Week 2 abstracts
Dan Abramovich: Artin fans
Artin fans are 0-dimensional algebraic stacks which encode the combinatorial structure of a subvariety of a toric variety, or more generally of a logarithmic structure. They are closely related to Olsson's stack of logarithmic structure. For tropical geometry, Artin fans have the appealing feature that "superabundance" disappears. I will explain how Artin fans were used in work of Chen, Gross, Marcus, Ranganathan, Siebert, Ulirsch and Wise.
Nick Addington: Exoflops
Consider a contraction pi: X -> Y from a smooth Calabi-Yau 3-fold to a singular one. (This is half of an "extremal transition;" the other half would be a smoothing of Y.) In many examples there is an intermediate object called an "exoflop" -- a category of matrix factorizations, derived-equivalent to X, where the critical locus of the superpotential looks like Y with a P^1 sticking out of it, and objects of D(X) that will be killed by pi_* correspond to objects supported at the far end of the P^1. I will discuss one or two interesting examples.
This is joint work with Paul Aspinwall.
Mina Aganagic: Instant counting, W-algebras and the little string
I will describe the relation between K-theoretic instanton counting for ADE type quiver gauge theories and q,t-deformed W-algebras of ADE type. I will also explain the origin of the correspondence from physics perspective: both describe a 6d string theory ("the little string”) compactified on a Riemann surface. The little string contains the theory X, the mysterious 6d QFT that has become omnipresent in mathematical physics. The bigger theory can be easier to understand, as this example shows. (This talk is based on http://arxiv.org/pdf/1506.04183.pdf.)
Omid Amini: Limit linear series and distribution of Weierstrass points
I will report on recent progress in constructing a general framework for the study of degenerations of linear series on degenerating families of smooth proper curves over a field of characteristic zero, generalizing the Eisenbud-Harris theory of limit linear series from the eighties to any semistable curve. I will then discuss an application to the problem of understanding the limiting behavior of Weierstrass points on such families. This leads to the following non-Archimedean version of a theorem of Mumford and Neeman: let X be a smooth proper curve over a non-Archimedean field of residue characteristic zero, and L an ample line bundle on X. The Weierstrass points of powers of L are equidistributed according to the Zhang measure on the dual graph of a semistable model.
The talk is partially based on joint works with M. Baker and with E. Esteves.
Dima Arinkin: Moduli of regular connections on the punctured disk
Connections with regular singularities are natural and important objects. From a classical (that is, analytic) point of view, regular connections on the punctured disk are easily classified: their isomorphism classes are given by their monodromy. The situation becomes much more interesting for algebraic families of connections, because monodromy is not an algebraic function.
The goal of my talk is to study (and to define) the moduli `space' of connections with regular singularities on the punctured disk and to see the non-trivial algebraic counterparts of classical analytic statements.
Matt Ballard: Orlov spectra in algebraic geometry and beyond
Here is a basic question. Take your favorite finite set A of n \times n matrices over \C. Call this an alphabet if every matrix can be written as a linear combination of products (words) in A. How long is the longest word? How about if we take the maximum over all A?
The Orlov spectrum of a triangulated category captures exactly this data when we use cones for products. It is notoriously difficult to compute thanks to a failure of additivity but existing results offer a testament to its appeal.
In this talk, I will introduce the Orlov spectrum, discuss some examples and conjectures. Themes to be touched on include: rationality, Hall algebras, and braid groups. Some results are joint with David Favero and Ludmil Katzarkov.
Arend Bayer: Stability and wall-crossing: applications to classical algebraic geometry
This talk will be a survey of applications of Bridgeland stability conditions and wall-crossing on surfaces, in particular K3 surfaces. Most of these application concern questions of classical flavour, for example birational geometry of moduli spaces. I will then explain the mechanism of such applications by way of reproving well-known results, such as the Brill-Noether theorem for curves on K3 surfaces.
Marcello Bernardara: Homological projective duality for determinantal varieties
In this talk I will present joint results with M.Bolognesi and D.Faenzi. Using homological projective duality for projective bundles and categorical resolution of singularities constructed by Buchweitz, Leuschke and Van den Bergh, we show homological projective duality for generalized determinantal varieties. As an application, we find derived equivalent (and birational) CY-manifolds and we show that the following question of A.Bondal: "for any variety X, does it exist a Fano variety Y and a full and faithful fucntor D(X) \to D(Y)?" has a positive answer in many cases, for example when X is a smooth plane curve.
Roman Bezrukavnikov: Geometric Langlands and Bridgeland stabilities
I will describe results and conjectures on interaction of the geometric Langlands type equivalences with natural
t-structures on the triangulated categories in question and an observed relation of those t-structures to Bridgeland stability conditions.
Lev Borisov: Equality of stringy E-functions of Pfaffian double mirrors and related results
I will talk about joint work with Anatoly Libgober on proving equality of stringy Hodge numbers of double mirror complete intersections in generalized Pfaffian varieties. In particular, I will explain the geometric construction involved in the proof that the class of affine line is a zero divisor in the Grothendieck ring of complex algebraic varieties.
