### Gerhard Hiss (RWTH Aachen University)

*Title* -- Two conjectures on the Weil representations of the finite symplectic and unitary groups* *

*Abstract* -- The Weil representations are distinguished complex representations of small degree of the finite symplectic and unitary groups. We begin by introducing the finite groups in question and some of their relevant subgroups. We also sketch a construction of the Weil representations. We then present two conjectures on the composition multiplicites of the tensor product of a Weil representation with an irreducible representation. Some evidence for the truth of these conjectures is given, as well as some motivation underlying the conjectures. This is joint work with Moritz Schroeer.

### Stephane Launois (University de Kent)

*Title* -- Total positivity is a quantum phenomenon: the grassmannian case

*Abstract* -- A real matrix is totally nonnegative if all its minors are nonnegative. This class of matrices has been studied in the past hundred years, and has connection with combinatorics, probability, etc. This notion of total positivity was generalised by Lusztig in the 1990s to arbitrary flag varieties. In particular, this led to the notion of totally nonnegative grassmannian. In 2006, Postnikov obtained groundbreaking results about a cell decomposition of the totally nonnegative grassmannians. In particular, he described various combinatorial objects parametrising this cell decomposition. Interestingly, this cell decomposition and the associated combinatorial objects have recently found key applications in Integrable systems (work of Kodama-Williams on the KP equation), and in Theoretical Physics (work of Arkani-Hamed and coauthors on scattering amplitudes).

### Sinead Lyle (University of East Anglia)

*Title*: On bases of Specht modules corresponding to 2-column partitions

*Abstract: *In this talk, we consider Specht modules and simple modules corresponding to 2-column partitions. Decomposition numbers for Specht modules corresponding to such partitions have long been known; as have homomorphims between them. For each such partition $\lambda$, we look at certain sets of standard $\lambda$-tableaux which can be defined combinatorially in terms of paths and which naturally label a basis of the simple module $D^\lambda$. We also prove that the $q$-character of $D^\lambda$ can be described in terms of this sets. We consider the extension to 3-column partitions.

### Laurent Rigal (Université Paris 13)

*Title*: Degeneration of quantum Schubert varieties

*Abstract*: The theory of quantum groups gives rise to non commutative versions of flag and Schubert varieties. In this talk, we will show how to study these analoguous versions in the context of non commutative algebraic geometry via toric degeneration. It is a joint work with Pablo Zadunaisky

### Sibylle Schroll (University of Leicester)

*Title*: On the Lie algebra structure of the first Hochschild cohomology

*Abstract*: The first Hochschild cohomology of a finite dimensional algebra equipped with the Gerstenhaber bracket is a Lie algebra. It is an interesting question to ask, which Lie algebras can actually appear in this way. In this talk we will analyse the structure of the first Hochschild cohomology as a Lie algebra and give some criteria for when it is solvable. As an application, we will show that for large classes of tame algebras the first Hochschild cohomology is solvable as a Lie algebra. This is joint work with Lleonard Rubio y Degrassi and Andrea Solotar.

### Bea Schumann (University of Cologne)

*Title*: A branching rule for the restriction of irreducible representations sl(2n,C) to sp(2n,C) via Littelmann paths

*Abstract*: Rules for decomposing the restriction of an irreducible representation into irreducible representations of a sub Lie algebra are called branching rules. Littelmann paths give a combinatorial tool to obtain such rules in many cases. However, there was no known branching rule in terms of Littelmann paths describing the restriction to a sub Lie algebra induced by a Dynkin diagram automorphism. In my talk I will explain how to do that for the case of the restriction from sl(2n,C) to sp(2n,C) based on joint work with Jacinta Torres.