Graduate
Advanced Mini-Courses
Fall 2024
Bernhard Keller (Université Paris Cité), October 21-25, 2024
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.
Lecture 2: Review on sheaves, constructible sheaves, and subanalytic sheaves.
Lecture 3: The regular Riemann-Hilbert correspondence in arbitrary dimension: statement and sketch of proof.
Lecture 4: The irregular Riemann-Hilbert correspondence.
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
Dates and times: April 27-May 1.
Cancelled due to the COVID-19 pandemic
Fall 2019
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.
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
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.