Abstract: I'll give a brief introduction to DT invariants (which count stable bundles and sheaves on a Calabi-Yau 3-fold X) and Joyce-Song's generalised DT invariants (which count semistable bundles and sheaves on X). Then 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 counting ideal sheaves of curves in X.
By the MNOP conjecture the latter invariants are equivalent to the Gromov-Witten invariants of X. Along the way we also show all these invariants are determined by 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. Talk VideoTalk Slides
Abstract: Nonabelian Hodge theory (NAHT) is a deep relationship between three different kinds of objects associated to a smooth projective variety: semistable Higgs bundles (Dolbeault), local systems (Betti), and connections (de Rham). The main results are equivalences of categories between the categories attached and homeomorphisms between the corresponding coarse moduli spaces. These homeomorphisms are not algebraic and, in fact, witness a much richer hyperkähler structure. They obviously also induce isomorphisms between the (singular) cohomologies and Borel-Moore homologies of the three coarse moduli spaces.
For curves, this relationship between the Dolbeault and Betti moduli spaces is already highly nontrivial and is the framework of the celebrated P=W conjecture, now proven twice. This conjecture states that the weight filtration on the cohomology of the Betti moduli space and the perverse filtration on the cohomology of the Dolbeault moduli space coming from the Hitchin map correspond to each other through the NAHT isomorphisms.
This talk will be concerned with the stacks of Higgs bundles and local systems on a smooth projective curve. One recovers the moduli spaces as the good moduli space of the corresponding stack. The Borel-Moore homologies of the stacks enjoy more algebraic structures, known as cohomological Hall algebras, making them particularly pleasant objects of study. I will explain how to construct a natural isomorphism between the Borel-Moore homologies of the stacks of semistable Higgs bundles and local systems using the theory of CoHAs. This isomorphism has consequences such as an equivalence between (still open) versions of the P=W conjecture for singular moduli spaces and for stacks. The proof involves the BPS algebras and preprojective algebras of quivers.
This is based on joint work with Ben Davison and Sebastian Schlegel Mejia, arXiv:2212.07668. Talk VideoTalk Slides
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 give an introduction to this circle of ideas and discuss a recent proof of the conjecture for GL_n (joint with Junliang Shen). Talk Video
Abstract: By $H_2$ we denote the Lie algebra of polynomial hamiltonian vector fields on the plane. We consider the moduli space of stable twisted Higgs bundles on an algebraic curve of given coprime rank and degree. De Cataldo, Hausel and Migliorini proved in the case of rank 2 and conjectured in arbitrary rank that two natural filtrations on the cohomology of the moduli space coincide. One is the weight filtration W coming from the Betti realization, and the other one is the perverse filtration P induced by the Hitchin map. Motivated by computations of the Khovanov-Rozansky homology of links by Gorsky, Hogancamp and myself, we look for an action of $H_2$ on the cohomology of the moduli space. We find it in the algebra generated by two kinds of natural operations: on the one hand we have the operations of cup product by tautological classes, and on the other hand we have the Hecke operators acting via certain correspondences. We then show that both P and W coincide with the filtration canonically associated to the $sl_2$ subalgebra of $H_2$. Based on joint work with Hausel, Minets and Schiffmann. Talk Video
Abstract: Consider the following problem, originally posed by Gizatullin: "Which automorphisms of a smooth quartic K3 surface in P^3 are induced by Cremona transformations of the ambient space?'' When S is a smooth quartic surface in P^3, the pair (P^3,S) is an example of a Calabi-Yau pair, that is, a mildly singular pair (X,D) consisting of a normal projective variety X and an effective Weil divisor D on X such that K_X+D= 0. In this talk, I will explain a general framework to investigate the birational geometry of Calabi-Yau pairs and how this can be applied to approach Gizatullin's problem. This is a joint work with Alessio Corti and Alex Massarenti. Talk Video
Abstract: This is mostly a report on work of Brown PhD students Veronica Arena and Stephen Obinna.
The Chow groups of a blowup of a smooth variety along a smooth subvariety are described in Fulton's book using Grothendieck's "key formula", involving the Chow groups of the blown up variety, the center of blowup, and the Chern classes of its normal bundle.
If interested in weighted blowups, one expects everything to generalize directly. This is in hindsight correct, except that at every turn there is an interesting and delightful surprise, shedding light on the original formulas for usual blowups, especially when one wants to pin down the integral Chow ring of a stack theoretic weighted blowup.
