The goal of this talk is to state and motivate Serre's modularity conjectures on Galois representations. The talk includes applications and, if time permits, some generalizations to real quadratic fields.
 

We will describe how arithmetic properties of some orders in quaternion algebras can be used solve Diophantine problems about representations of integers by some quaternary quadratic forms. Some open problems will be described along the way.

In commutative algebra, a major subject of investigation is the study of ideals in polynomial rings that are defined by graphs; the goal is to understand how the properties of the ideals relate to the properties of the associated graphs. One of the several ideals studied in this context is the Eulerian ideal of a graph G. This ideal is generated by binomials which identify Eulerian subgraphs of G with an even number of edges.

Partial representation theory is a relatively recent area of research, originating in the study of partial dynamical systems and C∗-algebras generated by partial isometries. This talk offers a brief tour of the field, highlighting some of its key ideas and recent developments: the globalisation problem and a new approach to it, the connections with inverse semigroups, the partial representation theory of finite groups and the isomorphism problem for partial group algebras, and – time permitting – some recent progress towards understanding partial (co)modules over Hopf algebras.

Towards a homological Kitaev model by Ulrich Krähmer (Dresden)

Abstract One of the key problems in building a quantum computer is error correction, and a key idea how to deal with it is Kitaev's toric code and its generalisations. I will explain this problem and this idea only assuming knowledge of basic linear algbera. At the end I will sketch the main results of recent work with Sebastian Halbig that extends the toric code to non-semisimple Hopf algebras.

sl2-crystals and duality in monoidal categories ( joint w. T. Zorman)
Sebastian Halbig (Marburg)

Abstract: A classical result of representation theory states that for a module M over a commutative ring R the following are equivalent:

(a) M is finitely-generated projective,

(b) M has a dual M* together with a canonical pairing and a "dual basis" subject to natural snake identities, and

(c) tensoring with M is left adjoint to tensoring with an object M*.

Hopf braces, related structures and their associated categories by Brais Ramos Pérez (Santiago de Compostela)

Abstract: Hopf braces are recent mathematical objects introduced by I. Angiono et al. [1] and obtained through a linearisation process from skew braces,  which give rise to non-degenerate, bijective and not necessarily involutive solutions of the Quantum Yang-Baxter Equation (see [4]), whose formulation is the following

(τ ⊗ idV ) ◦ (idV ⊗ τ ) ◦ (τ ⊗ idV) = (idV ⊗ τ ) ◦ (τ ⊗ idV ) ◦ (idV ⊗ τ ), (QYBE)

Abstract: In 1892, Klein’s Erlangen program proposed that all geometric problems should ultimately be studied through the lens of group theory. In the 1950s, Jacques Tits introduced coset geometries, a structure that bridges geometries and their automorphism groups, allowing properties of groups to be studied via geometry and vice versa. Coset geometries play a central role in establishing the one-to-one correspondence between regular polytopes and a class of groups known as string C-groups. Consequently, classifying polytopes becomes equivalent to classifying these groups.

Let $F$ be a family of finite groups closed under taking subgroups, quotients and finite direct products. Given an element $g$ of a profinite group $G$, the $F$-izer of $g$ in $G$ is the set of elements $x$ in $G$ such that $\langle g,x\rangle$ is a pro-$F$-group. Let $F(G)$ be the set of elements $g$ of $G$ such that the $F$-izer of $g$ in $G$ has positive Haar measure.

Numerical semigroups are the subsemigroups of the set of natural numbers that are cofinite and contain $0$. Let $S$ be a numerical semigroup and $c$ be the smallest number such that $S$ is the union of a finite subset of $[0,c]$ and the integer interval $[c,\infty)$. Wilf's conjecture states that the density of elements of $S$ in the interval $[0,c]$ is at least equal to $1/d$, where $d$ is the dimension of the numerical semigroup $S$.