Euler classes for special orthogonal bundles”. Joint work with Jeongseok Oh (KIAS).

Abstract: Matrix-resolvents are classical objects in integrable systems given by Lax pairs. Recently, they were used to compute logarithmic derivatives of tau-functions. In this talk, we explain this method for several cases (in particular for KdV) and present some applications for enumerative geometry. The talk is based on a series of joint works with Marco Bertola and Boris Dubrovin.

Abstract: Monstrous Moonshine emerged in the 1970s, from coincidences relating the largest sporadic simple group to moduli spaces of complex elliptic curves. More recently, manifestations of moonshine have materialized that relate sporadic simple groups to arithmetic invariants of elliptic curves over the rationals. In this talk I will describe forthcoming joint work with Cheng and Mertens that initiates a systematic approach to this phenomena. Our investigations yielded some unexpected results, including a connection between the congruent number problem of antiquity, and the smallest sporadic simple group.

Abstract: The quantum differential equation (qDE) is a rich object attached to a smooth projective variety X. It is an ordinary differential equation in the complex domain which encodes information of the enumerative geometry of X, more precisely its Gromov-Witten theory. Furthermore, the monodromy of its solutions conjecturally rules also the topology and complex geometry of X. In this talk I will introduce some analytic integral multitransforms of Borel-Laplace type, and I will use them to obtain Mellin-Barnes integral representations of solutions of qDEs. Based on arXiv:2005.08262.

Hurwitz theory is concerned with the enumeration of branched coverings of P^1 with given topology and constrained ramification. It can be approached/solved in at least three ways: integrable hierarchies coming from the representation theory of the symmetric (first unveiled by Okounkov and Pandharipande), intersection theory on the moduli space of curves (first seen in the Ekedahl-Lando-Shapiro-Vainshtein formula), and topological recursion (taking its roots in Bouchard-Marino conjecture). These three aspects have been established for many different type of Hurwitz problems, and after a brief review I will focus on double Hurwitz numbers where the three structures enrich each other: a joint work with Do, Karev, Lewanski and Moskowsky, we start from known representation-theoretic formulas for double Hurwitz numbers to prove a polynomiality result and topological recursion, which in turn implies an ELSV-like formula involving Chiodo classes and generalising a formula of Johnson-Pandharipande-Tseng, and proves along the way new vanishing properties of the Chiodo class.

Abstract: In 2014, Gross, Hacking, Keel and Kontsevich (GHKK) introduced theta functions on cluster varieties. They can be defined by considering scattering diagrams associated to skeletal curves in the analytification of the mirror cluster variety. A key insight of the work of GHKK is that theta functions on cluster varieties play a role analogous to the one played by the characters of an algebraic torus in toric geometry. In particular, they showed that one can consider “positive subsets” inside the scattering diagram to compactify a cluster variety. Various varieties arising in representation theory (such as flag varieties, double Bruhat cells, Grassmannians, certain Schubert varieties, etc.) fit this framework. The purpose of this talk is to present a geometric/combinatorial interpretation of positive sets. The main result I will talk about is the following: a set is positive if and only if it is “broken line convex”. This is joint work with Man-Wai Cheung and Timothy Magee.

Abstract: From a singular projective variety X_0, one can potentially obtain a smooth variety by smoothing or via a crepant resolution.  If X is a smoothing of X_0 and Y is a crepant resolution, we say that X and Y are related by extremal transition.  It is speculated that the moduli space of Calabi-Yau threefolds is connected via such transitions.  Therefore, understanding the behavior of Gromov-Witten Theory under extremal transitions has important applications to mirror symmetry.  In this talk I will describe a general procedure which produces extremal transitions between hypersurfaces in toric varieties and explain how their Gromov-Witten theories relate.  Joint with Rongxiao Mi.

Abstract: Homological projective duality (HPD) is a construction of a (noncommutative) projective variety that controls the derived categories of linear sections of a given projective variety and extends categorically the classical projective duality. I will describe the operation of a categorical join that extends categorically the classical join of two projective varieties and behaves well with respect to HPD --- the HPD of a categorical join is the categorical join of HPD. If time permits, I will also discuss some applications of this result. This work is joint with Alex Perry. 

Abstract: In this talk, I review the notions of d-critical loci, orientations, and motivic DT invariants following Joyce and collaborators.  I construct a d-critical locus structure on Hilb^n of local P^2 and provide it with an orientation using work of Shi.  The deduced motivic invariants are shown to agree with those defined by Behrend, Bryan, and Szendroi before a more general theory existed.  As a consequence, the generating function of motivic DT invariants of Hilb^n of local P^2 is computed.

