Valery Alexeev (University of Georgia, Athens, GA, USA) valery@uga.edu
Title: Compact moduli of K3 surfaces
Abstract: We discuss constructions of geometrically meaningful compactifications of moduli spaces of lattice polarized K3 surfaces.
Hanine Awada (Univ. Montpellier, France) hanine.awada@univ-nantes.fr
Title: Cubic fourfolds and associated K3 surfaces
Abstract: The rationality problem of smooth cubic hypersurfaces of dimension four is one of the most challenging open problem in algebraic geometry. According to a conjecture, based on the work of Hassett and Kuznetsov, rational cubic fourfolds are special and ought to have an associated K3 surface. Intersecting divisors parametrizing special cubic fourfolds in the moduli space of cubic fourfolds will allow us to construct families of cubic fourfolds with associated K3 surfaces of arbitrary rank; as an application, we will show that every divisor contains a countable infinity of 1-dimensional families of cubic fourfolds whose Chow motive is finite dimensional and of abelian type. This is a joint work with M.Bolognesi and C.Pedrini.
Vikraman Balaji (Chennai Math. Institute, Chennai, Siruseri Tamil Nadu, India) balaji@cmi.ac.in
Title : Torsors on semistable curves and the problem of degenerations
Abstract: Let G be an almost simple, simply connected algebraic group G over the field of complex numbers. In this talk I answer two basic questions in the classification of G-torsors on curves. The first one is to construct a flat degeneration of the moduli stack G-torsors on a smooth projective curve when the curve degenerates to an irreducible nodal curve. Torsors for a generalization of the classical Bruhat-Tits group schemes to two-dimensional regular local rings and an application of the geometric formulation of the McKay correspondence provide the key tools. The second question is to give an intrinsic definition of (semi)stability for a G-torsor on an irreducible nodal curve. The absence of obvious analogues of torsion-free sheaves in the setting of G-torsors makes the question interesting. This also leads to the construction of a proper separated coarse space for G-torsors on an irreducible nodal curve.
Pietro Beri (Univ. of Poitiers, France) Pietro.Beri@math.univ-poitiers.fr
Title: Birational involutions on Hilbert schemes of points on a K3 surface
Abstract: We present a complete classification of birational automorphisms of Hilbert schemes of n points on a very general K3 surface. This is a joint work with Alberto Cattaneo. Moreover, we study some linear systems of small dimension on Hilbert schemes of points admitting a birational involution. (Short communication.)
Marco Bertola (Concordia University, Montreal, Quebec, Canada) marco.bertola@concordia.ca
Title: Tyurin data, non-abelian Cauchy kernels and the Goldman bracket
Abstract: TBA
Alexander Braverman (University of Toronto, Waterloo, Ontario, Canada, and Skoltech, Moscow, Russia) braval@math.toronto.edu
Title: ТВА
Abstract: ТВА
Alexandr Buryak (HSE, Moscow, Russia), aburyak@hse.ru
Title: Dubrovin-Zhang hierarchies and relations in the cohomology of the moduli space of curves
Abstract: TBA
Gaia Comaschi (Univ. of Campinas, Campinas, Brazil)
Title: Instanton sheaves of low charge on Fano threefolds
Abstract: Let X be a Fano threefold of Picard number one and of index 2 + h, h = 0, 1. An instanton sheaf of charge k on X is defined as a semi-stable rank 2 torsion free sheaf F with Chern classes c1 = −h, c2 = k, c3 = 0 and such that F (−1) has no cohomology. Locally free instantons, originally defined on the projective space and later generalised on other Fano threefolds X, had been largely studied from several authors in the past years; their moduli spaces present an extremely rich geometry and useful applications to the study of curves on X. In this talk I will illustrate several features of non-locally free instantons of low charge on 3 dimensional quadrics and cubics. I will focus in particular on the role that they play in the study of the Gieseker-Maruyama moduli space MX(2;−h, k, 0) and describe how we can still relate these sheaves to curves on X. (Short communication.)
