Short Courses

Mohammed Abouzaid: Theory of bordisms

6th August, 2024

In this introductory lecture, which should be accessible to a general mathematical audience, I will review the classical bordism theory of manifolds, from its origin in Poincare's work, to the subsequent development by Pontryagin, Thom, Milnor, Wall, and Quillen among others.

Lecture 2: Bordism of orbifolds

An orbifold is a space with additional structure that describes it locally as the quotient of a manifold by a finite group. I will describe Pardon's recent result which reduces the study of orbifolds to the study of manifolds with Lie group actions. Then I will explain the relationship between equivariant and orbifold bordism, and formulation some structural properties of this theory.

Lecture 3: Bordism of derived orbifolds

The notion of a derived orbifold arises naturally in pseudo-holomorphic curve theory, and plays a central role in the emerging field of Floer homotopy. I will explain how it is related to the notion of "homotopical bordism" due to tom Dieck in the 1970s, and formulate some conjectures about its structure in the complex oriented case.

Ana Caraiani: A glimpse into the Langlands programme

20th May, 2024

The goal of this lecture series is to give you a glimpse into the Langlands programme, a central topic at the intersection of algebraic number theory, algebraic geometry and representation theory. In the first lecture, we will look at a celebrated instance of the Langlands correspondence, namely the modularity of elliptic curves. I will try to give you a sense of the different meanings of modularity and of the multitude of ingredients that go into establishing such a result. In the following lectures, I will focus on the more geometric ingredients, first in the special case of the modular curve and then for higher-dimensional Shimura varieties.

Charlotte Chan: Deligne-Lusztig theory: examples and applications

20th May, 2024

Geometry and representation theory are intertwined in deep and foundational ways. One of the most important instances of this relationship was uncovered in the 1970s by Deligne and Lusztig: the representation theory of matrix groups over finite fields is encoded in the geometry of a natural 'partition' of flag varieties. Recent developments have revealed rich connections between Deligne-Lusztig varieties and geometry studied in number-theoretic contexts. In this lecture series, we give an example-based tour of these ideas, focusing on how to extract concrete information from theory.

Peter Scholze: Real local Langlands as geometric Langlands on the twistor-ℙ1

4th March, 2024

In 2014, Fargues realized that one can formulate the local Langlands correspondence over p-adic fields as a geometric Langlands correspondence on the Fargues-Fontaine curve. This raises the question of a similar realization of the local Langlands correspondence over the real numbers. The goal of these lectures is to explain a possible formulation. As part of this, we will give a new perspective on the theory of variations of twistor structure, a generalization of the theory of variations of Hodge structure. This uses the theory of analytic stacks developed in our joint work with Clausen, of which we will give a brief overview.

Ivan Losev: Quantum category 𝒪

1st March, 2024

The representation theory of quantum groups including at roots of unity is an important part of Lie representation theory. In this talk, we will study one of categories of representations: the quantum category 𝒪, which is a suitable analogue of the classical Bernstein-Gelfand category 𝒪. We will relate it to a model representation category, the affine Hecke category, more precisely to the heart of the new t-structure on that category (all these terms will be defined in the lectures).

Tanja Hinderer: Analytical methods in general relativity

24th January, 2024

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Anton Ilderton: Strong-field and non-perturbative amplitudes

22nd January, 2024

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Quoos Luciane: Semigroups and algebraic geometry codes

6th November, 2023

A linear code is a vector subspace of 𝔽qn, where 𝔽q is a finite field with q elements. The family of linear error-correcting codes are specially important when one is attempting to transmit messages across a noisy communication channel. Data can be corrupted in transmission or storage by a variety of undesirable phenomenon, such as radio interference, electrical noise, scratch, etc.. It is useful to have a way to detect and correct such data corruption. An error-correcting code can correct more errors larger is its minimum distance. This course aims to introduce a family of error-correcting codes, the Algebraic Geometry Codes, and show how to use the theory of semigroups to improve the minimum distance of the code. This construction of codes make use of a function field in one variable over a finite field. We will show how the local information in one or two rational places, the knowledge of the semigroup in these places, can be used to improve the minimum distance of the code.

Maryna Viazovska: Fourier Uniqueness and Interpolation

25th October, 2023

Can we reconstruct a function by knowing only a subset of its values and a subset of the values of the function's Fourier transform?

How many values do we need to know for such a reconstruction? Can we interpolate a given subset of values? What are the possible applications of such interpolation? In this series of lectures, we will try to answer these questions.

In the first lecture, we will speak about the Cohn-Elkies linear programming bound for the sphere packing and how this bound's analysis led to the discovery of a Fourier interpolation formula. The second lecture will discuss explicit constructions of Fourier uniqueness sets and Fourier interpolation formulas. The third lecture will focus on analytic approaches to Fourier uniqueness and interpolation.

Sofia Tirabassi: Lieblich-Olsson deformation technique and applications

12th September, 2023

In this lecture series I will explain how one can use deformation theory to study derived categories in positive characteristic.

I will start by giving an overview on what does it mean to 'lift' something 'to characteristic 0' and when is this possible. Then I will present a baby example: the study of the Fourier-Mukai partners of products of elliptic curves over algebraically closed fields of characteristic at least 5. After that, I will present Lieblich-Olsson deformation technique which allows us to deform derived equivalence. This is a very versatile tools with many applications (not just in positive characteristic!). I will conclude the series by going over some of these applications in greater details.

