Previous Webinars

(For the VBAC video channel, please access https://educast.fccn.pt/vod/channels/whthltowl?locale=en)

Brill-Noether theory — 16 May 2022

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14:00 - 14:45  Jay Kopper (Penn State University): Developments in Brill-Noether theory for surfaces

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.

Talk Video

15:00 - 15:45  Hannah Larson (Stanford University and Clay Institute): Brill-Noether theory over the Hurwitz 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.

Talk Video

Stacks and Good Moduli Spaces — 28 March 2022

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14:00 - 14:45  Jochen Heinloth (Universität Duisburg-Essen): Proper moduli spaces for algebraic stacks

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.

Talk Slides Talk Video

15:00 - 15:45  Chiara Damiolini (University of Pennsylvania): Projectivity of moduli spaces of quiver representations

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.

Talk Slides Talk Video

Tropical Geometry — 24 January 2022

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14:30 - 15:15  David Jensen (University of Kentucky): Non-Abelian Brill-Noether Theory of Genus 13 Curves

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.

Talk slides Talk Video

15:30 - 16:15 Madeline Brandt (Brown University): Top Weight Cohomology of $A_g$

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.

Talk Slides Talk Video

Kuznetsov Components and Bridgeland Stability Conditions — 22 November 2021

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15:00 - 15:45 Soheyla Feyzbakhsh (Imperial College): Serre-invariant stability conditions and Ulrich bundles on cubic threefolds.

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.

Talk Slides: click here.

Talk Video:

16:00 - 16:45 Benjamin Schmidt (University of Hannover): A curious moduli space on cubic threefolds.

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.

Talk Slides: click here.

Talk Video:

A tribute to M. S. Narasimhan — 11 October 2021

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?

For further information, including videos and slides, see: A tribute to M. S. Narasimhan

Topological Reconstruction for Varieties — 13 September 2021

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14:00 - 14:45  János Kollár (Princeton University): The Zariski topology, linear systems, and algebraic varieties I.

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.

(Joint work with Max Lieblich and Will Sawin.)

Talk Slides: click here.

Talk Video:

15:00 - 15:45  Martin Olsson (University of California): The Zariski topology, linear systems, and algebraic varieties II.

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.

(Joint work with Max Lieblich and Will Sawin.)

Talk Slides: click here.

Talk Video:

Group photos

Geometry and Quantization of Moduli Spaces — 5,6 and 7 July, 2021

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Monday, July 5

15:00 - 16:00  Edward Witten (IAS, Princeton) - Quantization by Branes and Geometric Langlands

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)

Talk Slides: click here.

Talk Video:

16:15 - 17:00  Victoria Hoskins (Radboud University Nijmegen) - Motives of moduli spaces of bundles on curves

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.

Talk Slides: click here.

Talk Video:

17:15 - 18:00  William Mistegaard (IST, Austria) - Quantization of moduli spaces and TQFT

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.

Talk Slides: click here.

Talk Video:

Tuesday, July 6

15:00 - 16:00 Olivier Schiffmann (CNRS, Paris-Sud Orsay) - Cohomological Hall algebras of curves and surfaces

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.

Talk Slides: click here.

Talk Video:

16:15 - 17:00 Alessandro Malusà (University of Toronto) - A new quantisation scheme for hyperkähler manifolds with Sp(1) symmetry

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.

Talk Slides: click here.

Talk Video:

17:15 - 18:00 Jaime Silva (I.S.E. Lisboa) - Hodge and motivic structures on abelian character varieties

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.

Talk Slides: click here.

Talk Video:

Wednesday, July 7

15:00 - 16:00 Sergei Gukov (Caltech) - Brane quantization of SL(2,C) moduli spaces

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.

Talk Slides: click here.

Talk Video:

16:15 - 17:00 Gabriele Rembado (University of Bonn) - Quantisation of moduli spaces of meromorphic connections, and relations with (irregular) CFT

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.

Talk Slides: click here.

Talk Video:

17:15 - 18:00 Du Pei (Harvard University) - Verlinde Formula for PSL(2,C) Higgs bundles

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.

Talk Video:

Group photos

Variations of Hodge structure and Arakelov-Milnor inequalities — 17 May 2021

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15:00 - 15:45  Kang Zuo (Johannes Gutenberg-Universität Mainz): Arakelov Inequality for Families of Projective Manifolds

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.

Talk Slides: Zuo

Talk Video:

16:00 - 16:45  Brian Collier (University of California at Riverside): Maximal variations of Hodge structure and sl2-triples

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.

Talk Slides: Collier

Talk Video:

Non-reductive geometric invariant theory, 15 March 2021

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15:00 - 15:45  Frances Kirwan (Oxford): Non-reductive GIT and HKKP theory

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.

Talk Slides: Kirwan

Talk Video:

16:00 - 16:45  Eloise Hamilton (IMJ-PRG): Cohomology of Non-Reductive GIT quotients and unstable Higgs bundles of rank 2

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.

Talk Slides: Hamilton

Talk Video:

Cohomology of Higgs moduli spaces and the P = W conjecture, 18 January 2021

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15:00 - 15:45   Junliang Shen (MIT): Cohomology of the moduli of Higgs bundles

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.

Talk Slides: Shen

Talk Video:

16:00 - 16:45  Camilla Felisetti (Trento): P=W conjectures for character varieties with a symplectic resolution

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.

Talk Slides: Felisetti

Talk Video:

Group photos

K-stability, 23 November 2020

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15:00 - 15:45  Simon Donaldson (SCGP/Imperial): K-stability and scalar curvature

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.

Talk Slides: Donaldson

16:00 - 16:45  Chenyang Xu (Princeton): An algebraic construction of K-moduli space

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).

Talk Slides: Chenyang

Talk Video:

A tribute to C. S. Seshadri, 9 September 2020

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.

For further information, including videos and slides, see: A tribute to C. S. Seshadri