Graduate

Advanced Mini-Courses

Spring 2025

Jonathan Brundan (University of Oregon), January 13-17, 2025, Classical representation theory via categorification

Time and Place: (The lectures will be recorded and posted here.)

Lecture 1: Monday, Jan. 13, 2:45pm-4:15pm, Hastings Hall Room 210
Lecture 2: Wednesday, Jan. 15, 2:45pm-4:15pm, Hastings Hall Room 210
Lecture 3: Thursday Jan. 16, 2:45pm-4:15pm, Lake Hall Room 509
Lecture 4, Friday Jan. 17, 10:30am-12:00pm, Lake Hall Room 509

Abstract: The standard approach to representation theory of Lie type involves the combinatorics of the reflection representation of the underlying Weyl group, leading ultimately to Kazhdan-Lusztig polynomials and Soergel bimodules. In this minicourse I will talk about a non-standard approach within each of the families of classical groups (general linear, orthogonal, symplectic, ...). Perhaps it is less satisfactory because it has a somewhat different flavor depending on the particular family of classical group, but it is just as rich and has an equally long and distinguished history. The combinatorics that arises is that of integrable representations of certain affine Kac-Moody Lie algebras, connecting to crystal/canonical bases and 2-quantum groups. I will mainly focus on the GL family which is fully developed, with applications to representation theory of symmetric and general linear groups. There has been recent progress in the OSp, P and Q families, although lots of interesting questions remain to be investigated.

Tentative plan:

Lecture 1: Representations of symmetric groups and Heisenberg categorification. We will start by discovering the definition of Khovanov's Heisenberg category, which acts on the categories of representations of the symmetric groups. A generalization acts also on other abelian categories in the GL family. This leads to the general notion of a Heisenberg categorification, which gives a unifying framework for studying all such abelian categories.
Lecture 2: 2-quantum groups and Kac-Moody categorification. Heisenberg categorifications have a rich underlying structure which is best explained by passing from a categorical action of the Heisenberg category to a categorical action of an associated 2-quantum group. I will define the latter and outline the passage from Heisenberg to Kac-Moody established in [Brundan-Savage-Webster].
Lecture 3: The nil-Hecke algebra, Grassmannian bimodules, and the Chuang-Rouquier-Rickard complex. One of the most remarkable results about Heisenberg/Kac-Moody categorifications is the existence of derived equivalences between blocks associated to weights in the same Weyl group orbit. I hope to say something about how this is proved, reformulating the original approach of [Chuang-Rouquier] in terms of the nil-Hecke algebra and certain bimodules over the equivariant cohomology algebras of Grassmannians. This may be too ambitious but we can take the first steps together, at least!
Lecture 4: The nil-Brauer category and categorification of iquantum groups. So far the focus has been on GL families. At the end, I will say a bit about the OSp family where, as well as 2-quantum groups, certain categorifications of iquantum groups emerge; the simplest example of the latter is the nil-Brauer category which categorifies the split iquantum group of rank one. Or I might talk instead about the Q family, which is also miraculous in its own odd way.

Pre-requisites:

I am going to assume participants have seen some basic Lie theory, so that a root system or a Dynkin diagram defining a complex semisimple Lie algebra is not a total stranger. I think we will mainly be looking at type A root systems so it is not essential.

I will also assume that everyone has seen the definition of a monoidal category (knowing just about strict monoidal categories would be perfectly adequate) as I will be introducing several strict monoidal categories defined by generators and relations. I have made available some background material on monoidal categories and the string calculus which we will use to represent morphisms in them---see "Notes on monoidal categories" in the dropbox link (which is chapter 6 of an unpublished book). Do try to look at least at section 6.4 before the course begins---it explains my conventions for string calculus and what it means to define a strict monoidal category by generators and relations.

References:

[1] J. Brundan, On the definition of Kac-Moody 2-category, Math. Ann. 364 (2016), 353-372.

[2] J. Brundan, On the definition of Heisenberg category, Alg. Comb. 1 (2018), 523-544.

[3] J. Brundan, A. Savage and B. Webster, Heisenberg and Kac-Moody categorification, Selecta Math. 26 (2020).