Soheyla Feyzbakhsh (Imperial College, London, UK) s.feyzbakhsh@imperial.ac.uk
Title: Rank r DT theory from rank 1
Abstract: Fix a Calabi-Yau 3-fold X satisfying the Bogomolov-Gieseker conjecture of Bayer-Macrì-Toda, such as the quintic 3-fold. I will report on a joint project with Richard Thomas that aims to express Joyce’s generalised DT invariants counting Gieseker semistable sheaves of any rank r on X in terms of those counting sheaves of rank 1. By the MNOP conjecture, the latter are determined by the Gromov-Witten invariants of X. Our technique is to use wall-crossing with respect to weak Bridgeland stability conditions. (Short communication.)
Michael Gekhtman ( University of Notre Dame, Notre Dame, IN, USA) mgekhtma@nd.edu
Title: Generalized cluster structures on the Drinfeld double of SL_n
Abstract: TBA
Marcos Jardim (Univ. of Campinas, Campinas, Brazil)
Title: Asymptotic Bridgeland stability and instanton sheaves on threefolds
Abstract: We consider the geometric Bridgeland stability conditions for threefolds with Picard rank 1 as described by Bayer--Macri--Toda, which are parametrized by the upper half-plane H. We compare the moduli space of Bridgeland stable objects for large values of the parameters with the Gieseker moduli space of semistable sheaves, determining certain regions in H within with they coincide. As a case study, we look at instanton sheaves on the projective space and provide a full description of all walls and Bridgeland moduli spaces for certain unbounded paths in H.
Martijn Kool (Utrecht Univ., Utrecht, Netherlands)
Title: Proof of Magnificent Four
Abstract: Motivated by super-Yang–Mills theory on a Calabi–Yau 4-fold, Nekrasov and Piazzalunga assigned weights to r-tuples of solid partitions (4-dimensional piles of boxes) and conjectured a formula for their weighted generating function. We define K-theoretic virtual invariants of Quot schemes of 0-dimensional quotients of \O^r on affine 4-space by realizing them as zero loci of isotropic sections of orthogonal bundles. Using the Oh–Thomas localization formula, we recover Nekrasov–Piazzalunga’s weights. Applying ideas from Okounkov in the 3-dimensional case, we prove Nekrasov-Piazzalunga’s formula. Joint work with J. Rennemo.
Emanuele Macri (University Paris-Saclay, CNRS, Orsay, France ) emanuele.macri@universite-paris-saclay.fr
Title: Hyperkähler fourfolds and Fano manifolds
Abstract: I will report on joint work in progress with Laure Flapan, Kieran O'Grady, and Giulia Sacca on how to naturally associate to a polarized hyperkähler fourfold, of divisibility 2, a Fano manifold of dimension depending on the degree of the polarization. The basic case is the correspondence between cubic fourfolds and their varieties of lines with the Plücker polarization.
Paul Norbury (University of Melbourne, Parkville, VIC, Australia) norbury@unimelb.edu.au
Title: Enumerative geometry via the moduli space of super Riemann surfaces
Abstract: TBA
Kieran O'Grady (Sapienza University of Rome, Rome, Italy) ogrady@mat.uniroma1.it
Title: Fixed loci of anti-symplectic involutions of HK varieties
Abstract: This is a report of joint work with L. Flapan, E. Macri and G. Sacca. We study fixed loci of anti-symplectic involutions of HK varieties of Type K3^{[n]}. Motivations: one of our goals is to generalize the relation between a cubic 4fold and the corresponding family of lines, a HK 4fold of Type K3^{[2]}, to other families of polarized 4folds of Type K3^{[2]}; another goal is connected to the Beauville-Voisin conjectures.
Giorgio Ottaviani (University of Florence, Florence, Italy) giorgio.ottaviani@unifi.it
Title: Irreducible components of moduli space of instanton bundles
Abstract: Moduli spaces of instanton bundles on P^3 are known to be irreducible. We give evidence for the existence of at least two components on P^(2n+1) for n\ge 2.