As an application, one obtains a quick derivation of a formula, due to Di Lorenzo-Pernice-Vistoli and Inchiostro, of the Chow ring of the moduli space $\overline{M}_{1,2}$.. Talk Video
Abstract: I will discuss the Chow group of zero-cycles on the Fano variety of lines in a cubic fourfold from the perspective of certain distinguished surfaces contained in the Fano variety. In particular the surface of lines meeting a fixed line will be discussed in detail. The final aim is to establish an analogue of results of Beauville and Voisin for K3 surfaces. I will give some background on Chow groups, like Bloch-Beilinson filtration and work of Shen-Vial, Voisin and others, and stress the geometry of the situation. Talk Video
Abstract: One possible singular analogue of hyperkähler manifolds are irreducible symplectic varieties, mainly arising as moduli spaces of sheaves on trivial canonical surfaces and as partial resolution of symplectic quotients of hyperkähler manifolds. In this talk I will focus on the second class of examples, especially in the case of fourfolds. In order to produce new examples, we will start from the known hyperkähler fourfolds (Hilbert schemes and generalized Kummer) and act on them with natural automorphisms, for which a systematic analysis is possible. This is the content of a work in progress with Armando Capasso, Annalisa Grossi, Mirko Mauri and Enrica Mazzon. Talk Video
Cohomology of character varieties — 26 September 2022
Abstract: Mathematics have always served Physics along history of science, not only as a language but as a source of ideas. The very same is true about Physics as a source of inspiration for mathematical discoveries. The aim of this talk is to show how certain algebro-topological invariants, e-polynomials, of a G-character variety serve to construct a TQFT and, moreover, how this development allows to easily explore more general settings for G-character varieties, such as those related with singular curves. Talk Video
Abstract: With G a complex reductive group, let XrG denote the G-character varieties of free group Fr of rank r, and XirrG ⊂ XrG be the locus of irreducible representation conjugacy classes. Using the stratification of XrG by polystable type coming from affine GIT and the combinatorics of partitions, we show that the mixed Hodge structures on the cohomology groups of XrSLn and of XrPGLn and on the compactly supported cohomology groups of the irreducible loci XirrSLn and XirrPGLn are isomorphic, for any n,r ∈ N. In particular, this would imply their E-polynomials coincide, settling the question raised by Lawton-Muñoz. This is based on joint work with Carlos Florentino and Alfonso Zamora. Talk Slides
Abstract: I will discuss recent progress in Brill-Noether theory for vector bundles on surfaces, including "weak" Brill-Noether results describing the cohomology of general stable bundles, positivity results about global generation and ampleness, and strong Brill-Noether results about Brill-Noether loci in the moduli space.
Abstract: The main theorems of Brill-Noether theory describe the maps of general curves to projective space. In particular, for a general curve C, the space of degree d maps C —> P^r is known to be irreducible when its expected dimension is positive. However, in nature, curves C are often encountered already equipped with a map to some projective space, which may force them to be special in moduli. The simplest case is when C is general among curves of fixed gonality. For such curves, previous work has shown that the space of maps C —> P^r may have multiple components of varying dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will discuss joint work with Eric Larson and Isabel Vogt that explains these multiple components and proves analogs of all of the main theorems of Brill-Noether theory in this setting.
Abstract: As requested by the organizers, the main aim of the talk is to set the stage for the second talk by explaining the notions appearing in the existence theorem for good (resp. adequate) moduli spaces obtained in joint work with Jarod Alper and Daniel Halpern-Leistner, which provide necessary and sufficient conditions for a moduli problem to admit a proper moduli space. I will try to illustrate the notions in examples.
Abstract: In a recent work of Alper–Belmans–Bragg–Liang–Tajakka, the authors explore how the theory of good moduli spaces developed by Alper and Alper–Halpern-Leistner–Heinloth can be used to give an alternative proof of projectivity of the moduli space of vector bundles on a curve. In today's talk, we will see that a similar approach can be used to study projectivity of moduli spaces of representations of acyclic quivers. Analogies and differences with respect to the case of vector bundles over curves will be emphasized. This is based on ongoing work with Belmans, Franzen, Hoskins, Makarova and Tajakka.
Abstract: Non-abelian Brill-Noether theory is the study of higher rank vector bundles on algebraic curves. The theory of line bundles on curves has a rich history, with many important results including the celebrated Brill-Noether Theorem of Griffiths and Harris. In the late 90's, Bertram-Feinberg and Mukai independently provided conjectural analogues of the Brill-Noether theorem in the non-abelian setting. We verify this conjecture in the special case of genus 13 curves. As a consequence, we see that the moduli space of genus 13 Pryms is of general type. This is joint work with Gavril Farkas and Sam Payne.
