GQT Graduate School and Colloquium 2025

Graduate School abstracts

Annegret Burtscher (RU Nijmegen): Global Lorentzian Geometry

(Monday 25 August)

Abstract

In this minicourse we discuss some novel geometric approaches relevant for understanding (parts of) the universe based on Einstein's theory of general relativity. Our starting and end point is the Hawking singularity theorem from the 1960s which describes the occurrence of geodesic incompleteness in situations like a (mathematical) Big Bang. We first introduce the classical differential geometric ingredients for its formulation and proof, and then move on to a new approach via Lorentzian synthetic spaces. In the latter setting the background is no longer a smooth manifold but a metric measure space with a certain causal structure. Also curvature inequalities are no longer computed pointwise via tensor calculus but formulated globally using optimal transport theory.

References:

* J. Beem, P. Ehrlich, K. Easley. Global Lorentzian Geometry, second edition. CRC Press (1996)

* F. Cavalletti, A. Mondino. A review of Lorentzian synthetic theory of timelike Ricci curvature bounds. Gen. Relativity Gravitation 54 (2022), no. 11, Paper No. 137

* R. McCann. Displacement convexity of Boltzmann's entropy characterizes the strong energy condition from general relativity. Cambridge Journal of Mathematics 8:3 (2020) 609-681

* E. Minguzzi. Lorentzian causality theory. Living Reviews in Relativity 22 (2019), Paper No. 3

 

Francesca Arici (Leiden University): C^* algebras and their extensions

(Tuesday 26 August)

Abstract

This course introduces the theory of C*-algebras with an emphasis on their extensions and their algebraic and geometric aspects. We begin by presenting C*-algebras as noncommutative analogues of topological spaces, highlighting their role in noncommutative geometry, and reviewing their basic properties and representations. The second part of the course develops the theory of extensions, focusing on exact sequences and Busby invariants. We conclude with examples such as the Toeplitz and Cuntz algebras, emphasizing their connections with K-theory and noncommutative topology.

 

Pieter Belmans (Utrecht University): Quivers and Moduli spaces

(Wednesday 27 August)

Abstract

Quivers are directed multigraphs, and provide a flexible framework for encoding a wide range of linear algebraic problems. A representation of a quiver assigns vector spaces to vertices and linear maps to arrows, turning the combinatorial structure into an algebraic one. This perspective unifies and illuminates classical constructions in representation theory. Quiver representations thus offer a powerful setting in which questions about vector spaces, modules, and homomorphisms can be studied in a systematic and structured way.

 

This lecture series will introduce the basics of quiver representations and then briefly describe the geometric tools required for their classification. Time permitting, we will end with an overview of their similarities to other moduli spaces in algebraic geometry.

 

 

References:

 

* Markus Reineke, Moduli of representations of quivers, https://arxiv.org/abs/0802.2147

 

* Ryan Schiffler, Quiver Representations, https://doi.org/10.1007/978-3-319-09204-1

 

* Harm Derksen and Jerzy Weyman, An Introduction to Quiver Representations, https://bookstore.ams.org/gsm-184/