February 3rd, 2025

Davesh Maulik : $D$-equivalence conjecture for varieties of $K3^{[n]}$-type

Abstract: The $D$-equivalence conjecture of Bondal and Orlov predicts that birational Calabi-Yau varieties have equivalent derived categories of coherent sheaves. I will explain how to prove this conjecture for hyperkahler varieties of $K3^{[n]}$ type (i.e. those that are deformation equivalent to Hilbert schemes of $K3$ surfaces). This is joint work with Junliang Shen, Qizheng Yin, and Ruxuan Zhang.
Video.
Lecture Notes.

October 21st, 2024

Naoki Koseki : Gopakumar-Vafa invariants of local curves

Abstract: In the 1990s, two physisists, Gopakumar and Vafa, proposed an ideal way to count curves in a Calabi-Yau threefold, that is conjecturally equivalent to other curve counting theories such as Gromov-Witten theory. It is very recent that Maulik and Toda gave a mathematically rigorous definition of GV invariants. In this talk, I will review a recent progress on the GV theory, including $\chi$-independence for GV invariants on local curves and degree two GW/GV equivalence for some smooth curves with generic normal bundles. This is based on a joint work with T. Kinjo and another work with B. Davison.
Video.
Lecture Notes.

Past Schedule :

September 13th, 2021

Eric Zaslow : The proper Landau-Ginzburg potential is the open mirror map

Abstract: I will discuss toric Fano surfaces in the complement of a smooth anticanonical divisor and their mirror Landau-Ginzburg theories. I will focus on relations between open Gromov-Witten invariants of fibers of the Gross fibration, relative invariants, scattering diagrams and broken lines, tropical curves, superpotentials and wall crossing. This joint work with Tim Graefnitz and Helge Ruddat builds on works of Chan, Lau, Leung, Tseng; the wall crossing takes its cue from the work Gross, Pandharipande, Siebert. Lecture Notes.

September 20th, 2021

Renzo Cavalieri : The integral Chow ring of $M_{0,0}(\mathbb{P}^r,d)$

Abstract: We give an efficient presentation of the Chow ring with integral coefficients of the open part of the moduli space of rational maps of odd degree to projective space. A less fancy description of this space has its closed points correspond to equivalence classes of $(r+1)$-tuples of degree $d$ polynomials in one variable with no common positive degree factor. We identify this space as a $GL(2,\mathbb{C})$ quotient of an open set in a projective space, and then obtain a (highly redundant) presentation by considering an envelope of the complement. A combinatorial analyis then leads us to eliminating a large number of relations, and to express the remaining ones in generating function form for all dimensions. The upshot of this work is to observe the rich combinatorial structure contained in the Chow rings of these moduli spaces as the degree and the target dimension vary. This is joint work with Damiano Fulghesu.
Video.

October 4th, 2021

Richard Thomas : Higher rank DT theory from curve counting

Abstract: 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 express rank r DT invariants in terms of 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.
Video.

October 18th, 2021

Ben Davison : The decomposition theorem for 2-Calabi-Yau categories

Abstract: 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 $p:M\rightarrow N$ 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 $\mathbb{Q}_M$ isn't pure... As applications, I'll explain a proof of Halpern-Leistner's conjecture on the purity of stacks of coherent sheaves on K3 surfaces, and if time permits, a (partly conjectural) way to extend nonabelian Hodge theory to Betti/Dolbeault stacks.
Video.
Slides.

October 25th, 2021

Burt Totaro : Varieties of general type with doubly exponential asymptotics

Abstract: We construct smooth projective varieties of general type with the smallest known volume and others with the most known vanishing plurigenera in high dimensions. The optimal volume bound is expected to decay doubly exponentially with dimension, and our examples achieve this decay rate. We also consider the analogous questions for other types of varieties. For example, in every dimension we conjecture the terminal Fano variety of minimal volume, and the canonical Calabi-Yau variety of minimal volume. In each case, our examples exhibit doubly exponential behavior. (Joint work with Louis Esser and Chengxi Wang.)
Video.

November 1st, 2021

Dori Bejleri : Wall crossing for moduli of stable log varieties

Abstract: Stable log varieties or stable pairs $(X,D)$ are the higher dimensional generalization of pointed stable curves. They form proper moduli spaces which compactify the moduli space of normal crossings, or more generally klt, pairs. These stable pairs compactifications depend on a choice of parameters, namely the coefficients of the boundary divisor D. In this talk, after introducing the theory of stable log varieties, I will explain the wall-crossing behavior that governs how these compactifications change as one varies the coefficients. I will also discuss some examples and applications. This is joint work with Kenny Ascher, Giovanni Inchiostro, and Zsolt Patakfalvi.
Video.