Laura Pertusi (University of Milano, Milano, Italy) laura.pertusi@unimi.it
Title: Stability conditions on Gushel-Mukai varieties
Abstract: A generic Gushel-Mukai variety X is a quadric section of a linear section of the Grassmannian Gr(2,5). Kuznetsov and Perry proved that the bounded derived category of X has a semiorthogonal decomposition with exceptional objects and a non-trivial subcategory Ku(X), known as the Kuznetsov component. In this talk we will discuss the construction of stability conditions on Ku(X) and, consequently, on the bounded derived category of X. As applications, for X of even-dimension, we will construct locally complete families of hyperkaehler manifolds from moduli spaces of stable objects in Ku(X) and we will determine when X has a homological associated K3 surface. This is a joint work with Alex Perry and Xiaolei Zhao.
Nicolai Reshetikhin ( University of California, Berkeley, CA, USA) reshetik@math.berkeley.edu
Title: On completeness of spin Calogero-Moser systems
Abstract: TBA
Alejandra Rincon-Hidalgo (ICTP, Trieste, Italy) arincon@ictp.it
Title: Projectivity of moduli spaces of Bridgeland semistable holomorphic triples on curves
Abstract: Let TCoh(C ) be the abelian category of holomorphic triples on a smooth projective curve C with positive genus. In this talk we study Bridgeland stability conditions onD b (TCoh(C )) and the moduli spaces of Bridgeland semistable holomorphic triples. In particular, we shall prove that the moduli stacks are algebraic of finite type over C and they admit projective good moduli spaces. This is joint work with Dominic Bunnett.
Paolo Rossi (University of Padova, Padova, Italy) paolo.rossi@math.unipd.it
Title: Integrability and intersection theory on the moduli space of stable curves
Abstract: TBA
Vladimir Rubtsov (Univ. of Angers, Angers, France)
Title: Non-abelian Abel theorem, multiplicative kernels and Kontsevich polynomilals
Abstract: I shall review the title items based on the ongoing project with I. Gaiur, V. Golyshev and D.van Straten
Leonid Rybnikov (HSE University, Moscow, Russia) leo.rybnikov@gmail.com
Title: Gaudin model and crystals
Abstract: Drinfeld-Kohno theorem relates the monodromy of KZ equation to the braid group action on a tensor product of $U_q(\mathfrak{g})$-modules by R-matrices. The KZ equation depends on the parameter $\kappa$ such that $q=\exp(\frac{\pi i}{\kappa})$. We study the limit of the Drinfeld-Kohno correspondence when $\kappa\to 0$ along the imaginary line. Namely, on the KZ side, this limit is the Gaudin integrable magnet chain, while on the quantum group side the limit is the tensor product of $\mathfrak{g}$-crystals. The limit of the braid group action by the monodromy of KZ equation is the action of the fundamental group of the Deligne-Mumford space of real stable rational curves with marked points (called cactus group) on the set of eigenlines for Gaudin Hamiltonians (given by algebraic Bethe ansatz). On the quantum group side, the cactus group acts by crystal commutors on the tensor product of $\mathfrak{g}$-crystals. We construct a bijection between the set of solutions of the algebraic Bethe ansatz for the Gaudin model and the corresponding tensor product of $\mathfrak{g}$-crystals, which preserves the natural cactus group action on these sets. If time allows I will also discuss some conjectural generalizations of this result relating it to works of Losev and Bonnafe on cacti and Kazhdan-Lusztig cells.
Alessandra Sarti (University of Poitiers, Poitiers, France) sarti@math.univ-poitiers.fr
Title: Complex Reflection Groups and K3 surfaces
Abstract: We classify all K3 surfaces that one can obtain as quotients of surfaces by certain subgroups of finite complex reflection groups of rank 4. The K3 surfaces are mostly singular with A-D-E singularities. The proof of this fact avoids as much as possible a case-by-case analysis and involves the theory of finite complex reflection groups. Moreover we show that each family contains K3 surfaces with the maximum Picard number which is 20. This construction generalises a previous result by W. Barth and by A. Sarti. This is a joint work with C. Bonnafe.