Abstract: I will discuss a recent project in computing the top weight cohomology of the moduli space $A_g$ of principally polarized abelian varieties of dimension $g$ for small values of $g$. This piece of the cohomology is controlled by the combinatorics of the boundary strata of a compactification of $A_g$. Thus, it can be computed combinatorially. This is joint work with Juliette Bruce, Melody Chan, Margarida Melo, Gwyneth Moreland, and Corey Wolfe.
Abstract: After a brief introduction of Bridgeland stability conditions, we will discuss a general criterion that ensures a fractional Calabi-Yau category of dimension less than or equal to 2 admits a unique Serre-invariant stability condition up to the action of the universal cover of GL+(2, R). This result can be applied to a certain triangulated subcategory (called the Kuznetsov component) of the bounded derived category of coherent sheaves on a cubic threefold X. As an application, we will prove that the moduli space of Ulrich bundles of fixed rank r greater than or equal to 2 on X is irreducible. This is joint work with Laura Pertusi.
Abstract: The intermediate Jacobian J(X) of a cubic threefold X was introduced by Clemens and Griffiths in 1972 to prove irrationality of cubic threefolds. It is an abelian variety that can be interpreted as parametrizing degree zero cycles in dimension one up to rational equivalence. In this talk we will concentrate on its theta divisor \Theta. Clemens and Griffiths proved the Torelli theorem for cubic threefolds that says that the pair (J(X), \Theta) determines the cubic threefold. Shortly after, Mumford pointed out that X can be recovered from just the unique singular point of the theta divisor. Blowing up the singular point yields a resolution of singularities. We will construct this resolution as a moduli space of rank three vector bundles. This allows us to recover both the classical Torelli theorem and the so-called derived Torelli theorem. This is joint work with Bayer, Beentjes, Feyzbakhsh, Hein, Martinelli, and Rezaee.
14:00-14:30 - S. Ramanan: My collaborative work with Narasimhan.
14:30-15:00 - T. R. Ramadas: Narasimhan’s work on conformal blocks.
15:15-15:45 - O. Garcia-Prada: The theorem of Narasimhan and Seshadri and generalizations.
15:45-16:15 - G. Harder: Why is the Tamagawa number equal to one?
Abstract: We discuss the main steps of the proof that (with a few exceptions) the Zariski topology determines an algebraic variety. In the first talk we explain how to detect linear equivalence using the Zariski topology. Then in the second talk we show that knowing the Zariski topology plus linear equivalence determines the variety. From techniques of the 2 talks will be mostly independent of each other.
Abstract: We discuss the main steps of the proof that (with a few exceptions) the Zariski topology determines an algebraic variety. In the first talk we explain how to detect linear equivalence using the Zariski topology. Then in the second talk we show that knowing the Zariski topology plus linear equivalence determines the variety. From techniques of the 2 talks will be mostly independent of each other.
Abstract: In this talk, which is based on work with D. Gaiotto, Witten will explain a quantum field theory perspective on recent developments in the geometric Langlands program by P. Etinghof, E. Frenkel and D. Kazhdan (see their paper https://arxiv.org/abs/1908.09677)
Abstract: Various computations of cohomological invariants of moduli spaces of vector bundles and Higgs bundles on curves should be both unified and refined by working with motivic invariants, which encode finer invariants, like Hodge structures on cohomology groups and also algebro-geometric invariants such as Chow groups. In this talk, I will present joint work with Lie Fu and Simon Pepin Lehalleur, studying the rational Chow motives of various moduli spaces of vector bundles on curves with additional structure (such as a Higgs field or parabolic structure). After a short introduction to Chow motives, I will present some results which hold for bundles of arbitrary rank. Finally, I will give some explicit formulas in ranks 2 and 3.