November 8th, 2021

Aaron Pixton : Formulas for double-double ramification cycles

Abstract: The double-double ramification cycle parametrizes curves of genus g admitting two maps to the projective line with specified ramification profiles over two points. This cycle is easy to define in families of smooth curves, but much more subtle for general stable curves. I will discuss some special cases in which a formula for the double-double ramification cycle is now known. This is joint work with David Holmes, Sam Molcho, Rahul Pandharipande, and Johannes Schmitt.
Video.

November 22nd, 2021

Georg Oberdieck : From $K3$ surfaces to Hilbert schemes and back

Abstract: I will discuss the relationship between three different counting theories associated to a $K3$ surface $S$: (i) Gromov-Witten theory of the Hilbert scheme of points of $S$ with complex structure of the source curve fixed to be an elliptic curve $E$ with some rational tails, (ii) Donaldson-Thomas theory of $S\times E$, (iii) Virtual Euler characteristics of Quot schemes of stable sheaves on the $K3$ surfaces. This leads to new evaluations of these virtual Euler numbers, and to multiple cover formulas for all three theories.

December 13th, 2021

Samir Canning : The Chow rings of moduli spaces of elliptic surfaces

Abstract: For each nonnegative integer N, Miranda constructed a coarse moduli space of elliptic surfaces with section over the projective line with fundamental invariant N. I will explain how to compute the Chow rings of these moduli spaces when N is at least 2. The Chow rings exhibit many properties analogous to those expected for the tautological ring of the moduli space of curves: they satisfy analogues of Faber's conjectures, and they exhibit a stability property as N goes to infinity. When N=2, these elliptic surfaces are K3 surfaces polarized by a hyperbolic lattice. I will explain how the computation of the Chow ring confirms a special case of a conjecture of Oprea and Pandharipande on the structure of the tautological rings of moduli spaces of lattice polarized K3 surfaces. This is joint work with Bochao Kong.

January 10th, 2022

Borislav Mladenov : Degeneration of the Ext spectral sequence and formality of RHom DG algebras associated to holomorphic Lagrangian subvarieties

Abstract: Let $L$ be a complex Lagrangian in a holomorphic symplectic variety $X$. Consider the DG algebra $\operatorname{RHom}_X(\cal{L},\cal{L})$ a line bundle $\cal{L}$ on $L$. I will introduce the local-to-global spectral sequence computing its cohomology and describe the differential in terms of a deformation-obstruction class of Huybrechts-Thomas. I will then state the simplest versions of the degeneration and formality results and briefly explain the motivation coming from the work of Solomon-Verbitsky and Kapustin’s Seiberg-Witten duality. To conclude, I’ll give a quick reminder of Deligne’s degeneration of the Leray spectral sequence and explain how to adapt it to our setting. If time allows, I’d like to sketch a second proof via deformation quantisation without going into technicalities.

January 17th, 2022

Lothar Gottsche : Blowup formulas, strange duality and generating functions for Segre and Verlinde numbers of surfaces

Abstract:This is based in part on joint work with Martijn Kool, and is partially joint work in progress with Anton Mellit.
We formulate conjectural blowup formulas for Segre and Verlinde numbers of moduli spaces of sheaves on projective surfaces with $q=0$ and $p_g>0$. Combining these with a virtual version of the strange duality conjecture, we obtain a lot of information about the generating functions of these invariants, which allows to conjecturally determine them in many cases. In work in progress we give a complete conjectural determination of the Verlinde and Segre series of Hilbert schemes of points on surfaces, as well as a proof of this conjecture for surfaces with $K^2=0$.

January 24th, 2022

Dimitri Zvonkine : Gromov-Witten invariants of complete intersections

Abstract: We show that there is an effective way to compute all Gromov-Witten (GW) invariants of all complete intersections. The main tool is Jun Li's degeneration formula: it allows one to express GW invariants of a complete intersection from GW invariants of simpler complete intersections. The main difficulty is that, in general, the degeneration formula does not apply to primitive cohomology insertions. To circumvent this difficulty we introduce simple nodal GW invariants. These invariants do not involve primitive cohomology classes, but instead make use of imposed nodal degenerations of the source curve. Our work contains two main statements: (i) simple nodal GW invariants can be computed by the degeneration formula, (ii) simple nodal GW invariants determine all GW invariants of a complete intersection. The first statement is geometric; the second uses the invariance of GW invariants under monodromy and some representation theory. In this talk I will spend more time on part (ii) to complete more geometrically oriented talks by my co-authors: Hulya Arguz, Pierrick Bousseau and Rahul Pandharipande.