This talk is based on joint work with Yun Shi.

Abstract: We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3. 

Abstract: Exceptional collections in derived categories of coherent sheaves have a long history going back to the pioneering work of A. Beilinson. After recalling the general setup, I will concentrate on some very recent developments inspired by the homological mirror symmetry. Namely, I will define residual categories of Lefschetz decompositions and discuss a conjectural relation between the structure of quantum cohomology and residual categories. I will illustrate this relationship in the case of some isotropic Grassmannians. This is a joint work with Alexander Kuznetsov.

Abstract: By quantum K-theory, I mean the theory of Gromov-Witten invariants of K-theoretic nature. In the talk, I'll try not to focus on their detailed definitions, but rather outline the adelic structure of the Hirzebruch-Riemann-Roch formula connecting them with cohomological Gromov-Witten invariants, and then discuss some features of it, which look thought-provoking from the point of view of quantum-mechanics and its quasi-classical limit.

Abstract: We study a natural contact instanton (CI) equation on gauge fields over 7-dimensional Sasakian manifolds, which is closely related both to the transverse Hermitian Yang-Mills (tHYM) condition and the G_2-instanton equation. We obtain, by Fredholm theory, a finite-dimensional local model for the moduli space of irreducible solutions. We derive cohomological conditions for smoothness, and we express its dimension in terms of the index of a transverse elliptic operator. Finally we show that the moduli space of self dual contact instantons (ASDI) is Kähler, in the Sasakian case. As an instance of concrete interest, we specialise to transversely holomorphic Sasakian bundles over contact Calabi-Yau 7-manifolds, and we show that, in this context, the notions of contact instanton, integrable G_2-instanton and HYM connection coincide. 

Abstract: In this talk we will consider the action of symmetric group S_n on the rational cohomology of M_g,n the moduli space of genus g Riemann surfaces with  marked points. We will discuss how these cohomology groups stabilize in a precise representation-theoretic sense and what is the underlying algebraic structure that derives these stability patterns. 

Abstract: Motivated by the computation of certain Feynman amplitudes, Broadhurst and Roberts recently conjectured and checked numerically to high precision a set of remarkable quadratic relations between integrals of powers of Bessel functions. We interpret these integrals as coefficients of the period pairing between middle de Rham cohomology and twisted homology of symmetric powers of the Kloosterman connection. The talk will explain the general setup for quadratic relations between (possibly irregular) periods and deduce those for Bessel moments. Joint work with J. Fresán and J.-D. Yu.  

Abstract: In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime.  We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. Modular groups and modular forms arise in relation to these attractor points in a way that, to a physicist, is unexpected. 

This is a report on joint work with Mohamed Elmi and Duco van Straten.

Abstract: I will review work on the McKay correspondence in 3 dimensions and its differential-geometric version, relying on papers by Ito-Reid, Sardo Infirri, Craw-Ishi and others, and on some original results obtained with a group of collaborators. This is about the resolution of singularities of the type C^3/G, where G is a finite subgroup of GL(3,C) or SL(3,C). I will also discuss the chamber structure for the stability parameter of the GIT quotient. I will illustrate the general theory by means of a nontrivial but manageable example (C^33/Z_4 with Z_4 acting as a subrgoup of SL(3,C)). I will also hint at some physical motivations. 

Abstract: Plane curves are among the most classical objects in algebraic geometry. There is a solid understanding of them; individually as well as of many of their families. The family V(d,n)  parameterizing irreducible nodal plane curves of degree d  with precisely n  nodes is particularly interesting. This family has been extensively studied and many of its basic properties are known. A less explored aspect of this family V(d,n) is: What is the geometry described by the nodes of the curves in it? 

In this talk, I will report on progress about this question with applications to enumerative geometry. This is joint work with Tim Ryan (University of Michigan).  

Abstract: In the physics of superconformal field theories, certain class of boundary conditions, known as B-branes, can be identified with objects in algebraic geometry, such as coherent sheaves. I will explain how one can use physics to study the categories associated to these objects, how can one associate a natural local system to them and their behavior as we move on the moduli space of deformations of Kahler manifolds. If time permits I will present results for non-geometric points in the quantum (or extended) Kahler moduli  corresponding to LG orbifolds found in 2003.00182[hep-th]