[4] J. Brundan and N. Davidson, Categorical actions and crystals, Contemp. Math. 684 (2017), 116-159.

[5] J. Brundan and A. Kleshchev, Odd Grassmannian bimodules and derived equivalences for spin symmetric groups, https://arxiv.org/pdf/2203.14149.

[6] J. Brundan, W. Wang and B. Webster, Nil-Brauer categorifies the split iquantum group of rank one, https://arxiv.org/abs/2305.05877.

[7] J. Chuang and R. Rouquier, Derived equivalences for symmetric groups and sl_2-categorification, Ann. Math. 167 (2008), 245-298.

[8] M. Khovanov, Heisenberg algebra and a graphical calculus, Fund. Math. 225 (2014), 169-210.

[9] R. Rouquier, 2-Kac-Moody algebras, https://arxiv.org/pdf/0812.5023.

I have put all of these into a dropbox folder which can be accessed using this link.

Fall 2024

Ben Davison (University of Edinburgh), November 14-22, 2024, Yangians and BPS Lie algebras

Time and Place: All lectures will take place in 509 Lake Hall. They will be streamed at this zoom link, meeting ID: 950 7963 6701.

Lecture 1: Thursday, Nov. 14, 2:45-4:15pm, video recording, notes
Lecture 2: Friday, Nov. 15, 10:30am-12:00pm, video recording, notes
Lecture 3: Thursday, Nov. 21, 2:45-4:15pm, video recording, notes for lectures 3 and 4
Lecture 4: Friday, Nov. 22, 10:30am-12pm, video recording

Abstract: This course concerns two classes of algebras, and the comparison between them. The first are critical cohomological Hall algebras (CoHAs), with underlying vector space given by the vanishing cycle cohomology of moduli spaces of representations of Jacobi algebras. These first appeared in the work of Kontsevich and Soibelman, and provide a kind of categorification of refined BPS invariants in physics, as well as a connection between BPS state counting and quantum groups. The definition of the algebra depends on a choice of a quiver and potential (a linear combination of cyclic words in the arrows of the quiver). For certain choices, it is possible to describe the resulting algebra, but most of the time it is not. A very general result says that, for all choices of symmetric quiver with potential, there is a canonical subspace in the cohomological Hall algebra, closed under the Lie bracket, that generates it under the action of tautological classes and multiplication (a type of PBW theorem). This is the so-called BPS Lie algebra

The second class of algebras arise in the work of Maulik and Okounkov, in their book [4]. These are so-called Yangian algebras, which really do generalise the Yangians considered by Drinfeld in finite type, to arbitrary quivers. Via the theory of stable envelopes developed in [4], Maulik and Okounkov construct R matrices on tensor products of cohomologies of Nakajima quiver varieties. Via these R matrices, they are able to "reverse engineer" a Yangian-type algebra, for which the cohomology of Nakajima quiver varieties becomes a category of modules with their given R matrix providing a braiding (with spectral parameters). Just as in the case of CoHAs, Maulik and Okounkov prove a kind of PBW theorem, relating their Yangians to much "smaller" Lie subalgebras; in finite type, these Lie algebras are just the classical Lie algebras sitting inside the Yangian. It turns out that these Lie algebras are isomorphic to doubles of certain BPS Lie agebras, allowing us to transport many facts that are known on the critical CoHA side to the Maulik-Okounkov Yangian side, and vice versa.

Tentative plan:

Lecture 1: Cohomological Hall algebras and BPS Lie algebras. I will introduce the CoHA associated to a quiver, and the CoHA associated to a quiver with potential. There will be lots of examples, as well as the definition of the BPS Lie algebra and the integrality/PBW theorem.
Lecture 2: Preprojective CoHAs. Here we focus on a class of examples most suited to studying 2d geometry: preprojective CoHAs. While, in general, the BPS Lie algebra of a quiver with potential is very hard to calculate explicitly, we will see that in these examples there are many structural results that enable us to precisely characterise BPS Lie algebras.
Lecture 3: Maulik Okounkov stable envelopes and Yangians. I will move over to the "other side" and introduce the Maulik-Okounkov stable envelope construction. From this, we build new Yangian algebras, and the Lie algebras sitting inside them that generate them under the action of tautological classes under a PBW type theorem.
Lecture 4: Nonabelian stable envelopes and the comparison theorem. I will introduce the nonabelian stable envelope of Aganagic-Okounkov, and explain the first comparison theorem. This relates the nonabelian stable envelope to the CoHA multiplication, and provides the bridge for uniting the worlds of CoHAs and Yangians.