February 7th, 2022

Tudor Padurariu : Hall algebras in Donaldson-Thomas theory

Abstract: Kontsevich-Soibelman defined the cohomological Hall algebra (CoHA) of a quiver with potential. By a result of Davison-Meinhardt, CoHAs are deformations of the universal enveloping algebra of the BPS Lie algebra of the quiver with potential. My plan is to present the analogues stories in the categorical and K-theoretic contexts. I will introduce the categorical and K-theoretic Hall algebras of a quiver with potential and explain how to prove versions of the Davison-Meinhardt theorem in these contexts. These results have applications in categorical Donaldson-Thomas theory and in the study of Hall algebras of surfaces.
Video.

February 14th, 2022

Kai Behrend : Donaldson-Thomas theory of the quantum Fermat quintic

Abstract: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 mostly work of Yu-Hsiang Liu, with contributions by myself and Atsushi Kanazawa.

February 21st, 2022

Anders Buch : A Vafa-Intriligator formula for Fano varieties

Abstract:I will speak about a generalization of the Vafa-Intriligator formula that expresses the Gromov-Witten invariants of a Fano variety, for fixed markings and insertions of even degrees, in terms of eigenvalues of multiplication operators on the small quantum cohomology ring. This generalizes earlier known formulas for cominuscule flag varieties, including Grassmannians of type A and Grassmannians of maximal isotropic subspaces. In the Grassmannian cases, the relevant eigenvalues are explicitly known from the work of Rietsch and Cheong. The eigenvalues can also be described explicitly for Fano complete intersections. Applications include simple formulas for Tevelev degrees. This is joint work with Rahul Pandharipande.

February 28th, 2022 SPECIAL TIME: 8:45 am Pacific time / 5:45 pm Zurich time.

Tamas Hausel : Fixed point scheme as spectrum of equivariant cohomology and Kirillov algebras

Abstract: As an example of the multiplicity algebras of the Arnold school we will look at the equivariant cohomology of a Grassmannian and notice that it is inscribed in its regular fixed point scheme generalising observations of Brion-Carrell. In turn, we will describe it explicitely as the classical Kirillov algebra of a fundamental representation of $SL_n$, originally observed by Panyushev. These observations are motivated by mirror symmetry considerations originating in recent work with Hitchin on the mirror of very stable upward flows in the Hitchin system, which we will briefly indicate.

March 7th, 2022

Younghan Bae : Counting surfaces on Calabi-Yau fourfolds

Abstract: I will describe a sheaf theoretic way to enumerate surfaces on a smooth quasi-projective Calabi-Yau fourfold. By Borisov-Joyce/Oh-Thomas, a moduli space of sheaves or complexes on a Calabi-Yau fourfold has a virtual class. When we count two dimensional objects, this theory has to be modified to get nontrivial invariants. I will explain how to reduce the theory and deformation invariance along the Hodge loci. This can be related to the variational Hodge conjecture in some examples. Next I will explain two ways to compactly stable pairs, which we call PT0, PT1 pairs. Several conjectural correspondences between DT, PT0 and PT1 invariants will be discussed with evidences. This is a joint work in progress with Martijn Kool and Hyeonjun Park.

May 2nd, 2022

Arend Bayer : Kuznetsov categories of Fano threefolds

Abstract: I will give a survey of recent results on Kuznetsov categories of Fano threefolds of Picard rank one. These results give additional structure to their classification, and their moduli spaces. The techniques involved include moduli spaces of Bridgeland-stable objects, Brill-Noether statements, and equivariant categories (spiced with a pinch of derived algebraic geometry).

May 9th, 2022

Navid Nabijou : From orbifolds to logarithms via birational invariance

Abstract: Logarithmic and orbifold structures provide two different paths to the enumeration of curves with fixed tangencies to a normal crossings divisor. Simple examples demonstrate that the resulting systems of invariants differ, but a more structural explanation of this defect has remained elusive. I will discuss joint work with Luca Battistella and Dhruv Ranganathan, in which we identify birational invariance as the key property distinguishing the two theories. The logarithmic theory is stable under toroidal blowups of the target, while the orbifold theory is not. By identifying a suitable system of “slope-sensitive” blowups, we define a “limit orbifold theory” and prove that it coincides with the logarithmic theory. Our proof hinges on a technique – rank reduction – for reducing questions about normal crossings divisors to questions about smooth divisors, where the situation is much-better understood.