Sergey Shadrin (Korteweg-de Vries Institute for Mathematics, University of Amsterdam, Amsterdam, Netherlands) S.Shadrin@uva.nl
Title: Deformation theory and homotopy cohomological field theories
Abstract: TBA
Michael Shapiro (Michigan State University, East Lansing, MI, USA, and HSE University, Moscow, Russia.) mshapiro@msu.edu
Title: Noncommutative cluster integrability (joint w/ N.Ovenhouse and S.Arthamonov)
Abstract: We define a discrete dynamical system (non commutative pentagram map) and prove its noncommutative integrability. To prove integrability we define non commutative double quasi Poisson bracket on the space of non commutative arc weights of a directed graph on a cylinder which gives rise to the quasi Poisson bracket of Massuyeau and Turaev on the group algebra of the fundamental group of a surface. We show that the induced double quasi Poisson bracket on the boundary measurements can be described via non-commutative r-matrix formalism which gives a conceptual proof of the result by N.Ovenhouse that the traces of powers of Lax matrix form an infinity system of Hamiltonians in involution.
Paolo Stellari (University of Milano, Milano, Italy) paolo.stellari@unimi.it
Title: Categorical Torelli Theorems for Enriques surfaces
Abstract: We investigate a refined Derived Torelli Theorem for Enriques surfaces. Namely, we prove that two (generic) Enriques surfaces are isomorphic if and only if their Kuznetsov components are Fourier-Mukai equivalent. We analyze the similarities with analogous results for cubic fourfolds and threefolds and we show the applications of our techniques to a conjecture by Ingalls and Kuznetsov about the derived categories of Artin-Mumford quartic double solids. This is joint work with Li, Nuer and Zhao.
Alexander Tikhomirov (HSE, Moscow, Russia) astikhomirov@mail.ru, atikhomirov@hse.ru
Title: Geometry of moduli spaces of stable rank two bundles on Fano threefolds
Abstract: We produce new infinite series of moduli spaces of (semi)stable rank two bundles on some rational Fano varieties with Picard number 1. We give a complete description of certain spaces from these series considered as projective schemes. We discuss the problem of rationality of constructed moduli spaces. This is joint work with Danil Vassiliev from HSE, Moscow.
Yukinobu Toda ( Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba, Japan yukinobu.toda@ipmu.jp
Title: On categorical Donaldson-Thomas theory for local surfaces
Abstract: I will introduce the notion of categorical Donaldson-Thomas theories for moduli spaces of stable sheaves on the total space of a canonical line bundle on a smooth projective surface. They are constructed as certain gluings of locally defined triangulated categories of matrix factorizations, via the linear Koszul duality together with the theory of singular supports for coherent sheaves on quasi-smooth derived stacks. Several conjectures on wall-crossing formulas of categorical DT theories will be proposed.
Alexander Veselov (University of Loughborough, UK, and Moscow, Russia)
Title: Universal formula for Hilbert series of minimal nilpotent orbits
Abstract: Let $\mathfrak g$ be a complex simple Lie algebra and $\mathcal O_{min}$ be its minimal (in the sense of dimension) non-zero nilpotent orbit. We will show that the Hilbert series of the projectivisation $X=P(O_{min})$ is universal in the sense of Vogel: it will be written uniformly for all simple Lie algebras as certain generalized hypergeometric function of Vogel's parameters. We provide also a universal formula for the degree of $X$ in terms of classical Gamma function. The talk is based on a joint work with Atsushi Matsuo.
Angelo Vistoli (SNS, Pisa, Italy) angelo.vistoli@sns.it
Title: Integral intersection theory on stacks of curves
Abstract: Rational Chow rings on moduli spaces of smooth and stable curves has been studied very extensively. Following Totaro and Edidin-Graham, one can also define integral Chow rings of stacks of smooth and stable curves. In my talk I will survey know results, and introduce some new techniques in the subject, introduced in joint work with Andrea Di Lorenzo and Michele Pernice.
Kota Yoshioka (Kobe University, Kobe, Japan) yoshioka@math.kobe-u.ac.jp
Title: Moduli of stable sheaves on Enriques surfaces
Abstract: I will explain recent developements such as non-emptyness, some topological invariants, and the birational geometry on the moduli of stable sheaves on Enriques surfaces.