References:

[1] Tommaso Botta and Ben Davison, Okounkov's conjecture via BPS Lie algebras

[2] Ben Davison, Lucien Hennecart and Sebastian Schlegel Mejia, BPS algebras and generalised Kac-Moody algebras from 2-Calabi-Yau categories

[3] Ben Davison and Sven Meinhardt, Cohomological Donaldson-Thomas theory of a quiver with potential and quantum enveloping algebras

[4] Davesh Maulik and Andrei Okounkov, Quantum groups and quantum cohomology

[5] Maxim Kontsevich and Yan Soibelman, Cohomological Hall algebra, exponential Hodge structures and motivic Donaldson-Thomas invariants

Suggested introductory and further reading:

For getting to grips with the stable envelope+Yangian side of the story, [4] above is not just the canonical reference, but a great introduction. For those that have never seen a CoHA before, the first section of [5] is a good place to start, and then [3] is a relatively self-contained guide through the rest of what we need.
For a friendly and short introduction to many of the key components on both sides of the course, Michael McBreen's PhD thesis "Quantum Cohomology of Hypertoric Varieties and Geometric Representations of Yangians" is excellent.
For a self-contained exposition of preprojective CoHAs along with their relation to their critical counterparts, I like "On two cohomological Hall algebras" by Yaping Yang and Gufang Zhao. This relation goes via "dimensional reduction"; to get a head-start on this, you may like to consult the appendix of "The critical CoHA of a quiver with potential"
For a rather different approach to the comparison theorem relating the Yangian to the CoHA, there is "Cohomological Hall algebras of quivers and Yangians" by Schiffmann and Vasserot.
Kac polynomials will run through these talks; part of the motivation for the above work was to prove Okounkov's conjecture, which is expressed in terms of these polynomials. For a little introduction to these miraculous polynomials, as well as their role in geometric representation theory and beyond, I recommend Schiffmann's ICM talk "Kac polynomials and Lie algebras associated to quivers and curves"

Fall 2024

Bernhard Keller (Université Paris Cité), October 21-25, 2024, Towards Higgs categories for cluster algebras in higher
Teichmuller theory

Time and Place:

Lecture 1: Monday, Oct. 21, 4:00-5:30pm, 509 Lake Hall, (video recording, notes)
Lecture 2: Wednesday, Oct. 23, 4:00-5:30pm, 509 Lake Hall (video recording, notes)
Lecture 3: Thursday, Oct. 24, 2:45-4:15pm, 509 Lake Hall (video recording, notes)
Lecture 4: Friday, Oct. 25, 10:30am-12pm, 509 Lake Hall (video recording, notes)

Abstract: Higher Teichmuller spaces are associated with pairs (G,S) consisting of a split semi-simple Lie group G and a marked surface S. The classical case is the one where G is the group PGL(2,R). In the 2000s, Fock and Goncharov have developed a cluster-theoretic approach to higher Teichmuller theory for groups of type A. Their work was extended to all classical groups by Ian Le and to arbitrary split semi-simple groups by Goncharov-Shen in 2019 (preprint). In particular, they construct a cluster algebra A(G,S) for each pair (G,S). In this lecture series, we will give an introduction to the tools used in the ongoing efforts aiming at the additive categorification of these cluster algebras. For G=PGL(2,R), this aim was reached thanks to the work of Fomin--Shapiro--Thurston, Derksen--Weyman--Zelevinsky and Labardini-Fragoso. An alternative approach, based on Kapranov-Schechtman's idea of perverse schober and techniques from higher category theory, is due to Merlin Christ. The category he constructs is equivalent, as an extriangulated category, to the Higgs category (in the sense of Yilin Wu) associated to the ice quiver with potential constructed by Labardini-Fragoso (at least if the surface does not have punctures). Christ's construction admits a natural conjectural generalization to (simply-laced) groups G of higher rank. In the case where the surface is a triangle, recent work by Miantao Liu will likely confirm that Christ's conjectural category is the Higgs category associated with Goncharov-Shen's ice quiver endowed with a natural potential. In particular, it would contain a canonical cluster-tilting object with the expected endomorphism algebra. If this holds, the conjectural categorification will follow if one can 1) prove a glueing theorem for cluster-tilting objects and 2) check that the glued objects are infinitely mutable.

