This seminar has ended in this form, because Raphaël and I are leaving for new positions.
Past talks
The answer is: I have no idea! :-) Which is the point of this talk. First I’ll give a very pedestrian introduction to holonomic functions and holonomic ideals/modules for the Weyl algebra. Second I’ll introduce the notion of Airy structures, as originally introduced by Kontsevich and Soibelman in 2017. As you will see, the two notions share many similarities: are they actually the same, or special cases thereof (in one direction or the other)? This is what we will try to answer together!
After the Jones polynomial was invented in early 1980s, Kauffman realized that one can obtain the Jones polynomial by replacing each crossing with no crossing in two ways. This idea has since been generalized vastly in different directions. We will see one such generalization, namely skein modules, and discuss how they provide deformation quantizations of character varieties of 3-manifolds.
Using a toy-model of Turok & Boyle, for expanding universes where matter and radiation are averaged out, a quantum theory for the scale-factor of Friedman-LeMaître-Robertson-Walker (FLRW) metrics can be defined. We will introduce that theory, and under some genericity assumptions, write the resurgence relations of the corresponding Schrödinger eigenvalue equations, relating the behaviour of the scale-factor’s wave-function at early and late times.
In this talk, we describe a method to find modular equations which helps determine the asymptotical behaviour of Fourier coefficients of modular forms. We start by discussing the arithmetic properties of the Eisenstein series, which are the main tools of this method. Then we present some applications of these results. These include counting the number of representations of a positive integer by a quadratic form (refinements of Siegel’s formula), counting the number of certain types of partitions, and comparisons between the number of representations by triangular numbers and the number of representations by squares. Bonus: If we have time, we give one more application in which we use extensions of Rogers-Ramanujan reciprocal identities to determine a new formula for the class number in terms of a limit of Fourier coefficients of modular forms which seems to be converging fast.
In the 1920s Dirac proposed an equation to describe fermionic particles. In order to explain the solutions with negative energy, Dirac proposed a model in 1930 called Dirac sea, in which “the vacuum is not void”. This model provisions the existence of positrons, which was experimentally detected in 1932. The mathematical model underlying Dirac sea is a semi-infinite version of the fermion Fock space. In this talk I will briefly review the stories of Dirac sea and introduce some semi-infinite (a.k.a. Tate) linear algebra if time permitting.
The coherent-constructible correspondence (CCC) for toric varieties is a version of homological mirror symmetry where the Fukaya category is formulated using constructible sheaves. The proof of CCC ultimately leads to investigations of questions in algebraic geometry and commutative algebra using symplectic methods. In this talk, I will explain the basic mechanism of CCC and how to use it to prove a conjecture of Orlov regarding the dimension of the derived category of toric varieties. I will connect this result to a geometric construction that universally provides wall-crossing functors connecting the CCC mirrors of all toric varieties constructed from the same linear GIT problem.
Conformal field theory has its origins in the early 1970s via an operator algebra formalism used to study the Ising model, and the seminal 1984 paper of Belavin, Polyakov, and Zamolodchikov cemented the role of CFT as a uniquely special type of QFT. This talk will describe aspects of the interplay between CFT and random geometry. Introduced by Schramm in the early 2000s, random geometry uses tools from probability theory and complex analysis to study scaling limits of two-dimensional statistical mechanics models. This approach allowed mathematicians to rigorously verify many calculations made by physicists using CFT. More recently, ideas from CFT have allowed mathematicians to bring novel insight into their understanding of the underlying random geometry.
W-algebras are vertex algebras obtained as quantum Hamiltonian reductions of affine vertex algebras. Because of this construction, they are characterized as the (universal) kernel of screening operators acting on certain free-field algebras. In this talk, we discuss these screenings from several viewpoints: the classical one for integrable systems and the quantum one for the logarithmic Kazhdan-Lusztig correspondence for representation theory.
KdV hierarchy is one of the most famous infinite dimensional integrable systems found originally as the non-linear PDE which describes solitary waves in the shallow surface. It is now a more or less ubiquitous family of PDEs appearing in enumerative geometry. In this talk, I will review one of its algebro-geometric aspects related to W-algebras appearing in conformal field theory, following an old work of Feigin and Frenkel.