Tom Bridgeland: Stability and wall-crossing
Over the last ten years we have learnt how to use our understanding of wall-crossing phenomena in derived categories of Calabi-Yau threefolds to prove results about generating functions for Donaldson-Thomas (DT) invariants. Examples include Toda's formula describing the effect of a flop on the DT curve-counting invariants, and the DT/PT correspondence between reduced DT curve-counting invariants and the stable pair invariants of Pandharipande-Thomas. The crucial ingredient in these developments is the theory of motivic Hall algebras, as developed by Joyce and Kontsevich-Soibelman. In the first lecture I will give an introduction to Hall algebras. The second lecture will discuss the results mentioned above together with the Kontsevich-Soibelman wall-crossing formula. The last lecture will discuss some conjectural relations between stability conditions and cluster varieties, inspired by the work of Gaiotto-Moore-Neitzke.
Jim Bryan: Elliptically fibered Calabi-Yau threefolds, Jacobi-Forms, and the topological vertex
Several recent conjectures suggest that curve counting partition functions for elliptically fibered Calabi-Yau threefolds are governed by Jacobi forms. We survey the various conjectures and describe recent progress in Donaldson-Thomas theory which has led to proofs of some of the conjectures. Namely, by combining motivic and toric techniques, we compute the partition functions for these geometries in terms of the topological vertex. Unexpected connections between the topological vertex and Jacobi forms arise.
Andrei Căldăraru: Algebraic proofs of degenerations of Hodge-de Rham complexes
I shall present a new perspective relating proofs of degenerations of Hodge-de Rham spectral sequences in characteristic p (Deligne-Illusie, Ogus-Vologodsky) to problems of formality of derived intersections in Azumaya spaces. This is joint work with Dima Arinkin and Marton Hablicsek.
Lucia Caporaso: Degenerations of line bundles on algebraic curves: new methods and results
It has been well known for a long time that degenerations of line bundles when smooth curves specialize to singular ones can be very hard to describe. The talk will describe new methods and results inspired by tropical geometry, and describe a selection of open problems.
Dustin Cartwright: Combinatorial tropical surfaces
A tropical complex is a Delta-complex, together with some additional integral data. I will talk about tropical surfaces, meaning 2-dimensional tropical complexes, and their combinatorial properties which are analogs of basic results on algebraic surfaces. In particular, on a tropical surface, the Picard group modulo the connected component of zero is finitely generated and has a non-degenerate intersection pairing satisfying a Hodge index theorem.
Melody Chan: Topology of the tropical moduli spaces of curves
The moduli space of n-marked, genus g tropical curves is a cell complex that was identified in work of Abramovich-Caporaso-Payne with the boundary complex of the complex moduli space M_{g,n}. I will give results on the topology of tropical M_{1,n}, joint with Galatius and Payne, and on tropical M_{2,n}, obtaining as corollaries new calculations on the top-weight cohomology of the complex moduli spaces M_{1,n} and M_{2,n}.
Giulio Codogni: Schottky problem, quadratic forms and Satake compactifications
We prove some results about the singularities of Satake compactifications of classical moduli spaces; this will give an insight into the relation among solutions of the Schottky problem in different genera. In the case of the moduli space of curves, we show that there are no stable solutions of the Schottky problem. In particular, given two inequivalent positive even unimodular quadratic forms, the difference of the associated theta series does not vanish on the Torelli locus when the genus is big enough; we are able to give an effective bound on the genus just in the rank 24 case. This is joint work with N. Shepherd-Barron.
María Angélica Cueto: Repairing tropical curves by means of linear tropical modifications
Tropical geometry is a piecewise-linear shadow of algebraic geometry that preserves important geometric invariants. Often, we can derive classical statements from these (easier) combinatorial objects. One general difficulty in this approach is that tropicalization strongly depends on the embedding of the algebraic variety. Thus, the task of funding a suitable embedding or of repairing a given "bad" embedding to obtain a nicer tropicalization that better reflects the geometry of the input object becomes essential for many applications. In this talk, I will show how to use linear tropical modifications and Berkovich skeleta to achieve such goal in the curve case. Our motivating example will be plane elliptic cubics defined over a non-Archimedean valued field. This is joint work with Hannah Markwig.
Olivia Dumitrescu: From Cellular Graphs to TQFT
We present a new axiomatic formulation of a 2D TQFT. Our formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). We assign to each cellular graph an element of a symmetric tensor algebra of a Frobenius algebra. We show a surprising result that the edge-contraction axioms make this assignment graph independent, and that it is equivalent to the TQFT corresponding to the Frobenius algebra. The edge-contraction operations are used in enumeration problems, such as Hurwitz numbers and lattice points on the moduli space of curves. Examples of our theory contain the Gromov-Witten and Hurwitz theory of the classifying space of a finite group. This is joint work with Motohico Mulase.
Carel Faber: Teichmüller modular forms
Vector valued Siegel modular forms may be viewed as sections on a toroidal compactification of A_g of the bundles obtained by applying a Schur functor for GL(g) to the Hodge bundle. Similarly, Teichmüller modular forms are sections on M_g or its Deligne-Mumford compactification of the pullbacks of those bundles via the Torelli morphism. I will first recall several results of Ichikawa on scalar valued Teichmüller modular forms, of genus three especially. Then I will report how joint work with Bergström and Van der Geer indicates the existence of many vector valued Teichmüller modular forms of genus three.
John Francis: Poincaré/Koszul duality
I'll describe a duality between factorization homology of formal moduli problems and factorization homology of augmented n-disk algebras. The terms in this duality expression specialize to Hochschild homology, Poincaré duality for manifolds, and Koszul duality for associative algebras. I'll interpret this result in terms of quantization of topological sigma models.