Tentative plan:

Lecture 1: From indecomposable representations to cluster variables. We will explain Caldero-Chapoton's formula which allows to compute the
cluster variables of a (cluster-finite acyclic) quiver from the geometry of its indecomposable representations.
Lecture 2: Introduction to additive categorification of cluster algebras without coefficients. We will introduce the cluster category of an acyclic quiver and explain how it allows to lift the combinatorics of the associated cluster algebra to the level of a triangulated category. We will then show how these constructions generalize from acylic quivers to 2-acyclic quivers with potential following Derksen-Weyman-Zelevinsky and Amiot.
Lecture 3: Cluster algebras with coefficients and Higgs categories. We will introduce Yilin Wu's Higgs category of an ice quiver with potential and
show how it generalizes the category of finite-dimensional representations of a preprojective algebra used in the work of Geiss-Leclerc-Schroer. We will use the Higgs category to extend the theory presented in the previous lecture to the case of cluster algebras with coefficients, which we will illustrate on geometric examples.
Lecture 4: Higgs categories for Goncharov-Shen's basic triangles and the glueing problem. We will present Miantao Liu's uniform approach to the construction of Goncharov-Shen's basic triangle associated with a split simple Lie group. We will discuss the problem of glueing
the corresponding Higgs categories and its solution by Merlin Christ in the case of the group PSL(2,R).

References:

Merlin Christ, Cluster theory of topologigal Fukaya categories

Alexander Goncharov and Linhui Shen, Quantum geometry of moduli spaces of local systems and representations, 233 pages.

Bernhard Keller, Cluster algebras, quiver representations and triangulated categories, London Math. Soc. Lecture Note Ser., 375, 2010, 76–160.

Yilin Wu, Relative cluster categories and Higgs categories, Adv. Math. 424 (2023), Paper No. 109040, 112 pp.

Spring 2024

Andrea d’Agnolo (Università degli studi di Padova), The Riemann-Hilbert correspondence, May 6-9, 2024

Time and place: All lectures will take place in 509 Lake Hall. They will also be streamed over Zoom.

Lecture 1: Monday, May 6, 3:00 - 4:30
Lecture 2: Tuesday, May 7, 3:00 - 4:30
Lecture 3: Wednesday, May 8, 3:00 - 4:30
Lecture 4: Thursday, May 9, 11:00 - 12:30 (Please note the different time!)

Zoom link: https://northeastern.zoom.us/j/94036166453

Abstract: This course is an introduction to the Riemann-Hilbert correspondence.

The original statement appears as the 21st problem of the famous list that Hilbert established in 1900. It asks to show that there always exists a Fuchsian (a.k.a. regular) differential operator on the complex line, with given singularities and monodromy.

In 1970, Deligne reformulated it on complex manifolds of arbitrary dimension. He established a correspondence between, on the one hand, regular flat meromorphic connections with prescribed pole locus and, on the other hand, local systems on the complement of that locus.

In 1984, Kashiwara vastly generalized Deligne's result by considering arbitrary systems of linear PDEs instead of flat meromorphic connections. More precisely, he provided a correspondence between regular holonomic D-modules and perverse sheaves (roughly, complexes of sheaves whose cohomologies are local systems along the components of a stratification). He also constructed an inverse correspondence, nowadays naturally expressed in the languange of subanalytic sheaves by Kashiwara and Schapira.

