# PhD position in Combinatorics and Representation Theory

University of Geneva | Geneva, Switzerland

Classification: Combinatorics and Algebra

A four-year PhD position is available at the math institute of the University of Geneva https://www.unige.ch/math/fr/ in Combinatorics and Representation Theory, funded by the Eccellenza Grant "Partition identities and interactions" of Pr. Jehanne Dousse (the grant will start in September 2022). The successful applicant will join the group of Jehanne Dousse and work on a broad project which aims at combining the combinatorial theory of partition identities with representation theory and computer algebra. In particular, the future PhD will work on the interaction between partition identities and the representation theory of affine Lie algebra. According to the successful applicant's tastes, the research problems can be chosen to be more combinatorial or more representation theoretic. Summary of the PhD project: A partition of a positive integer n is a non-increasing sequence of natural numbers whose sum is n, the partitions of 3 being 3, 2+1, and 1+1+1. Partition identities, such as the Rogers-Ramanujan identities, are theorems of the form "for every integer n, the number of partitions of n satisfying some conditions is equal to the number of partitions of n satisfying some other conditions". They are central objects in combinatorics and number theory, and have connections with several other fields such as representation theory, computational mathematics, algebraic geometry, theoretical physics, and probability theory. The goal of the project is to understand the interplay between partition identities and representation theory. The exciting connection between these two fields was revealed in the 1980’s, when Lepowsky and Wilson interpreted the Rogers-Ramanujan identities in terms of characters for representations of affine Lie algebras. This led to the development of vertex operators, and triggered the discovery of new partition identities which were still unknown to combinatorialists. Many of these identities are still conjectural, as proving them amounts to showing that certain sets of vectors are bases of representations, which is very difficult. A starting project for the future PhD student will be to prove some these conjectural identities using combinatorial and/or representation theoretic methods. Further projects will then be elaborated jointly with the PhD student, with the goal of developing new connections between partition identities and representation theory. Duties: Research on the themes of the project, teaching of one exercise class per week (in mathematics, most likely at the undergraduate level). Starting date: Negotiable, no earlier than September 2022. Gross salary: around 47'000 CHF per year Requirements: Candidates must have a master's degree in mathematics or equivalent at the start of the appointment, with background in Combinatorics and/or Representation Theory. Application: The application should be sent by email to dousse@math.cnrs.fr and should contain the following material: - CV - cover letter - transcript with grades from the master's degree - name and email address of two references (including the master's thesis advisor, if applicable) Deadline: June 15th 2022 If you have any question, do not hesitate to contact dousse@math.cnrs.fr .

Last updated: 14 May 2022