Abstract: The Reshetikhin-Turaev topological quantum field theory (TQFT) was motivated from physics by Witten's work on quantum Chern-Simons. In Witten's work quantization of moduli spaces of flat connections and conformal field theory (CFT) was presented as two equivalent approaches to construct the Hilbert space associated to an oriented two-manifold. Both approaches depend a priori on a choice of complex structure on the two-manifold, although the topological nature of the theory suggests that the Hilbert space should be independent of this choice, and support a projective linear action of the mapping class group. On the CFT side this topological invariance and the existence of a mapping class group action was proven by Tsuchia, Ueno and Yamada. On the quantization side it was proven for some two-manifolds independently by Hitchin and Axelrod, Della Pietra and Witten. Laszlo proved mathematically that the CFT approach and the quantization approach of Hitchin are equivalent. Finally, Andersen and Ueno have established that the CFT representations of the mapping class groups are isomorphic to the Reshetikhin-Turaev TQFT mapping class group action. In this talk, we will; 1) partly review the above story, 2) review how quantization was used to prove important results in quantum topology, and 3) present work in progress joint with Andersen, which constructs the TQFT-representations of the mapping class groups from the quantization approach in some of the remaining (parabolic cases), not previously dealt with by Hitchin or Axelrod-Della Pietra and Witten.
Abstract: We will survey some recent developments on the computations of various cohomological Hall algebras associated either to (smooth projective) curves, or to a pair consisting of a curve inside a smooth surface. The latter case is related to various types of affine yangians. Based on joint work with E. Diaconescu, F. Sala and E. Vasserot.
Abstract: It is often the case, with many (complex) moduli problems, that the resulting spaces come with hyperkähler structures and symmetries that act non-trivially on the corresponding families of Kähler forms, rather than preserving them individually. This makes it delicate to approach their quantisation, as a preferred symplectic structure may not be given or the group action to be quantised may not be symplectic with respect to it. The U(1)-action on the Hitchin moduli spaces is an example of this.
In an ongoing joint work with Andersen and Rembado, we approach this problem under the assumption that the symmetry group is an extension of Sp(1) with a transitive action on CP^1, identified with the associated space of complex structures. This is the case for known examples such as linear spaces, the Taub-NUT space, nilpotent orbits of complex Lie groups, the moduli spaces of framed SU(r)-instantons on R^4, and the Atiyah-Hitchin manifolds of monopoles on R^3. We propose a new hyperkähler quantisation scheme by assuming given a smooth equivariant family of pre-quantum line bundles, and by defining a collection of quantum Hilbert spaces parametrised by CP^1. The quantisation of the symmetry group may then be addressed in terms of actions on this family and compatibility with naturally defined connections. These, however, turn out to not be flat in general, even projectively, but we obtain group of representations on spaces of holomorphic sections of the family of Hilbert spaces, rather than flat ones, and we therefore propose this space of holomorphic sections as the relevant quantization of these SP(1)-symmetric Hyper-Kähler manifolds.
Abstract: In this talk, I will make an overview about my work on the mixed Hodge structures and motives of abelian character varieties.
I will start by giving a brief account of Hodge structures on character varieties, and how those relate to the topic of non-abelian Hodge theory. Illustrating this topic, I will cover my results on the mixed hodge structures of free abelian character varieties and how those illustrate some predictions related to mirror theory.
Afterwards, I will talk about more recent results on the motives of character varieties. In this, I will talk about our attempt to adapt our previous work on Hodge structures of finite quotients by using a motive that allows for a suitable equivariant version - the so-called equivariant Chow motives. This is joint work with C. Florentino.
Abstract: The problem of quantization of symplectic manifolds and the Fukaya category side of mirror symmetry start with the same input data. Therefore, it is natural to wonder whether the answer to the former may be contained in some form of the latter. The goal of this talk will be to illustrate how this approach, often called "brane quantization," can help with understanding certain aspects of the quantization of the moduli space of flat SL(2,C) connections.
Abstract: The geometry and quantisation of moduli spaces of unitary flat connections on Riemann surfaces have been widely studied in the past: as the complex structure on the surface is deformed the moduli spaces assemble into a local system of symplectic manifolds, and Kähler quantisation turns it into a projectively flat vector bundle.
The complexified version brings about holomorphic connections and hyperkähler manifolds, requiring new ideas in Kähler quantisation; deformation quantisation on the other hand has been carried out in greater generality, namely for moduli spaces of meromorphic connections with irregular singularities.
In this talk we will briefly review this story and phrase the singular case in the same geometric language of the nonsingular one, involving flat symplectic fibre bundles: their bases provide an intrinsic approach to isomonodromic deformations, and their quantisation provides a mathematical approach to irregular singularities in the Wess-Zumino-Novikov-Witten model.
Abstract: In this talk, I will discuss how to obtain the Verlinde formula for G-Higgs bundles when G is not simply connected. I will also mention some of its applications to mirror symmetry and brane quantization.