More recently, Kashiwara and I managed to eliminate the regularity condition. To do so, we had to enhance the target category of perverse sheaves in order to account for the appearance of Stokes phenomena. Our enhancement made use of two ingredients. Subanalytic sheaves, on the one hand, to treat holomorphic functions with tempered growth. Adding an extra real variable, on the other hand, as Tamarkin originally did in the context of symplectic geometry. In our context, it provides a framework for tracing the different growths of solutions at singular points. A key ingredient of our proof is the description of the structure of flat connections obtained by Mochizuki and Kedlaya, independently.

The lectures are aimed at graduate students, particularly those interested in analysis, algebraic geometry, topology, and representation theory, and the presentation will be interspersed with concrete illustrative examples.

References:

Pierre Deligne, Équations différentielles à points singuliers réguliers, Lecture Notes in Math. 163, Springer (1970), iii+133 pp.
Masaki Kashiwara, The Riemann-Hilbert problem for holonomic systems, Publ. Res. Inst. Math. Sci. 20 (1984), no. 2, 319–365.
Andrea D'Agnolo and Masaki Kashiwara, Riemann-Hilbert correspondence for holonomic D-modules, Publ. Math. Inst. Hautes Études Sci. 123 (2016), no. 1, 69–197.

Tentative plan:

Lecture 1: The regular Riemann-Hilbert correspondence in dimension one. Review on D-modules. (Video and Notes)
Lecture 2: Review on D-modules, continued. Perversity of solutions to holonomic D-modules. The regular Riemann-Hilbert correspondence in arbitrary dimension: statement. (Video and Notes)
Lecture 3: The regular Riemann-Hilbert correspondence in arbitrary dimension: sketch of proof via subanalytic sheaves. (Video and Notes)
Lecture 4: The irregular Riemann-Hilbert correspondence by concrete illustrative examples. (Video and Notes)

Suggested reading on preliminary material: A very good presentation of sheaves and their derived category may be found in the first three chapters of [1]. (Constructible and perverse sheaves are also treated in later chapters, leaning on the powerful microlocal point of view invented by the authors. A beautiful treatment demanding a rather steep learning curve.) A lively introduction to perverse sheaves is given in [7].

An excellent reference for D-modules is [2]. The papers [5, 6] provide a more focused and elementary introduction.

A self-contained presentation of subanalytic sheaves is given in [4]. (Subanalytic sheaves were introduced in [3] as a particular case of ind-sheaves. This requires a rather steep learning curve.)

Masaki Kashiwara and Pierre Schapira, Sheaves on Manifolds, Grundlehren der Mathematischen Wissenschaften 292, Springer (1990), x+512 pp.
Masaki Kashiwara, D-modules and Microlocal Calculus, Translations of Mathematical Monographs 217, American Math. Soc. (2003), xvi+254 pp.
Masaki Kashiwara and Pierre Schapira, Ind-sheaves, Astérisque 271 (2001), 136 pp.
Luca Prelli, Sheaves on subanalytic sites, Rend. Semin. Mat. Univ. Padova 120 (2008), 167–216.
Claude Sabbah, Introduction to the theory of D-modules. Notes and slides from a course in Nankai (2011).
Jean-Pierre Schneiders, An introduction to D-modules, Bull. Soc. Royale Sci. Liège (1995), 223–295.
Geordie Williamson, Illustrated guide to perverse sheaves, Notes and exercises of a course given in Pisa (2015).

Spring 2023

David Hernandez (Université Paris Cité), Cluster algebras and quantum affine algebras

Time and place: All lectures will take place in 509 Lake Hall

- Lecture 1: Monday, March 20, 2:45 - 4:15
- Lecture 2: Tuesday, March 21, 2:45 - 4:15
- Lecture 3: Wednesday, March 22, 2:45 - 4:15
- Lecture 4: Thursday, March 23, 2:45 - 4:15

Abstract: Quantum affine algebras are important examples of Drinfeld-Jimbo quantum groups. They can be defined as quantizations of affine Kac-Moody algebras or as affinizations of finite type quantum groups (Drinfeld Theorem). The representation theory of quantum affine algebras is very rich. It has been studied intensively during the past thirty years from different point of views, in particular in connections with various fields in mathematics and in physics, such as geometry (geometric representation theory, geometric Langlands program), combinatorics (crystals, positivity problems), theoretical physics (Bethe Ansatz, integrable systems)...In particular, the category of finite-dimensional representations of a quantum affine algebra is one of the most studied object in quantum groups theory. Relatively recently, it was discovered that these representations can be studied from the point of view of cluster algebras (remarkable commutative algebras with distinguished set of generators obtained from inductive processes). The aim of these lectures will be to explain this connection, and some of the developments in this direction.

