# 6 PhD projects

### Vacancies

The KdVI currently has vacancies for 6 PhD positions in Mathematics.

### Project descriptions

#### Project 1: Frobenius manifolds and integrable systems

The general topic will involve a number of questions in geometry and mathematical physics coming from the theory of Frobenius manifolds and integrable systems, and their interactions with cohomological field theory, combinatorics, geometry of moduli spaces. More specific topics which will be covered are: relations between Gromov-Witten type invariants and integrable hierarchies; Hamiltonian structures for integrable hierarchies and their deformation theory; infinite dimensional Frobenius manifold theory and integrable hierarchies in higher number of spatial variables; interaction of the Givental group action and the explicit formulation of tau-symmetric integrable hierarchies, in particular their quadratic Hirota equations. Affinity with mathematical physics is a plus.

The student is expected to spend roughly half of the time in Amsterdam and half in Dijon.

Supervisors: Prof. Sergey Shadrin (Universiteit van Amsterdam) and Prof. Guido Carlet (Université de Bourgogne).

#### Project 2: Lie Algebroids and deformation quantization

Project 2 centers around the theory of Lie algebroids. Lie algebroids are certain geometric objects encoding symmetries, and occur in a wide variety of situations. As such, they can be studied from many different points of view, bringing together techniques from differential/algebraic geometry, representation theory and noncommutative geometry. The aim of this project is to compare these different approaches in the study of representations of Lie algebroids and use the setting of Lie algebroids to explore recent new ideas in the theory of deformation quantization, making a connection to the theory of Frobenius manifolds.

Supervisor: Dr Hessel Posthuma

#### Project 3: Frobenius manifolds and their relation to homological mirror symmetry

The research topic will be Frobenius manifolds and their relation to homological mirror symmetry. In particular, the candidate will investigate how Frobenius manifolds appear both in the deformation theory and the theory of stability conditions of categories that come from mirror symmetry and tie this to singularity theory and the theory of Riemann surfaces.

Supervisor: Dr Raf Bocklandt.

#### Project 4: Infinite-dimensional affine and polynomial preserving processes

This project concerns infinite-dimensional SDEs and their practical applications in financial markets. Topics of interest include stochastic partial differential equations, weak approximation, affine stochastic processes, pricing options, energy derivatives, and hedging approaches.

Applicants should have a strong background in analysis and/or probability theory. Ideally speaking the candidate has written an MSc thesis on a topic in stochastic analysis. Furthermore, a profound interest to combine theory with practical applications is expected.

Supervisors: Dr Asma Khedher and Dr Sonja Cox.

#### Project 5: Zeta functions of the Newton strata of Shimura varieties

The Langlands-Kottwitz method has been developed in order to computer the cohomology of Shimura varieties and to construct many of the Galois representations predicted by the Langlands program. In this project we propose to adapt the Langlands-Kottwitz method to compute the cohomology of Newton strata of Shimura varieties.

Background in algebraic geometry and number theory is required.

Supervisor: Dr Arno Kret.

#### Project 6: Harmonic analysis on affine symmetric pairs and boundary correlation functions

This project is at the interface of representation theory and theoretical physics. It aims to describe correlation functions for integrable quantum field theories with boundaries using representation theory of quantum affine Lie algebras. Affinity with theoretical physics is a plus.

Supervisors: Prof. Jasper Stokman and Prof. Nicolai Reshetikhin.