Potential PhD candidates interested in starting in the fall of 2024 should apply as soon as possible, and preferably before July 31, 2024. A second round of applications will be organized in the spring of 2025 for a start in the fall of 2025.
The following PhD positions will be funded through GRACE:
Computational number theory and cryptographic applications
Post-quantum cryptography and fully-homomorphic encryption
Supervised by Jean-Sébastien Coron, with connections to Gabor Wiese
The project will consider the construction and analysis of fully-homomorphic encryption schemes based on lattice problems.
Abelian varieties and their applications to cryptography
Supervised by Antonella Perucca
The research project concerns an explicit and algorithmic study of elliptic curves and, more generally, abelian varieties, defined over number fields and finite fields. Advances in this direction are welcome for both classical and post-quantum cryptography (elliptic curve cryptography and isogeny-based cryptography).
Exploration of modular forms and Galois representations
Supervised by Gabor Wiese, with connections to Antonella Perucca
The project supervised evolves around computational and theoretical methods in number theory with possible focus on Galois representations, modular forms, number fields and potential relations to cryptography.
Modelling and computational experiments in geometry
Three-dimensional conformally flat Lorentzian manifolds through experimentation
Supervised by Karin Melnick
The general area of this project is group actions on manifolds. Potential topics include conformal transformations of Lorentzian or other pseudo-Riemannian manifolds as well as parabolic geometries with locally submaximal symmetry. Projects will include the use of computational tools to search for examples with special local or global properties and to aid in classification.
Computational experiments with 3-dimensional hyperbolic volume
Supervised by Jean-Marc Schlenker, with connections to Karin Melnick
Convex co-compact 3-dimensional hyperbolic manifolds have finite topology but infinite volume. However they have a well-defined renormalized volume, first defined in the physics literature, and then quickly introduced in the mathematics literature.
The renormalized volume is closely related to the AdS/CFT correspondence, a central tenet of quantum gravity, which asserts that the partition function of a CFT on a certain “boundary” manifold Y is equal to the sum of the exponentials of minus the actions of certain superstring theories on all suitable manifolds X with ∂X = Y (or in fact on the products of those fillings by “small” manifolds which disappear in a certain limit). When Y has dimension 2 and one considers a certain highly simplified limit, the action limits to the renormalized volume of X.
This project will investigate, including by computational methods, several questions on the renormalized volume of 3-dimensional hyperbolic manifolds which are motivated by the AdS/CFT correspondence.
Algebraic geometry, representation theory and logic
Representation-theoretic methods in algebraic geometry
Supervised by Karin Melnick and Pieter Belmans
The candidate will work on a project combining algebraic geometry and representation theory. Algebraic groups and quivers allow for explicit calculations which in turn give us tools to understand associated algebro-geometric objects, from numerical invariants to derived categories.
Explicit calculations of elliptic cohomology for toric varieties and beyond
Supervised by Sarah Scherotzke, with connections to Pieter Belmans
Toric varieties represent an important class of algebraic varieties. Their purely combinatorial definition using so-called fans makes them a big class of examples which can explicitly be studied using mathematical software. Note that many important algebraic varieties of interest in representation theory are toric varieties.
We propose to study the elliptic cohomology and elliptic genus of toric varieties. Witten's construction of a universal elliptic genus showed that these objects have deep connections with quantum field theory. Today, elliptic cohomology is a central object of intense research in homotopy theory.
Expressivity of modal languages for spatial model checking
Supervised by Bruno Teheux
The candidate will work on a project revolving around the connections between topology, geometry and logic. Possible areas include, but are not limited to, topological and polyhedral semantics for modal or epistemic logics and their use in computer science, categories for logic, and related topics.