In the 1980’s Moore and Seiberg showed that a conformal field theory gives rise to a mathematical structure called a modular fusion category. Since then, modular fusion categories, and more generally tensor categories, have been shown to have deep connections to modular representation theory, quantum groups and topological phases of matter. A major conjecture in the area is whether every modular fusion category comes from a conformal field theory. In this talk I will explain one strategy to solving this conjecture by having topological defect lines act on vertex operator algebras.
Hurwitz numbers count covers of Riemann surfaces with given ramification, or equivalently, decompositions in symmetric groups with given cycle types. Since the start of the century, they have been connected to many other interesting things as well: integrable hierarchies, Gromov--Witten theory, and topological recursion. I will give a short introduction to some of these relations, also dipping in the tropical formulation of Hurwitz numbers.
After giving a clean geometric interpretation of the infamous renormalisation procedure of theoretical physics in terms of the classical theory of envelopes, we will show how to apply it where it seems to not be needed but comes to save the day. Namely, the recursive analysis of singularly perturbed linear differential systems known as the WKB method.
Talks from the previous run, Winter 2021-22
Airy structures are too. Which is the topic of this talk. I’ll tell you all the cool stuff about Airy structures (first introduced by Kontsevich and Soibelman in 2017, as an algebraic reformulation of topological recursion): what they are, and what makes them interesting. It will be very introductory. I’ll go through the definition, and the details of the main proof of existence and uniqueness which is the foundation of the theory. Then I’ll tell you a little bit about the connection with VOAs, highest weight vectors, and Whittaker vectors. Fun stuff.
The term “loop equations” was coined by Migdal in the early 90’s in the context of quark confinement to refer to certain sets of equations describing the propagation of test particles in quantum field theory. They have since appeared many times at the boundary between mathematics and physics. From statistical spectroscopy to enumerative geometry via integrable hierarchies, they encode conformal features of various problems in an analytic way. The prototypical situation where loop equations are valuable is that of matrix integrals, with applications ranging from all the aforementioned topics to number theory and even artificial neural networks. A seemingly antipodal approach to matrix integrals is that of orthogonal polynomials, unveiling a subtle relationship between loop equations and the geometry of meromorphic connections. This bridge can be abstracted in terms of W-algebras and we shall cross it towards a quark confinement heuristics in terms of vertex operators.
Cohomological field theories (CohFTs) are a generalisation of two-dimensional topological quantum field theories (TFTs) that are also an axiomatic version of Gromov-Witten theory. Moreover, semisimple CohFTs can be reconstructed from an underlying TFT together with some extra data via Givental-Teleman theory. This last formalism turns out to be almost equivalent to topological recursion. I will give an introduction to all of these points of view, and if time permits relate this to Hurwitz numbers.
Vertex operator algebras (VOAs) are one of the main mathematical approaches to conformal field theories. Being such important objects, understanding their representation theory has been of great interest to mathematicians over the last few decades. The simplest examples, from the viewpoint of representations, are called Holomorphic VOAs. While a Holomorphic VOA has a simple representation theory, the VOA formed by the fixed points of a group action will not. VOAs constructed in this manner are called Orbifolds, and in this talk I will explain how by looking at “Twisted” modules of a Holomorphic VOA you can understand the representation theory of the Orbifold VOA. In particular, I will explain how these twisted modules can be used to prove the Dijkgraaf-Witten Conjecture (and go much further).
The Hindus and Buddhists argue that “Everything is its opposite”. Inspired by this koan, we want to relate the intersection theory on the moduli space of open and closed curves. Such a relation is given by a formula of Buryak that relates the two generating functions. I will present a matrix model proof of this formula.
Math has been profoundly influenced by our nonmathematical experience. The planet we evolved on, especially the senses we're stuck with, is for better or worse the starting point of our math – our notions of space and quantity and symmetry etc. And to be honest our math hasn’t really moved far from its prehistoric beginnings in that sense. Our math I fear is still that of bipedal apes frolicking on the savannah. But the quantum gifts us a very different reality. I am interested in the hints quantum field theory gives us about where our math should be heading. About how parochial our math still is. How classical is the quantum world? Not classical in the physics sense, but classical in the mathematical sense. In particular, this talk will ask: how naive is our notion of symmetry?
