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 $S$.

The classification of tuples of matrices up to simultaneous conjugation is a classical problem in linear algebra and representation theory. While the case of a single matrix leads to the well-known Jordan normal form, the situation becomes far more intricate for tuples, where the problem is considered "wild" and therefore not classifiable in any reasonable sense. Geometric Invariant Theory (GIT) provides a powerful framework for studying such spaces through quotient varieties, which parametrize semisimple representations of free associative algebras.

A numerical semigroup $S$ is a cofinite submonoid of the aditive monoid $(\mathbb{N},+)$. The (finite) complement of $S$ in $\mathbb{N}$ is the genus of $S$.

I plan to recall a well-known process of exploring the classical numerical semigroups tree as a means to count numerical semigroups by genus. The process is easily adapted to verify properties: one verifies the property at each explored node.

How much does the universal enveloping algebra of a Lie algebra remember about the Lie algebra itself? This is the statement of a classical problem in algebra called ``the isomorphism problem for enveloping algebras". In this talk, I'll explain how modern tools from algebraic homotopy theory and deformation theory allow us to make new progress on this problem. Along the way, we'll see that this is closely related to the following intriguing question: in the realm of homotopy theory, is being a commutative algebra simply a property, or is it an additional structure?

In this talk, we will present the bijective correspondence between Young diagrams and proper numerical sets. Then we will use this correspondence to introduce Young diagram decompositions for symmetric numerical sets (and semigroups). We will extend these decompositions to almost symmetric numerical semigroups.

There will be a coffee break after the seminar. 

References