## The Section

# Open Positions

# Postes vacants

## Postdoc position in mathematics (representation theory, combinatorics or computer algebra)

#### Job description

**Any questions please contact Professor Jehanne Dousse directly via email:**

We are welcoming applications for a 2-year postdoc position at the University of Geneva, Switzerland.

The successful candidate will conduct research in the group of Professor Jehanne Dousse, as part of the Eccellenza grant "Partition identities and interactions" of the SNSF (see detailed summary below). They will work in one or more of the following fields:

- representation theory (in particular on the representation theory of affine Lie algebras and crystal bases),

- enumerative and algebraic combinatorics, q-series,

- computer algebra and functional equations.

Applications from all three areas are welcome, but priority will be given to excellent candidates with background in representation theory.

The candidate will also take part in the department’s teaching activities as an assistant. Their main duties will be teaching exercises sessions and grading exams.

The candidate must have defended their PhD by – and no longer than 5 years before – the date of employment.

**Starting date **

September 1st, 2024 (negotiable)

**Any questions please contact Professor Jehanne Dousse directly via email:**

**Applications**

Applications with the following documents must be sent by email to **before April 15, 23h59, Geneva time**.

- CV including list of publications,

- research statement containing a summary of past work and projects,

- copy of the PhD diploma (if the PhD has already been defended, otherwise master’s diploma and a certificate from the PhD advisor with the expected date of PhD defence),

- name and email address of two references, who may be contacted to obtain more information about the candidate.

**Summary of research project**

A partition of a positive integer n is a non-increasing sequence of integers (the parts) whose sum is n. For example, the partitions of 3 are 3, 2+1 and 1+1+1. A partition identity is a theorem stating that for all n, two different types of partitions of n have the same cardinality. For example, the famous Rogers-Ramanujan identities state that for all n, there are as many partitions of n into parts congruent to 1 or 4 mod 5 (for n=4, we have 4 and 1+1+1+1) as partitions of n such that two consecutive parts differ by at least 2 (for n=4, we have 4 and 3+1). Partition identities play key roles in combinatorics and number theory, but are also related to several other fields such as representation theory, computational mathematics, algebraic geometry, theoretical physics, and probability theory. The goal of this project is to explore the interplay between partition identities, representation theory, and computer algebra, and to build new bridges between them by combining techniques from all three fields.

The first goal is to understand the interplay between partition identities and representation theory. The connection between these two fields was revealed in the 1980’s, when Lepowsky and Wilson interpreted the Rogers-Ramanujan identities in terms of 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 combinatorial project would be to prove these conjectural identities. Another approach, started by Primc in the 1990’s and developed by the PI and her coauthors in the past few years, consists in studying crystal bases of representations of affine Lie algebras to obtain character formulas which can be related to partitions. Using the two approaches above, several new partition identities or character formulas have been obtained, but the connection between the combinatorial and representation theoretic aspects of partition identities is still far from being completely understood. The goal of the project is to understand these connections more deeply, leading to new results on partitions, Lie algebra representations, and better understand the connection between the vertex operator algebra and crystal approaches.

The second part of the project, which is interconnected with our first objective, focuses on algorithmic and computational aspects of partition identities. We aim to provide automatic methods to prove partition identities in general, and those related to representation theory in particular. The goal is to build an automatic method to prove and refine families of partition identities. The idea is to automatically solve recurrences and q-difference equations satisfied by the partition generating functions. The method works in many particular cases, and understanding its exact field of applicability and making it fully automatic would have great impact on all the aspects of this project.