Daniel Huybrechts: Brauer groups and twisted sheaves on K3 surfaces

11th September, 2023

This will be a gentle introduction into Brauer groups and twisted sheaves. The emphasis will be on geometric aspects and eventually on moduli spaces of twisted sheaves on K3 surfaces. We will study the different ways to think about Brauer groups as groups of Azumaya algebras, Brauer-Severi varieties, twisted sheaves, 𝔾m-gerbes... How to translate from one to the other, how to define Chern classes, how to split Brauer classes, etc.

Christian Lehn: Lagrangian Fibrations of Holomorphic Symplectic Varieties

11th September, 2023

By Matsushita's fundamental results, Lagrangian fibrations are essentially the only morphisms on irreducible holomorphic symplectic varieties with positive fibre dimension. We will start by reviewing these results and discuss their validity also for singular symplectic varieties. We will study singular fibres and some of the fundamental conjectures. Towards the end of the course, we will turn to some of the fascinating recent developments in the Hodge theory of Lagrangian fibrations.

Viktor Todorov: Non-parametric Methods for Short-Dated Options

20th April, 2023

There has been a recent dramatic increase in trading on exchanges of short-dated options, i.e., options with very short time to expiration. This workshop covers non-parametric methods for extracting information from these options. Formal non-parametric econometric analysis of derivatives data has proved difficult. The complications arise from the highly non-linear dependence of option prices on state variables and parameters as well as the possible dependence in the option observation errors. The short-dated options allow to aggregate option data in ways that facilitate the practical application of asymptotic expansions for option maturities approaching zero.

We first start by introducting various model-free measures of spot volatility. These measures separate true spot volatility from the price jump component (and its pricing) as well as the volatility mean-reversion effects present in option prices. Following that, we introduce measures of risk-neutral jump variation and jump tails as well as methods for studying anticipated event risks. Empirical illustrations of the methods will be presented along with various applications for studying volatility forecasting, return predictability via option measures and analysis of risk premia.

Gigliola Staffilani: The Study of Wave Interactions: Where Beautiful Mathematical Ideas Come Together

25th January, 2023

Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge mathematicians have developed interdisciplinary approaches that are powerful and beautiful. In the first lecture, which will be colloquium style, I will describe some of these approaches and show for example how the need to understand certain multilinear and periodic wave interactions provided also the tools to prove a famous conjecture in number theory, or how classical tools in probability gave the right framework to still have viable theories behind certain deterministic counterexamples. In the second and third lecture I will open a small window into the concept of weak wave turbulence. I will start with the deterministic approach of Bourgain, involving the study of long time asymptotic of higher Sobolev norms of solutions of dispersive equations, and I will end with the rigorous derivation of a 3-wave kinetic equation.

Asilata Bapat: Triangulations, rigid motions, and applications to representation theory

27th June, 2022

The course will begin with a brief survey of the theory of triangulations of a convex n-gon, which appear in several different places in mathematics. This course will focus on one such, possibly unexpected, appearance, namely in the theory of rigid motions of points in the plane. With this perspective, we will move to non-convex arrangements of points, and discuss the appropriate replacement of a triangulation. Finally, we will say a word about how these constructions are related to representation theory, via configuration spaces of points in the plane and a certain category of quiver representations. I will also mention some open questions in this direction.

Geordie Williamson: Kazhdan-Lusztig Polynomials: Representation, Geometry and Combinatorics

27th June, 2022

This will be a course on the representation theory of algebraic groups, and relations to the representation theory of symmetric groups. Reductive algebraic over finite fields and their algebraic closures are fascinating objects: one the one hand they look like Lie groups, but on the other hand they look like finite groups. Thus they mix two very different areas of mathematics. I will outline some of the basic theory, and then move on to questions of current interest.

Ting Xue's lectures in the previous week will provide essential background. I will aim to point out connections to the modular representation theory of finite groups. Although not essential, some background in algebraic geometry (e.g. the first three chapters of Hartshorne's Algebraic Geometry) will help with understanding latter parts of the course.

Uri Onn: Representation zeta functions

20th June, 2022

Algebraic groups are fundamental objects in representation theory and number theory. The course will discuss the structure theory of linear algebraic groups over algebraically closed fields. Topics include tori, parabolic subgroups and Borel subgroups, Lie algebras, root data, Weyl group, and classification of simple algebraic groups. If time permits, we will briefly discuss relation between compact Lie groups and algebraic groups.

Ting Xue: Introduction to linear algebraic groups

20th June, 2022

Algebraic groups are fundamental objects in representation theory and number theory. The course will discuss the structure theory of linear algebraic groups over algebraically closed fields. Topics include tori, parabolic subgroups and Borel subgroups, Lie algebras, root data, Weyl group, and classification of simple algebraic groups. If time permits, we will briefly discuss relation between compact Lie groups and algebraic groups.

Caucher Baukar: Classification Theory of Algebraic Varieties

16th February, 2022

The classification of algebraic varieties is at the heart of algebraic geometry. With roots in the ancient world the theory saw great advances in dimensions one and two in the 19th century and the first half of 20th century. It was only in the 1970-80s that a general framework was formulated, and by the early 1990s a satisfactory theory was developed in dimension 3. The last 30 years has seen great progress in all dimensions.