Mark Gross: Mirror Symmetry
In 2001, Bernd Siebert and I began a program to understand mirror symmetry by constructing mirrors of Calabi-Yau varieties as degenerations, describing the degenerations via explicit smoothings of the central fibre encoded in terms of tropical geometry. These constructions have led, at this point in time, to at least an intuitive understanding of how a mirror partner encodes information about rational curves on the original geometric object. I will attempt to explain this intuition at an elementary level. Further, I will touch on current ideas involving theta functions (a generalization of theta functions on abelian varieties, currently being developed with Hacking, Keel and Siebert) and a logarithmic version of Gromov-Witten invariants, (whose theory is being developed with Abramovich, Chen and Siebert). These ideas should lead to making this intuition rigorous.
Walter Gubler: Skeletons and Tropicalizations
A generalization of Berkovich's skeletons will be introduced which is suitable for comparing with tropicalizations. In fact, we will see that such a skeleton can be faithfully embedded into a suitable tropicalization. In this way, we can compare arithmetic and geometric information of the underlying algebraic variety. A rational function induces a piecewise linear function on the skeleton which satises a slope formula. The latter is a non-archimedean analogue of the Poincaré-Lelong formula in complex analysis. This is joint work with Joseph Rabinoff and Annette Werner.
Jack Hall: Coherent Tannaka duality
I will describe a version of Tannaka duality for algebraic stacks with affine stabilizers. Then I will touch on some applications to algebraization questions. This is joint work with David Rydh (KTH).
Dan Halpern-Leistner: \Theta-reductive moduli problems, stratifications, and applications
I will discuss a structural framework for moduli problems which generalizes structures appearing in geometric invariant theory and in the moduli of coherent sheaves (as well as moduli of objects in derived categories). The idea is that for a locally finite type algebraic stack, a "solution" to the corresponding moduli problem should come in the form of a stratification which admits a nice modular interpretation, where the big open stratum admits a good moduli space and the remaining strata fiber over moduli problems which also admit good moduli spaces. For a class of moduli probems which we call \Theta-reductive, such stratifications can be described by specifying certain cohomology classes on the stack. Along the way I will discuss a new structure parameterizing degenerations of points in a stack which generalizes the fan of a toric variety and the spherical building of a semi-simple group. I will also discuss applications of these stratifications to derived categories of coherent sheaves and virtual localization formulas.
Tamás Hausel: Arithmetic of wild character varieties
We start with some conjectures, originating both in arithmetic and physics, on the mixed Hodge polynomials and perverse Hodge polynomials of tame character varieties and moduli of parabolic Higgs bundles on Riemann surfaces. We will then study the arithmetic of one class of Boalch's wild character varieties using the character theory of Yokonuma-Hecke algebras yielding a conjecture on their mixed Hodge polynomials. We will comment on how refined link invariants show up in our formulas, and also how the pure part is related to the t-adic geometry of some quiver varieties. These are joint works with Martin Mereb, Michael Wong and Dimitri Wyss.
Hiroshi Iritani: Constructing mirrors via shift operators
In this talk, I will explain a mirror construction for big equivariant quantum cohomology of toric varieties via shift operators of equivariant parameters. Shift operators in equivariant quantum cohomology have been introduced in the work of Braverman, Maulik, Okounkov and Pandharipande and can be viewed as an equivariant lift of the Seidel representation. These operators naturally define a mirror Landau-Ginzburg potential together with a primitive form and yield an almost tautological proof of toric mirror symmetry. They are also closely related to the Gamma structure in quantum cohomology.
David Jensen: Tropical Independence and the Maximal Rank Conjecture for Quadrics
The maximal rank conjecture, which has roots in the work of Noether and Severi in the late 19th and early 20th centuries, predicts the Hilbert function of the general embedding of a general curve. In recent joint work with Sam Payne, we show that this conjecture holds for the Hilbert function evaluated at m=2, meaning that such a curve is contained in the expected number of independent quadrics. From this we deduce that the general curve of genus g and degree d in projective space of dimension r is projectively normal if and only if (r+2)(r+1)/2 is at least 2d-g+1. Our proof uses techniques from tropical and nonarchimedean geometry.
Yunfeng Jiang: Quantum cohomology of hypertoric DM stacks and Monodromy conjecture
We calculate the equivariant quantum cohomology of hypertoric DM stacks defined by Jiang and Tseng. We relate it to the monodromy conjecture of Braverman-Maulik-Okounkov in the case of symplectic resolution of hypertoric DM stacks and argue how the Crepant Transformation Conjecture may imply the Monodromy conjecture. This is a joint wok with H.-H. Tseng.
Jesse Kass: What is the universal theta divisor, really?
The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. A basic problem is to understand how these objects extend over the compact moduli space of stable curves. In this talk, I will show how to construct extensions of the universal Jacobian using a stability parameter and then prove a wall-crossing formula describing how the class of the theta divisor depends on that parameter. Time permitting, I will discuss some relations to work by Samuel Grushevsky, Richard Hain, Fabian Müller, and Dmitry Zakharov. This work is joint with Nicola Pagani.
Sheldon Katz: BPS invariants of elliptically fibered Calabi-Yau threefolds and Jacobi forms
Inspired by string theory, we conjecture, with evidence, that a generating function of BPS invariants of elliptically fibered Calabi-Yau threefolds is a Jacobi form of a precise type, rendering it exactly computable given only a finite amount of algebro-geometric data. This talk is based on joint work with Minxin Huang and Albrecht Klemm.