- Lecture 1 : Motivating example : representations of quantum affine sl_2 (video recording)
  - We explore the first examples in the category of finite-dimensional representations of quantum affine sl_2. We see how the cluster algebra structures emerge.
- Lecture 2 : Monoidal categorification of cluster algebras (video recording)
  - We explain how cluster algebras can be categorified using representations of general quantum affine algebras. We give examples and explain how (cluster) exchange relations are obtained from R-matrices.
- Lecture 3 : Baxter relations, category O and spectra of quantum integrable models (video recording)
  - We explain categorification of cluster algebras in terms of category O of Borel quantum affine algebras. In particular we will discuss cluster theoretic interpretation of Baxter relations of the relevant quantum integrable models. If time permits, we will discuss application to Bethe Ansatz equations.
- Lecture 4 : Quantum Grothendieck rings and quantum cluster algebras (video recording)
  - We discuss the interplay of cluster algebra theory with deformations of Grothendieck rings called quantum Grothendieck rings. If time permits, we will discuss very recent applications to the Kazhdan-Lusztig conjecture for non simply-laced quantum affine algebras.

References:

D. Hernandez and B. Leclerc, Quantum affine algebras and cluster algebras(avec B. Leclerc) in Progress in Mathematics 337 (2021), arxiv.org/pdf/1902.01432.pdf.
D. Hernandez, Avancées concernant les R-matrices et leurs applications (d'après Maulik-Okounkov, Kang-Kashiwara-Kim-Oh...): Sém. Bourbaki 1129, Astérisque 407 (2019), 297--331, https://arxiv.org/pdf/1704.06039.pdf.
M. Kashiwara, Crystal bases and categorifications—Chern Medal lecture. Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, 249–258, https://arxiv.org/pdf/1809.00114.pdf.
D. Hernandez and B. Leclerc, Quantum Grothendieck rings and derived Hall algebras (avec B. Leclerc) : PDF, Crelle - Journal für die reine und angewandte Mathematik 701 (2015), 77--126, https://arxiv.org/pdf/1109.0862.pdf.
B. Leclerc, Cluster algebras and representation theory. Proceedings of the International Congress of Mathematicians. Volume IV, 2471–2488, Hindustan Book Agency, New Delhi, 2010, https://arxiv.org/pdf/1009.4552.pdf.

Spring 2020

Mario Salvetti (Università di Pisa) and Giovanni Paolini (Caltech), The K(π,1) conjecture for affine Artin groups

Dates and times: March 30-April 3.
Cancelled due to the COVID-19 pandemic

David Hernandez (Université Paris-Diderot), Title TBA.

Dates and times: April 27-May 1.
Cancelled due to the COVID-19 pandemic

Fall 2019

Gennadi Kasparov (Vanderbilt), K-theory of group C*-algebras and applications.

Dates and times:

Lecture 1: Tuesday, October 22, 1:15 - 2:35.

Lecture 2: Wednesday, October 23, 1:15-2:35.

Lecture 3: Thursday, October 24, 1:15-2:35.

Abstract:

The rapid development of the K-theory of group C*-algebras started about 40 years ago. To a large extent this was due to deep relations of this theory with the topology and geometry of smooth manifolds, with the representation theory of Lie groups, and with the index theory of elliptic operators.

In this lecture series I will give a general review of the Novikov conjecture on higher signatures from the initial statement of 1969 to most recent results. Another topic will be the Baum-Connes conjecture which is deeply related with the representation theory of Lie groups and also with the representation theory of discrete groups. Applications to discrete series representations of semi-simple groups and to the Kadison-Kaplansky conjecture will be discussed.