Abstract: The Arakelov inequality for families of algebraic curves and abelian varieties goes back to the works by Arakelov-Parshin, Faltings and Deligne (sharp form), and for systems of Hodge bundles is due to the works by Green-Griffiths-Kerr, Jost-Zuo, Peters, Viehweg-Zuo. A very recent work by Biquard-Collier-Garcia-Prada-Toledo is making a further progress on Arakelov-Milnor inequalities.
In my talk I shall briefly report on my recent work joint with Xin Lu and Jinbang Yang. We show the Arakelov inequality holds STRICTLY for canonical heights of families of $n$-folds of general type. We also show it is asymptotically sharp in a sense. Note that this Arakelov inequality can become actually an equality for families of abelian varieties, in which case they are precisely Shimura families.
Abstract: In this talk we will discuss holomorphic maps from the upper half space into certain homogeneous spaces (period domains) which are equivariant with respect to a representation of the fundamental group of a closed surface. Such maps arise from Higgs bundles on a Riemann surface which are fixed points of a C* action. When the target is a hermitian symmetric space, the Toledo invariant provides an integer invariant which is bounded in absolute value. Moreover, representations which maximize the Toledo invariant satisfy certain rigidity phenomena and arise from the uniformizing representation of the Riemann surface. We will discuss how to generalize such an invariant for arbitrary period domains, explain how this invariant is bounded and describe how the rigidity phenomena which occur when the invariant is maximized are related to sl2 triples. This is based on joint work with Biquard, Garcia-Prada and Toledo.
Abstract: In a recent paper F. Haiden, L. Katzarkov, M. Kontsevich and P. Pandit study notions of (semi-)stability and Harder-Narasimhan filtrations in polarised lattices and weight filtations for modular lattices, proving existence and uniqueness theorems in these very general settings. The aim of this talk is to explore the relationship of their theory with recent extensions of geometric invariant theory to linear algebraic group actions by non-reductive groups with graded unipotent radicals.
Abstract: Non-Reductive GIT is a generalisation of GIT which enables the construction of new moduli spaces. In particular it can be used to construct moduli spaces for unstable Higgs/vector bundles on a smooth projective curve. The aim of this talk is to describe a method for computing the Poincare series of Non-Reductive GIT quotients when the initial variety is smooth (analogous to the existing method in classical GIT), and to show how it can be applied in practice in the case of unstable Higgs bundles of rank 2.
Abstract: The moduli space of Higgs bundles and Hitchin's integrable system lie at the crossroads of mathematics physics, representation theory, and geometry. In this talk, we focus on cohomological structures of these moduli spaces from the aspects of non-abelian Hodge theory, hyper-kaehler geometry, and mirror symmetry. We will discuss recent progress on the P=W conjecture as well as connections to some other open conjectures concerning Higgs moduli spaces. Based on joint work with Mark de Cataldo and Davesh Maulik.
Abstract: Character varieties parametrise representations of the fundamental group of a curve. In general these moduli spaces are singular, therefore it is customary to slightly change the moduli problem and consider smooth analogues, called twisted character varieties. In this setting, the P=W conjecture by de Cataldo, Hausel, and Migliorini suggests a surprising connection between the topology of Hitchin systems and Hodge theory of character varieties. In joint work with M. Mauri we establish (and in some cases formulate) analogous P=W phenomena in the singular case . In particular we show that the P=W conjecture holds for character varieties which admit a symplectic resolution, namely in genus 1 and arbitrary rank and in genus 2 and rank 2.
Abstract: This will be an overview talk about existence results in complex differential geometry connected to the notion of K-stability. We will explain the analogies with the corresponding results, going back to Narasimhan and Seshadri, for holomorphic vector bundles and outline some strategies of proofs that have been employed. We will illustrate the general with a discussion of the case of toric manifolds.
Abstract: K-stability of Fano varieties has become a fast developed topic in algebraic geometry. One major output is the construction of moduli spaces of K-(semi,poly)-stable Fano varieties, which resolves a number of pathological issue for families of general Fano varieties. The purely algebraic construction is built on a systematical study of K-stability using higher dimensional geometry, including a more comprehensive understanding of the notion of K-stability (for Fano varieties).
13:00-13:40 M. S. Narasimhan: Reminiscences of my association and work with Seshadri
13:50-14:30 T. Gómez: Parabolic Bundles
14:40-15:20 V. Balaji: C. S. Seshadri, my mentor, a mathematician and a remarkable person
15:30-16:00 Time for those who want to share their reminiscences and appreciation of Seshadri.