Maxim Kontsevich: Mirror symmetry: new definitions
I'll review some recent advances in (homological) mirror symmetry:
a) an approach to the definition of Fukaya categories via singular Lagrangians,
b) mirrors to Calabi-Yau varieties over nonarchimedean fields, relation to canonical bases and mutations.
Maxim Kontsevich: Iterated stability
I will describe a general hypothetical procedure producing Bridgeland stability structures on triangulated categories in an iterative manner (in a sense by induction in dimension). As an example, one obtains a complete description of semistable objects on the product of an arbitrary number of generic elliptic curves.
The idea is motivated by the consideration of Fukaya categories for collapsing Kahler manifolds.
Kobi Kremnizer: Towards a p-adic Riemann-Hilbert correspondence
I will describe a new approach to analytic geometry in the framework of relative algebraic geometry. Using this approach I will give a definition of infinite order differential operators on an analytic manifold. I will then describe work in progress to show that the solution functor into topological sheaves is fully faithful (which is known in the Archimedean case by work of Prosmans and Scneiders). This is joint work with Oren Ben-Bassat.
Eric Larson: Interpolation for curves in projective space
We aim to address the following: When is there a (smooth) curve of degree d and genus g passing through n general points in P^r? We will start by relating these statements to a property of the normal bundle of curves in projective space, and look at several important examples. Time permitting, we will sketch some of the ingredients that go into resolving this problem to arbitrary nonspecial curves (the range when is d greater or equal than g + r). This is joint work with Atanas Atanasov and David Yang.
Y.P. Lee: Birational Transformation and degeneration in Gromov--Witten theory
The functoriality of Gromov--Witten theory under flops, flips, blowups and transitions (degenerations followed by small resolutions) will be discussed to various degrees of depth. This talk is based on a (long-term) joint project with Hui-Wen Lin and Chin-Lung Wang (TIMS and National Taiwan University).
Naichung Conan Leung: Witten deformation and scattering diagram in A-model
We discuss Witten deformation for A-infinity structure on deRham complex and explain an application to scattering diagram in quantum deformation of symplectic structures.
Jun Li: Mixed-Spin-P fields and algorithm to evaluate GW and FJRW invariants of quintic CY manifolds
The theory of Mixed-Spin-P fields is a geometric theory realizing the wall crossing between GW and FJRW invariants of the Fermat quintic. Using cosection localized virtual cycles, MSP invariants are defined, along with a collection of vanishings.
Using virtual localization, these collection provides polynomial relations among all genus GW invariants of the quintic CY threefolds and the FJRW invariants of the Fermat quintic. These relations are conjectured to determine the GW invariants of the quintics and the FJRW of the Fermat quintic. This is a joint work with HL Chang, WP Li and CC Liu.
Melissa Liu: On the remodeling conjecture for toric Calabi-Yau 3-orbifolds
The remodeling conjecture proposed by Bouchard-Klemm-Marino-Pasquetti relates Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-orbifold to Eynard-Orantin invariants of the mirror curve of the toric Calabi-Yau 3-fold.
It can be viewed as a version of all genus open-closed mirror symmetry. In this talk, I will describe results on this conjecture based on joint work with Bohan Fang and Zhengyu Zong.
Jason Lo: t-structures on elliptic fibrations
Given an elliptic fibration that comes with a "dual" fibration, we can construct various t-structures on the derived category of coherent sheaves on the total space, using the geometry of the fibration structure as well as the associated Fourier-Mukai transform. In this talk, I will describe how, by understanding these t-structures, we can understand better moduli spaces of sheaves (and perhaps complexes) on the total space.
Jacob Lurie: Cohomology Theories and Commutative Rings
One of the primary goals of algebraic topology is to obtain information about topological spaces X by studying algebraic invariants of X, such as the cohomology H^*(X; R) with coefficients in a commutative ring R. For many applications, it is useful to consider more exotic invariants given by "extraordinary" cohomology theories (such as K-theory). In this talk, I'll give an informal introduction to extraordinary cohomology theories, emphasizing their algebraic role as generalized commutative rings.
Jacob Lurie: Representation Theory in Intermediate Characteristic
Let G be a finite group. One can study representations of G over any field k: that is, vector spaces over k equipped with an action of G. In general, such representations behave very differently in characteristic zero (where all representations are completely reducible) and in characteristic p (where, if G is a p-group, there are no irreducible representations other than the trivial representation). In these talks, I will discuss representation theory over more exotic "fields" known as Morava K-theories, which in some sense interpolate between fields of characteristic zero and fields of characteristic p, and share many pleasant features of both.
Jacob Lurie: Roots of Unity in Intermediate Characteristic
In classical algebraic geometry, there is often a stark difference between the behavior of fields of characteristic zero (such as the complex numbers) and fields of characteristic p (such as finite fields). For example, the equation x^p = 1 has p distinct solutions over the field of complex numbers, but only one solution over any field of characteristic p. In this talk, I'll give an introduction to K(n)-local homotopy theory, which in some sense interpolates between characteristic zero and characteristic p, and describe some curious behavior of roots of unity in these intermediate regimes.
Emanuele Macrì: Bridgeland stability conditions on higher dimensional varieties
I will present recent results and work in progress on the construction of Bridgeland stability conditions on higher dimensional varieties.
The key ingredient will be an analogue of the Bogomolov inequality for stable sheaves.
This is based on joint works with Bayer, Bertram, Schmidt, Stellari, and Toda.