In the process I will explain some basic facts about KK-theory which played a key role in most results on these two conjectures obtained so far. And since technical tools also include index theory of elliptic operators, I will touch this area as well.

The lectures will be accessible to graduate students. (Some knowledge of Hilbert space theory will be required.)

Fall 2018

Vikraman Balaji (Chennai Mathematical Institute), Parabolic bundles, parabolic torsors and representations of Fuchsian groups.

Time and rooms:

Lecture 1: Monday, November 5, 11:00 AM – 12:00 PM.

Lecture 2: Wednesday, November 7, 12:00 PM – 1:00 PM.

Lecture 3: Thursday, November 8, 12:00 PM – 1:00 PM.

Abstract:

In this mini course I will work over the ground field C; I will begin by recalling the classical theorem of Mehta-Seshadri on parabolic vector bundles on curves and their relation with representations of Fuchsian groups. I will then shift attention to more general structure groups G which are semisimple and simply connected. I will introduce the concept of parahoric subgroups of G(C[[t]]) and relate them to the so-called Bruhat-Tits group schemes over Spec C[[t]]. I will then indicate how to get a self-contained construction of the Bruhat-Tits group schemes in the setting of orbifold bundles. I will then work with X, an irreducible smooth projective algebraic curve of genus g ≥ 2. I will introduce the notion of semistable and stable parahoric torsors under a certain Bruhat-Tits group scheme G on X and construct the moduli space of semistable parahoric G-torsors; we also identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of G. I will briefly discuss Heinloth’s uniformization results for parahoric torsors and indicate the computation of the Picard groups of the parahoric moduli stacks.

Owen Gwilliam (University of Massachusetts Amherst), Chern-Simons theory & quantum groups approached through factorization algebras.

Time and rooms:

Lecture 1: Tuesday, October 23, 10:30 AM – 12:00 PM.

Lecture 2: Wednesday, October 24, 4:00 PM – 5:30 PM.

Lecture 3: Thursday, October 25, 1:00 PM – 2:30 PM.

Abstract:

Over the last few decades, various mathematical communities have studied the three-dimensional topological field theory called Chern-Simons theory. Recently, ideas and methods from higher algebra have been applied to it. These lectures will give an introduction to such work, using it as an excuse to discuss some sophisticated ideas, like En algebras, factorization homology, or filtered Koszul duality. (We will not assume comfort with higher algebra on the part of the audience.) The focus will be on joint work with John Francis and Kevin Costello, which directly connects the Feynman diagrammatic approach (think: Vassiliev invariants of knots) to the quantum group approach (think: HOMFLYPT polynomials). We will also relate our approach to the recent work of Ben-Zvi, Brochier, and Jordan on “integrating quantum groups” and of Calaque, Pantev, Toen, Vaquie, and Vezzosi on categorical deformation quantization.

Lecture 1: Braided monoidal categories, Drinfeld’s classification of quantum groups with formal parameter, and the overall strategy. The aim is to articulate the central question addressed in joint work with Costello and Francis: how is perturbative Chern-Simons theory related to quantum groups? The talk is thus devoted to introducing Chern-Simons theory, a 3-dimensional topological field theory on oriented 3-manifolds that depends on a Lie group G, and to introducing the notion of a quantum group, which we approach via its braided monoidal category of representations and so depends, for us, on a Lie algebra and a choice of invariant symmetric bilinear form. We will gesture at the knot invariants produced by these two notions, such as the Bott-Taubes integrals versus HOMFLYPT polynomials, but will not discuss them in detail.
Lecture 2: Chern-Simons theory, its quantization, its factorization algebra, and a one-dimensional defectThe aim is to explore algebraic structures produced by the perturbative quantization of Chern-Simons theories (CS). We will introduce factorization algebras and describe in detail the classical observables of CS in these terms. We will then outline of what perturbative quantization means and how it deforms the factorization algebra. A key ingredient will be a higher dimensional generalization of the Hochschild-Kostant-Rosenberg theorem, due to Calaque-Willwacher, which will put us in dialogue with the CPTVV approach to categorical deformation quantization.
Lecture 3: The tangle hypothesis and the bridge by Koszul duality between the quantizations. The goal here is to indicate how (a class of) line defects can be encoded by stratified factorization algebras, and how this gives us a bridge to braided monoidal categories. Quantization of CS will thus determine a deformation of braided monoidal categories. We will discuss key tools—a filtered Koszul duality and a charged fermion—that will let us identify this deformation with a quantum group.