June 20th, 2022

Marc Levine : Equivariant localization in quadratic enumerative geometry

Abstract: We explain how to adapt the classical Atiyah-Bott torus localization methods to the computation of invariants, such as degrees of Euler classes, in the Grothendieck-Witt ring of quadratic forms, These quadratic enumerative invariants lift the usual integer-valued ones via the rank function, and for fields contained in the reals, the signature gives invariants for problems over the reals. Examples include quadratic counts of twisted cubic curves on hypersurfaces and complete intersections, this last is joint work with Sbarina Pauli.

September 26th, 2022

Jim Bryan : A theory of Gopakumar-Vafa invariants for orbifold Calabi-Yau threefolds

Abstract: 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.

Lecture Notes. Video.

October 3rd, 2022

Woonam Lim : Virasoro constraints in sheaf theory and vertex algebras

Abstract: In enumerative geometry, Virasoro constraints first appeared in the context of moduli of stable curves and maps. These constraints provide a rich set of conjectural relations among Gromov-Witten descendent invariants. Recently, the analogous constraints were formulated in several sheaf theoretic contexts; stable pairs on 3-folds, Hilbert scheme of points on surfaces, higher rank sheaves on surfaces with only (p,p)-cohomology. In joint work with A. Bojko, M. Moreira, we extend and reinterpret Virasoro constraints in sheaf theory using Joyce's vertex algebra. This new interpretation yields a proof of Virasoro constraints for curves and surfaces with only (p,p) cohomology by means of wall-crossing formulas.

Lecture Notes. Video.

October 24th, 2022

Dhruv Ranganathan : Logarithmic enumerative geometry for curves and sheaves

Abstract:I will discuss ongoing work with Davesh Maulik in which we formulate a generalization of the GW/DT conjectures to the setting of simple normal crossings pairs. When the divisor in the pair is smooth, the formulation of the conjecture necessitates the study of the cohomology of the Hilbert scheme of points on a surface. The formulation of the logarithmic GW/DT conjecture requires new geometry coming from a logarithmic Hilbert scheme of points. We prove a strengthened logarithmic degeneration formula on both sides of the correspondence and prove that the new conjectures are compatible with the old ones via degeneration. I’ll discuss this circle of ideas, and explain which parts of the conjectures are within reach.

Lecture Notes. Video.

November 7th, 2022

Davesh Maulik : The $P=W$ conjecture for $GL_n$

Abstract: The $P=W$ conjecture, first proposed by de Cataldo-Hausel-Migliorini in 2010, gives a link between the topology of the moduli space of Higgs bundles on a curve and the Hodge theory of the corresponding character variety, using non-abelian Hodge theory. In this talk, I will explain this circle of ideas and discuss a recent proof of the conjecture for $GL_n$ (joint with Junliang Shen).

Lecture Notes. Video.

November 21st, 2022

Zhiyu Liu :Castelnuovo bound and Gromov-Witten invariants of the quintic 3-fold

Abstract: One of the most challenging problems in geometry and physics is to compute higher genus Gromov-Witten invariants of compact Calabi-Yau 3-folds, such as the famous quintic 3-fold. I will briefly describe how physicists compute Gromov-Witten invariants of the quintic 3-fold up to genus 53, using five mathematical conjectures. Three of them have been already proved, and one of the remaining two conjectures has been solved in some genus. I will explain how to prove the last open one, called the Castelnuovo bound, which predicts the vanishing of Gopakumar-Vafa invariants for a given degree at sufficiently high genus. This talk is based on the joint work with Yongbin Ruan.

Lecture Notes. Video.

March 20th, 2023

Longting Wu :All-genus WDVV recursion, quivers, and BPS invariants.

Abstract: Let $D$ be a smooth rational ample divisor in a smooth projective surface $X$. In this talk, we will present a simple uniform recursive formula for (primary) Gromov-Witten invariants of $\mathcal{O}_X(-D)$. The recursive formula can be used to determine such invariants for all genera once some initial data is known. The proof relies on a correspondence between all-genus Gromov–Witten invariants and refined Donaldson–Thomas invariants of acyclic quivers. In particular, the corresponding BPS invariants are expressed in terms of Betti numbers of moduli spaces of quiver representations. This is a joint work with Pierrick Bousseau.

Lecture Notes. Video.

May 1st, 2023

Sam Payne :Cohomology groups of moduli spaces of curves

Abstract: Algebraic geometry endows the cohomology groups of moduli spaces of curves with additional structures, such as (mixed) Hodge structures and Galois representations. Standard conjectures from arithmetic, regarding analytic continuations of L-functions attached to these Galois representations, lead to striking predictions, by Chenevier and Lannes, about which such structures can appear. I will survey recent results unconditionally confirming several of these predictions and studying patterns in the appearances of motives of low weight. The latter are governed by the operadic structures induced by tautological morphisms and the cohomology of graph complexes.

Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.

Lecture Notes. Accompanying Tables. Video.

September 11th, 2023

John Pardon :Universally counting curves in Calabi--Yau threefolds

Abstract: Enumerating curves in algebraic varieties traditionally involves choosing a compactification of the space of smooth embedded curves in the variety. There are many such compactifications, hence many different enumerative invariants. I will propose a "universal" (very tautological) enumerative invariant which takes values in a certain Grothendieck group of 1-cycles. It is often the case with such "universal" constructions that the resulting Grothendieck group is essentially uncomputable. But in this case, the cluster formalism of Ionel and Parker shows that, in the case of threefolds with nef anticanonical bundle, this Grothendieck group is freely generated by local curves. This reduces the MNOP conjecture (in the case of nef anticanonical bundle and primary insertions) to the case of local curves, where it is already known due to work of Bryan--Pandharipande and Okounkov--Pandharipande.

Lecture Notes. Video.

October 2nd, 2023

Georg Oberdieck :Curve counting on the Enriques surface and the Klemm-Marino formula

Abstract: An Enriques surface is the quotient of a K3 surface by a fixed point-free involution. Klemm and Marino conjectured a formula expressing the Gromov-Witten invariants of the local Enriques surface in terms of automorphic forms. In particular, the generating series of elliptic curve counts on the Enriques should be the Fourier expansion of (a certain power of) Borcherds automorphic form on the moduli space of Enriques surfaces. In this talk I will explain a proof of this conjecture. The proof uses the geometry of the Enriques Calabi-Yau threefold in fiber classes. If time permits, I will also discuss various conjectures about non-fiber classes.

Lecture Notes. Video.

February 12, 2024

Felix Thimm : The 3-fold K-theoretic DT/PT vertex correspondence

Abstract: Donaldson-Thomas (DT) and Pandharipande-Thomas (PT) invariants are two curve counting invariants for 3-folds. In the Calabi-Yau case, a correspondence between the numerical DT and PT invariants has been conjectured by Pandharipande and Thomas and proven by Bridgeland and Toda using wall-crossing. For equivariant K-theoretically refined invariants, the DT/PT correspondence reduces to a DT/PT correspondence of equivariant K-theoretic vertices. In this talk I will explain our proof of the equivariant K-theoretic DT/PT vertex correspondence using a K-theoretic version of Joyce's wall-crossing setup. An important technical tool is the construction of a symmetized pullback of a symmetric perfect obstruction theory on the orginial DT and PT moduli stacks to a symmetric almost perfect obstruction theory on auxiliary moduli stacks. This is joint work with Henry Liu and Nick Kuhn.

Lecture Notes. Video.

Monday,April 15th, 2024

Miguel Moreira : The cohomology ring of moduli spaces of 1-dimensional sheaves on $\mathbb{P}^2$

Abstract: The cohomology of moduli spaces of 1-dimensional sheaves, together with a special filtration called the perverse filtration, can be used to give an intrinsic definition of (refined) Gopakumar-Vafa invariants. While there are methods to calculate the Betti numbers of these moduli spaces in low degree, understanding the perverse filtration is more challenging. One way to compute it is to fully determine the cohomology ring. In this talk I will explain an approach to describing the cohomology rings of moduli spaces and moduli stacks in terms of generators and relations, which allowed us to determine them for curve classes up to degree 5 (for moduli spaces) and 4 (for moduli stacks). I will explain the $P=C$ conjecture, which is an analogue of the $P=W$ conjecture in a Fano and compact setting. This is joint work with Yakov Kononov, Woonam Lim and Weite Pi.

Lecture Notes. Video.

Monday, May 13th, 2024

Ian Cavey :Verlinde Series for Hirzebruch Surfaces

Abstract: Verlinde series are generating functions of Euler characteristics of line bundles on the Hilbert schemes of points on a surface. Formulas for Verlinde series were determined for surfaces with $K=0$ by Ellingsrud, Göttsche, and Lehn. More recently, Göttsche and Mellit determined Verlinde series for surfaces with $K^2=0$, and gave a conjectural formula in the general case. In this talk, I will give a formula for the Euler characteristics of line bundles on the Hilbert schemes of points on $\mathbb{P}^1 \times \mathbb{P}^1$, and a combinatorial (but less explicit) formula for ample line bundles on the Hilbert schemes of points on Hirzebruch surfaces. By structural results of Ellingsrud, Göttsche, and Lehn, this determines the Verlinde series for all surfaces. The proof is based on a new combinatorial description of the equivariant Verlinde series for the affine plane.

Lecture Notes. Video.