Travis Mandel: Tropical curve counting and canonical bases
Gross, Hacking, Keel, and Kontsevich recently constructed certain canonical bases for cluster algebras. The construction is combinatoric, but the bases are conjecturally controlled by the Gromov-Witten theory of the mirror cluster variety. I will discuss a new construction of these bases in terms of certain tropical curve counts which one expects to correspond to the predicted holomorphic curve counts. I will also discuss a refinement of the tropical counts which produces quantized versions of the canonical bases.
Eyal Markman: A survey of hyperholomorphic bundles in hyperkahler geometry
We will survey the role hyperholomorphic vector bundles and sheaves play in the proof of the following three results:
1) The proof of the Shafarevich conjecture (by N. Buskin, extending work of S. Mukai) that every rational isometry between two K3 surfaces is algebraic.
2) Proof of the standard conjectures for projective hyperkahler varieties of K3^[n]-type (joint with F. Charles).
3) The geometric construction of global non-commutative and gerby deformations of the derived categories of K3 surfaces (joint with S. Mehrotra).
Hannah Markwig: Tropicalizing rational relative Gromov-Witten theory of P^1
We show that the relative stable map compactification of M_0,n (for maps to P^1, relative to two points) is a tropical compactification. Furthermore, the tropicalization of the open part equals the tropical space of relative stable maps to P^1. Consequently, the Chow ring of the relative stable map space can be computed by means of tropical intersection theory in an intuitive way. The correspondence theorem for Hurwitz numbers, and more generally, genus 0 relative Gromov-Witten invariants, is now an easy corollary of this strong connection on the level of moduli spaces.
This is joint work with Renzo Cavalieri and Dhruv Ranganathan.
Cristian Martinez: Change of polarization for moduli spaces of sheaves as Bridgeland wall-crossing
The notion of stability for pure sheaves on a smooth projective complex surface S depends on the choice of an ample class H. The moduli spaces M_H(v) of H-Gieseker semistable sheaves with Chern character v are projective and can be constructed via GIT. There is a wall and chamber decomposition of the ample cone of the surface such that moduli spaces are isomorphic for polarizations in the same chamber. It is well known (in the case of 2-dimensional sheaves) that when crossing a wall, the moduli spaces M_H(v) pass through a sequence of Thaddeus flips in the category of moduli spaces of twisted sheaves. We give an interpretation of this result in terms of stability conditions. Indeed, every wall in Amp(S) corresponds to a finite sequence of Bridgeland walls, each producing a single Thaddeus flip of the corresponding moduli space. As a consequence of our construction, we find polyhedral regions in the ample cone of the moduli spaces, parametrized by ample divisors on S. This is joint work with Aaron Bertram.
Dave Morrison: Periods, Gromov-Witten invariants, and the Mukai pairing
The Gromov-Witten invariants of a Calabi-Yau manifold can famously be predicted by computing periods of the mirror manifold. The author interpreted this computation long ago in terms of a limiting mixed Hodge structure for the mirror family. But the periods contain more data related to zeta values, and it has only recently been realized that as a result, the mixed Hodge structure fails to spli over the integers.
We will explain the role that zeta values play in extracting Gromov- Witten predictions from periods. The key tools are the ""gamma class"" discovered by Libgober and rediscovered by Iritani and by Katzarkov-Kontsevich-Pantev, and a modification of the Mukai isometry relating the Mukai pairing to the ""vertical"" cohomology.
Based on joint work with Halverson, Kumar, Jockers, Lapan, and Romo.
David Nadler: Singular Lagrangians
Singular Lagrangians arise as the classical support of objects in mirror symmetry and sheaf theory. We'll present a solution to the analogy
smooth function : Morse function :: singular Lagrangian : ???
We'll introduce a class of elementary Lagrangian singularities, called arboreal singularities, and show that every Lagrangian singularity admits a deformation to a singular Lagrangian with arboreal singularities. Time permitting, we'll explain how to use this to calculate microlocal sheaves in parallel to the Morse calculation of cohomology.
Tom Nevins: D-modules on stacks from the GIT point of view
Equivariant D-modules, and more general D-modules on algebraic stacks, play many roles in algebraic geometry and geometric representation theory. I will explain how ideas and tools from geometric invariant theory can shed light on vanishing theorems, the derived category of D-modules on a stack, and the topology of GIT quotients. The talk is based on joint works with G. Bellamy, C. Dodd, and K. McGerty.
Bao Châu Ngô: Singularities in formal arc spaces and harmonic analysis over non-archimedean fields
Singularities in formal arc spaces may be relevant in formulating answers to certain important problems in harmonic analysis. I will discuss a sample of such problems.
Johannes Nicaise: Refined curve counting and Hrushovski-Kazhdan motivic integration
Motivated by mathematical physics, Block and Göttsche have defined "quantized" versions of Mikhalkin's multiplicities for tropical curves. In joint work with Sam Payne and Franziska Schroeter, we propose a geometric interpretation of these invariants as chi_y genera of semi-algebraic analytic domains over the field of Puiseux series. In order to define and compute these chi_y genera, we use the theory of motivic integration developed by Hrushovski and Kazhdan and we explore its connections with tropical geometry.