Spring 2018

Samuel Grushevsky (Stony Brook University), Mirzakhani's recursion for Weil-Petersson volumes of moduli spaces.

Time and rooms:

Tuesday, April 17, 10:30am-12:00pm, Room: Lake Hall -- LA 509 (LA is building 34 on this map).

Wednesday, April 18, 10:30am-12:00pm, LA 509.

Thursday, April 19, 10:30am-12:00pm, Richards Hall -- RI 458 (RI is building 42 on this map).

Abstract:

The Weil-Petersson metric is a natural Kaehler metric on the moduli space of Riemann surfaces. We will start by outlining the proof of Wolpert's formula that expresses the metric in terms of the "length and twist" parameters for cutting up a Riemann surface into a collection of spheres with three holes (i.e. in terms of Fenchel-Nielsen coordinates). We will then state Mirzakhani's recursive formula for the Weil-Petersson volumes of the moduli spaces of bordered Riemann surfaces, and explain some ingredients of the proof, and also how this recursion implies the Witten's conjecture that the intersection numbers of tautological classes on moduli satisfies the KdV integrable hierarchy.

Alex Küronya (Goethe-Universität Frankfurt), Geometric aspects of Newton-Okounkov bodies.

Time and rooms:

Monday, March 12, 2:30-3:30pm. Room: Lake Hall -- LA 509 (LA is building 34 on this map).

Tuesday, March 13, 10:30-11:30am, LA 509.

Wednesday, March 14, 2:30-3:30pm, LA 509.

Thursday, March 15, 10:30-11:30am, LA 509.

Abstract:

Recent years have witnessed a new way to introduce convex geometric methods to areas of mathematics around algebraic geometry: based on earlier works of Newton and Okounkov, Kaveh-Khovanskii and Lazarsfeld-Mustata defined convex bodies (so-called Newton-Okounkov bodies), which capture the vanishing behaviour of sections of line bundles.

As a first approximation, the theory of Newton-Okounkov bodies is an attempt to create a correspondence between line bundles and convex bodies known from toric geometry, except that in the absence of a large torus action, one has to make do with an infinite collection of bodies for every line bundle.

This point of view has been fairly successful in that Newton-Okounkov bodies has been shown to encode positivity of line bundles, and they also serve as targets for completely integrable systems analogous to moment maps. The theory has exciting connections with symplectic geometry, representation theory, and combinatorics for instance, nevertheless, in these lectures we will focus on its applications to projective geometry.

After reviewing fundamental notions of positivity for line bundles and introducing Newton-Okounkov bodies along with their basic theory, we will discuss the case of surfaces, where there is a particularly satisfying theory, and the connection to (local) positivity of line bundles. If time permits, we will look at how to define interesting functions on Newton-Okounkov bodies which yield an interesting connection to Diophantine approximation.

Fall 2017

François Charles (Université Paris-Sud), Algebraic cycles and Arakelov geometry.

Time and rooms:

Tuesday, October 10, 9-10:30am, Behrakis 204

Wednesday, October 11, 5-6:30pm, Cargill 097

Friday, October 13, 10-11:30am, Ryder 155

Abstract:

Arakelov geometry gives a way to work geometrically with schemes defined over the integers. We will discuss some applications of Arakelov geometry to some problems in algebraic cycles and periods, trying to emphasize how geometric ideas can be translated in the setting of arithmetic geometry. The plan is:

Lecture 1: general setting of Arakelov geometry, relationship to geometry of numbers.

Lecture 2: application of arithmetic intersection theory to isogenies of elliptic curves.

Lecture 3: application to transcendance problems, theta-invariants.

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