Jan-Willem van Ittersum (UvA)
Abelian differentials
Large genus asymptotics of Masur-Veech volumes
In this lecture series, we study Masur–Veech volumes of strata of Abelian differentials, which parametrize pairs (X,ω) consisting of a Riemann surface X and a holomorphic 1-form ω with prescribed orders of zeros. These moduli spaces refine the moduli space of curves and carry natural period coordinates with respect to which the Masur–Veech measure is defined. Their volumes encode rich geometric and dynamical information and are closely related to billiard dynamics in rational polygons.
A central idea is that these volumes can be expressed in terms of Hurwitz numbers of Riemann surfaces, which count branched coverings with specified ramification data. This connection translates geometric questions into combinatorial ones and makes it possible to study large-genus asymptotics using representation-theoretic tools, such as representations of the (infinite) symmetric group and shifted symmetric functions. In the study of these asymptotics, quasimodular forms naturally appear as generating series of Hurwitz numbers. Modular forms are explicit functions on the complex upper half plane, which can be viewed as sections of natural line bundles over the moduli space of elliptic curves. Quasimodular forms are natural extensions of modular forms.
The emphasis throughout is on the interplay between geometry, combinatorics, and representation theory.
Literature
- Renzo Cavalieri and Eric Miles, Riemann surfaces and algebraic curves: A first course in Hurwitz theory. London Math. Soc. Stud. Texts 87, Cambridge Univ. Press, Cambridge, 2016, xii+183 pp.
- Spencer Bloch, Andrei Okounkov, The character of the infinite wedge representation. Adv. Math. 149 (2000), no. 1, 1–60.
- Alex Eskin and Andrei Okounkov, Asymptotics of numbers of branched coverings of a torus and volumes of moduli spaces of holomorphic differentials. Invent. Math. 145 (2001), no. 1, 59–103.
- Dawei Chen, Martin Möller, Don Zagier, Quasimodularity and large genus limits of Siegel-Veech constants. J. Amer. Math. Soc. 31 (2018), no. 4, 1059–1163.
Bas Janssens (TU Delft)
Unitary representations of infinite-dimensional Lie groups
Continuous symmetries are modelled by Lie groups, and unitary representation theory governs how these symmetries appear in quantum theory. For finite-dimensional Lie groups, our understanding of the representation theory has reached an amazing level of refinement over the course of the 20th century. However, it turns out that many of the the groups of symmetries/transformations in foundational physical theories are infinite-dimensional -- think of diffeomorphisms in General Relativity, or gauge transformations in Yang-Mills theory. This course will be a gentle introduction to the unitary representation theory of infinite-dimensional Lie groups, with an emphasis on general features (central extensions, cohomology, positive energy conditions) and the way in which they play out in classes of examples.
Prerequisites: I will assume some familiarity with Lie theory (Lie groups, Lie algebras) and differential geometry (line bundles, Chern classes).
Steffen Sagave (Radboud)
Higher categories and stable homotopy theory
Higher categories are a generalization of ordinary categories that allows one to systematically keep track of the way in which different mathematical objects are equivalent to each other, including higher level coherence data such as equivalences between equivalences of objects. Based on work by Joyal and Lurie, this subject area underwent a rapid development during the last two decades. It now serves as a powerful foundation for research in various directions within pure mathematics, including algebraic topology, but for example also parts of representation theory and algebraic geometry. In this course, I will give an introduction to higher category theory, with a focus on motivating ideas and central examples, rather than proofs of foundational results.
Prerequisites: This course is meant to be accessible to PhD candidates in topics related to the themes of the GQT-cluster. I will assume familiarity with basic notions in ordinary category theory (categories, functors, natural transformations, limits and colimits) and topology (topological spaces, continuous maps, homotopies).