Andrei Okounkov: Enumerative geometry and representation theory
My goal in this lectures is to explain how geometric representation theory of certain interesting infinite-dimensional algebras interacts with Donaldson-Thomas theory, which is an enumerative theory of sheaves on a nonsingular threefold X. Enumerative information may be collected in many different formats, and our focus will be on computations in equivariant K-theory, with equivariant
cohomology providing a blueprint. In the first lecture, my plan is to motivate the interest in K-theoretic enumerative geometry by discussing a conjectural relation between K-theoretic DT counts to counts of curves in certain associated Calabi-Yau 5-folds, as well as certain remarkable dualities that K-theoretic counts are expected to satisfy. In the second lecture, I will review the structure of DT theory and relate its building blocks to representation theory of certain infinite-dimensional quantum groups related to double loop algebras. The third lecture will be about the geometry behind the construction of these quantum groups. The central notion here will be the notion of a stable envelope in equivariant cohomology, equivariant K-theory, and, eventually, equivariant elliptic cohomology. My lectures will be based on joint work with Mina Aganagic, Roman Bezrukavnikov, Davesh Maulik, Nikita Nekrasov, and Andrei Smirnov.
Rahul Pandharipande: Cycles on the moduli space of curves
I will discuss recent progress in the study of the tautological ring of the moduli space of curves: Pixton's relations, r-spin curves, double ramification cycles, and the Chern characters of the Verlinde bundle. The associated formulas, related to the Givental-Teleman classification of semisimple CohFTs, point to a new calculus of the moduli of curves.
Tony Pantev: Shifted deformation quantization
I will review shifted symplectic and Poisson structures in derived geometry and will explain how these structures can be constructed on moduli stacks. I will discuss several explicit examples and will show how formal geometry helps in relating non-degenerate shifted Poisson structures and shifted symplectic structures. I will review the quantization problem in derived geometry and will explain how shifted symplectic structures with non-zero shifts can always be quantized.
Sam Payne: Tropical methods in Brill-Noether theory
Riemann-Roch theory for graphs, as developed in the work of Baker—Norine, points to the existence of deep combinatorial structures related to the classical algebraic geometry of linear series on algebraic curves. In this talk I will present joint work with Cools—Draisma—Robeva, and with Jensen, pursuing this line of reasoning to give tropical proofs of the Brill—Noether and Gieseker—Petri theorems, which describe the dimension and local structure of the moduli spaces parametrizing linear series of given degree and rank on a general curve.
Aaron Pixton: Ranks of tautological rings
The tautological ring of the moduli space of smooth curves of genus g is the subring of its Chow ring generated by the kappa classes. There are two conjectures giving conflicting descriptions of the structure of this ring. I will review these two conjectures and then discuss some interesting features of the ranks predicted by each of them.
J.P. Pridham: A concrete approach to higher and derived stacks
Although it is conventional nowadays to describe derived algebraic geometry in terms of infinity-sheaves, there is a far more down-to-earth definition, originally due to Grothendieck. Derived algebraic n-stacks can be given by diagrams of affine schemes, like Cech complexes for schemes. I will explain this approach and several results which it motivates.
Joe Rabinoff: Uniform bounds on rational points via p-adic integration and Berkovich skeletons
The Mordell conjecture, famously proved by Faltings in 1983, states that a Q-curve X of genus g >= 2 has finitely many rational points. Recently, Stoll proved that the number of rational points on a hyperelliptic curve of Mordell--Weil rank at most g-3 is bounded by a number depending only on the genus. He used a Chabauty--Coleman integration argument as applied to a decomposition of X into open discs and annuli. We extend Stoll's methods, reformulating the problem in terms of skeletons of Berkovich curves and using the Baker--Norine theory of linear systems on the resulting metric graphs. We obtain a uniform bound on the number of rational points on any curve of Mordell--Weil rank at most g-3. Importantly, our methods also allow one to bound geometric torsion points lying on a curve, giving a uniform Manin-Mumford type result for curves of certain highly degenerate reduction types. This work is joint with Eric Katz and David Zureick-Brown.
Nick Rozenblyum: Algebro-geometric aspects of higher quantization
A basic example of quantization is the ring of algebraic differential operators on a
variety as a deformation of functions on the cotangent bundle. From a certain point of
view, this is an example of one-dimensional quantization. I will describe, in geometric
terms, higher (and lower) dimensional versions of such a construction. In the two
dimensional case, the story is closely related to the theory of vertex algebras and has
interesting representation theoretic applications. I will also describe some potential
geometric applications of these ideas.
Yongbin Ruan: A mathematical theory of gauged linear sigma model (GLSM)
GLSM was a physical model proposed by Witten in early 90's to give a physical derivation of Landau-Ginzburg (LG) /Calabi-Yau (CY) correspondence. Since then, it found application in many area of geometry and physics such as mirror symmetry. In the talk, I will describe a mathematical theory of gauged linear sigma model by constructing a variety of new moduli spaces and their virtual cycles.
The construction is purely algebraic and applies to so called the sector of compact type. Then, Witten's physical derivation of LG/CY correspondence can be interpreted as a series of wall-crossing formula of GLSM to connect FJRW-theory to GW-theory. If the time is permitted, other applications will be touched. This is a joint work with Huijun Fan and Tyler Jarvis.
Helge Ruddat: Canonical Calabi-Yau families
Starting with combinatorial degeneration data, Gross and Siebert found a canonical formal smoothing of the associated degenerate Calabi-Yau space. Siebert and I compute certain period integrals for these families and show that the mirror map is trivial. As a consequence, the formal families lift to analytic families. We introduce tropical 1-cycles that we turn in to ordinary n-cycles whose periods have a simple log pole. We show that such cycles generate the dual of the tangent space to the CY moduli space.