Abstract: In the late 1980's Edward Witten coined the term "topological quantum field theory" and is oft-quoted as comparing the subject to the more classical group theory both in importance and structure. In a particularly small dimension, and with suitable definitions, we now know that a topological quantum field theory is equivalent to a "modular tensor category", a less topological and very algebraically-structured object. In many ways these are the finite groups of the future. To organize the futuristic finite groups, modular tensor categories possess a categorical notion of dimension akin to computing the dimension of a representation of a finite group by taking the trace of the identity matrix. The difference is that categorical dimensions are not necessarily (rational) integers. Trivial exercises from your undergraduate class on finite groups now become research papers. In this talk I'll discuss my pandemic-age classification of modular tensor categories of prime categorical dimension contained at the very end of this paper: doi.org/10.1016/j.jalgebra.2020.10.014
Oftentimes, we are interested in studying continuous families of objects parametrised by some space (for example, a smoothly-varying family of one-dimensional vector spaces over a manifold, which is better known as a line bundle). If you allow the parameter space to vary, the study of such families can become quite overwhelming, unless you can somehow find a "universal" representative family. The challenge of doing this is called a moduli problem. In general, these problems cannot be solved---at least, not with traditional spaces, so the notion of stacks were introduced to remedy the situations where universal families were impossible to find. In this talk, I will give a high-level (and hopefully friendly) overview of the basics of algebraic stacks, and briefly describe some ways of constructing stacks to solve moduli problems.
Join me as we go on an adventure. We’ll see the lay of the land, and plot our course. Along the way we will pass such sites as the foothills of symmetry, and the rolling planes of modular forms. Terrors shall be braved like the fearsome differential equation, and character of a VOA. We’ll witness the majestic beauty of the vector-valued modular form in its natural habitat. And should we dare (or time permits) we’ll scale all the way to the top of mount Multi-vectored Jacobi form and take in its view. We’ll see these sights and many more with me as your guide as we adventure to modular forms and beyond!
Links and knots inside R^3 can be visualized by projecting them on a sheet of paper while keeping track of over and under crossings. Thickening the resulting graphs one obtains ribbon graphs. Ribbon graphs are composed out of elementary pieces. These elementary pieces can be identified with morphisms in a ribbon category and so every ribbon graph and hence link can be identified with a morphism. Isotopic links necessarily have the same associated morphisms, so these provide a good invariant. I will introduce both ribbon categories and ribbon graphs in some detail. Next I will explain subtleties and solutions arising if categories fail to be semisimple and where to find such categories.
The Virasoro algebra of conformal transformations of the plane admits a q-deformation appearing in toy models as well as natural phenomena. In this talk I will describe how the topological recursion method can be continued from computing the large-weight asymptotics of conformal blocks of the Virasoro algebra to cater to that of its deformed counterpart.
In the mid 2000's multiple researchers discovered that the same recursive formula kept appearing in different matrix models. Then, in 2007, Eynard and Orantin had the idea to generalise this recursion, making it a definition of meromorphic n-differentials on an algebraic curve that were computed from the initial data of a spectral curve. It has since been conjectured, and proven in some cases, that these meromorphic n-differentials may be used to construct solutions of a quantisation of the original spectral curve; these quantisatons are the so-called quantum curves. In this talk we will introduce the theory of quantum curves and topological recursion, before briefly mentioning potential applications and opportunities for future research.
When we feel overwhelmed in life it is often because we are unable to let go of things, whether it be literal objects, or intangibles like love, performance, or trauma. Here we apply this yoga to Conformal Field theory, and try to let go of as much of the structure as possible (as quickly as possible) and we will arrive at Fusion Rings and their Categorifications. In fact, we have already seen examples of these in Reinier's talk last week in the form of the representation categories of symmetric groups and their corresponding character rings. We will look at lots of examples which don't come from finite groups and see how rich the mathematical theory is, even after letting go of so much of the motivation.
Symmetric functions are a classical subject in mathematics, going back centuries. They can be used to express coefficients of polynomials in terms of their roots, and as such have been instrumental in Galois theory and the theory of characteristic classes. Other symmetric polynomials encode the representation theory of symmetric groups and general linear algebras, or other related algebraic objects. Both of these aspects come together in their importance as a building block for matrix models and solutions of integrable hierarchies, as can be seen in the interpretation of either fermionic or bosonic Fock space as a highest weight module.