David Rydh: Local structure of Artin stacks
A point on a Deligne–Mumford stack has an étale neighborhood that is the quotient of a scheme by the stabilizer group. I will present a similar étale-local description of Artin stacks at points with linearly reductive stabilizer.
The proof uses Tannaka duality, complete stacks and equivariant Artin algebraization. Applications include generalizations of classical theorems by Luna and Sumihiro and results on actions by tori on Deligne–Mumford stacks. This is joint work with Jarod Alper (ANU) and Jack Hall (ANU).
Giulia Saccà: Symplectic singularities and quiver varieties
The aim of the talk is to study a class of singularities of moduli spaces of sheaves on K3 surfaces by means of Nakajima quiver varieties. The singularities in question arise from the choice of a non generic polarization, with respect to which we consider stability, and admit natural symplectic resolutions corresponding to choices of general polarizations. By considering appropriate Fourier-Mukai transforms, we show that these moduli spaces are, locally around a singular point, isomorphic to a quiver variety in the sense of Nakajima and that, via this isomorphism, the natural symplectic
resolutions correspond to variations of GIT quotients of the quiver varieties. This is joint work with E. Arbarello.
Vivek Shende: Legendrian knots and moduli spaces of microlocal sheaves
To a Legendrian knot in the cocircle bundle over a surface, we associate the category of sheaves with microsupport in this knot. Moduli spaces of objects in this category behave in many ways like character varieties of surfaces. We will explain their relation to knot homology and to cluster algebra.
Nick Sheridan: Counting curves using the Fukaya category
In 1991, string theorists Candelas, de la Ossa, Green and Parkes made a startling prediction for the number of curves in each degree on a generic quintic threefold, in terms of periods of a holomorphic volume form on a `mirror manifold'. Givental and Lian, Liu and Yau gave a mathematical proof of this version of mirror symmetry for the quintic threefold (and many more examples) in 1996. In the meantime (1994), Kontsevich had introduced his `homological mirror symmetry' conjecture and stated that it would `unveil the mystery of mirror symmetry'. I will explain how to prove that the number of curves on the quintic threefold matches up with the periods of the mirror via homological mirror symmetry. I will also attempt to explain in what sense this is `less mysterious' than the previous proof.
Artan Sheshmani: On the proof of the S-duality modularity conjecture for the quintic threefold
I will talk about recent joint works with Amin Gholampour, Richard Thomas and Yukinobu Toda, on an algebraic-geometric proof of the S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin, regarding the modularity of DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold. Our strategy is to first use degeneration and localization techniques to reduce the threefold theory to a certain intersection theory over relative Hilbert scheme of points on surfaces and then prove modularity; More precisely, together with Gholampour we have proven that the generating series, associated to the top intersection numbers of the Hibert scheme of points, relative to an effective divisor, on a smooth quasi-projective surface is a modular form. This is a generalization of the result of Okounkov-Carlsson for absolute Hilbert schemes. These intersection numbers, together with the generating series of Noether-Lefschetz numbers, will provide the ingredients to prove modularity of the above DT invariants over the quintic threefold.
Paolo Stellari: Uniqueness of dg enhancements in geometric contexts and Fourer--Mukai functors
The quest for a (possibly unique) lift of exact functors and triangulated categories to dg analogues is highly non-trivial and rich of subtle aspects, already in geometric contexts. This is made clear by the most recent developments concerning Fourier--Mukai functors, which will be reviewed in this talk. On the other hand, it was a general belief and a formal conjecture by Bondal, Larsen and Lunts that the dg enhancement of the bounded derived category of coherent sheaves or the category of perfect complexes on a (quasi-)projective scheme is unique. This was proved by Lunts and Orlov in a seminal paper. In this talk we will explain how to extend Lunts-Orlov's results to several interesting geometric contexts. Namely, we care about the category of perfect complexes on noetherian separated schemes with enough locally free sheaves and the derived category of quasi-coherent sheaves on any scheme. This is a joint work with A. Canonaco.
David Swinarski: Vector partition functions for conformal blocks
It has long been known that ranks of sl_2 conformal blocks on \bar{M}_{0,n} are vector partition functions: that is, the rank is equal the number of lattice points in a polytope that depends on the input data. We exhibit vector partition functions for degrees of conformal blocks on \bar{M}_{0,4} and intersection numbers between conformal block divisors and F-curves, investigate vector partition functions for conformal blocks for other Lie algebras, and discuss applications.
Richard Thomas: Homological projective duality
We cannot have a section on derived categories without Sasha Kuznetsov. Since he is unable to make it, I will pretend to be him.
I will review Beilinson's theorem, Orlov's family version, and then how this leads naturally to Kuznetsov's wonderful theory of Homological projective duality. Basically I am advocating reading his paper backwards to make the theory easier to understand (for me).
HPD is a generalisation of classical projective duality that relates derived categories of coherent sheaves on different algebraic varieties. I will explain some applications that occur in joint work with Addington, Calabrese and Segal.
Yukinobu Toda: Non-commutative thickening of moduli spaces of stable sheaves
Non-commutative deformation theory of a module over an algebra was introduced by Laudal in 2002, which was later developed by Eriksen, Segal and Efimov-Lunts-Orlov. In 2013, Donovan-Wemyss used the universal non-commutative deformation algebras of floppable curves inside 3-folds to construct twist functors which describe flop-flop derived equivalences. In the last year, I described the dimensions of Donovan-Wemyss’s algebras in terms of Katz’s genus zero Gopakumar-Vafa invariants. This phenomena suggest that there might be an interesting Donaldson-Thomas type theory which captures non-commutative deformations of sheaves, and has some relations to the usual DT theory. In order to realize this story, one has to construct more or less global non-commutative moduli spaces of stable sheaves, and non-commutative virtual cycles. In this talk, I show that the moduli spaces of stable sheaves have a quasi-NC structure which generalizes Kapvanov’s NC structure. If there is no higher obstruction space, this result is applied to construct non-commutative virtual structure sheaves on the moduli spaces of stable sheaves.
Gabriele Vezzosi: Recent directions in Derived Geometry
I will give a non-exhaustive overview of recent directions in derived geometry,
with special emphasis on symplectic and Poisson derived algebraic geometry,
and on derived analytic geometry.
Filippo Viviani: Fourier-Mukai and autoduality for compactified Jacobians
To every reduced (projective) curve X with planar singularities one can associate many fine compactified Jacobians, depending on the choice of a polarization on X, which are birational (possibly non-isomorphic) singular Calabi-Yau projective varieties, each of which yields a modular compactification of a disjoint union of copies of the generalized Jacobian of X. We define a Poincaré sheaf on the product of any two (possibly equal) fine compactified Jacobians of X and show that the associated integral transform is an equivalence of their derived categories, hence it defines a Fourier-Mukai transform. This generalizes the classical result of S. Mukai for Jacobians of smooth curves and the more recent result of D. Arinkin for compactified Jacobians of integral curves with planar singularities, and it provides further evidence for the classical limit of the geometric Langlands conjecture (as formulated by Donagi and Pantev). As a corollary, we prove that there is a canonical isomorphism (called autoduality) between the generalized Jacobian of X and the connected component of the identity of the Picard scheme of any fine compactified Jacobian of X and that algebraic equivalence and numerical equivalence coincide on any fine compactified Jacobian. This is joint work with M. Melo and A. Rapagnetta.
Michael Wemyss: Aspects of the Homological Minimal Model Program
I will talk about the homological version of the minimal model program, with applications to VGIT and group actions on derived categories. The story begins with an analogy between flops and cluster theory, and lifts this analogy to a functorial isomorphism between the flop functor and a mutation functor, suitably interpreted. Then, to actually run the program, the key new ingredient is the theory of noncommutative deformations, which at any given stage allows us to detect which curves are floppable in an easy way, in a manner that (as yet) has no counterpart in cluster theory.
The key point is that all this technology allows us to iterate, and hence jump between the minimal models in a canonical way. Tracking all this information back to the original space/algebra has many applications, and I will describe some of these, mainly to VGIT, chamber structures, and autoequivalences.
Annette Werner: Sections of tropicalization maps
Let X be a closed subvariety of a toric variety over a non-Archimedean field. We look at the tropicalization map from the Berkovich space associated to X to the Kajiwara-Payne tropicalization of the toric variety. On the locus of tropical multiplicty one there exists a natural section of this map, which is given by the Shilov boundary points in the fibers of tropicalization. We investigate continuity of this section map. This is joint work with Walter Gubler and Joseph Rabinoff.
Chris Woodward: Quantum K-theory of geometric invariant theory quotients
(Work in progress with E. Gonzalez) Given an action of a complex reductive group G on a
smooth polarized projective variety X, there is a canonical "quantum Kirwan" map from the equivariant quantum K-theory QK_G(X) to the quantum K-theory of the git quotient QK(X // G). As a sample computation, I will discuss a presentation for the quantum K-theory of a smooth toric Deligne-Mumford stack with projective coarse moduli space, at a canonical bulk deformation.
Tony Yue Yu: First steps of non-archimedean enumerative geometry
The motivation comes from the study of mirror symmetry, especially from the non-archimedean approach suggested by Kontsevich and Soibelman in 2000. I will present several new results in this direction, concerning the moduli spaces of non-archimedean stable maps and their tropical geometry. As an application, I will talk about the enumeration of curves in log Calabi-Yau surfaces. We obtain new geometric invariants. An explicit example for a del Pezzo surface will be discussed in detail, which verifies the expected wall-crossing formula. If time permits, I will explain how non-archimedean enumerative geometry can shed new light on classical algebraic geometry.
Zhiwei Yun: Intersection numbers of cycles on the moduli of Shtukas
In joint work with Wei Zhang, we prove a generalization of
Gross-Zagier formula in the function field setting. Our formula
relates self-intersection of certain cycles on the moduli of Shtukas
for GL(2) to higher derivatives of L-functions.
Xinwen Zhu: The geometric Satake isomorphism for p-adic groups
The geometric Satake isomorphism (for a (split) reductive group over an equal characteristic local field) is a fundamental theorem in the geometric Langlands program. However, for certain arithmetic applications, it would be desirable to establish a similar statement for (split) p-adic groups. It turns out that this is possible, if one works in a world of algebraic geometry built from perfect rings in characteristic p. Time permitting, I will also explain some potential arithmetic applications of this new geometric Satake.
Dimitri Zvonkine: Double ramification cycles
Given n integers a_1, ..., a_n with vanishing sum, the double ramification cycle DR_g(a_1, ..., a_n) is the locus of stable curves (C, x_1, ..., x_n) such that the divisor sum a_i x_i is principal. We present an explicit formula for the homology class represented by the double ramification cycle.