The commuting graph of a finite non-commutative semigroup $S$ is the simple graph whose vertices are the non-central elements of $S$ and where two distinct vertices $x$ and $y$ are adjacent if and only if $xy=yx$.

I will review some classical examples of groups whose strictly descending chains of subgroups are finite.  Some recent constructions of such groups (having large cardinality) will also be explained. Set-theoretic consistency will play a role in some of the stated results. This is joint work with Saharon Shelah.

Kruskal’s uniqueness theorem gives a simple criterion ensuring that a 3-way tensor admits a unique expression as a sum of r product tensors. This talk will present a geometric proof of Kruskal's theorem, and show that rank and border rank coincide for tensors which satisfy Kruskal's uniqueness condition. The talk will also serve as an introduction to the notions of tensor rank and border rank. We will briefly touch on applications of tensor rank decompositions in the sciences.

Cantor famously used two versions of Diagonalization for his fundamental results in set theory. First, he used it to prove the uncountability of the set of real numbers. Second, he used a more general version to prove that the power set of a set A has a higher cardinality than A. In this talk we will discuss how Diagonalization is the core of many of delimitation theorems, such as Gödel's (First) Incompleteness Theorem or Turing's Undecidability Result.

The Hasse principle is the idea that a Diophantine equation over the rational numbers should have a rational solution if and only if it has solutions in all of its completions, namely, the real numbers and all p-adic fields. In recent work of Lorenzo and Vullers, they give twists of the modular curve X(7) that are counterexamples to the Hasse principle. In this talk, we will discuss generalizations of their result, for example, that there are infinitely many counterexamples to the Hasse principle that are twists of the modular curve X(p) for primes p congruent to 1 mod 4.

Quasigroupoids and weak Hopf quasigroups are non-associative generalizations of groupoids and weak Hopf algebras. In this talk, we will establish their main properties and an equivalence between the category of finite quasigroupoids and that of pointed cosemisimple weak Hopf quasigroups over a field K. As an immediate consequence, we obtain a categorical equivalence between quasigroups (in the sense of Klim and Majid, i.e., loops with the inverse property) and pointed cosemisimple Hopf quasigroups over K.

Polynomial expansion concerns the heuristic expectation that, for a typical polynomial P in n variables over a field F and subsets A₁,...,A_n of F, the image P(A₁,...,A_n) is substantially larger than each of the individual sets A_k. We establish new expansion results for certain classes of polynomials over finite fields, including a classification result for ternary quadratic polynomials. Our methods rely on spectral bounds for certain graphs arising from incidence geometry. This is joint work with Sam Chow.

Quasi-hereditary algebras are a class of finite-dimensional associative algebras that appear frequently in representation theory of associative algebras, but also of algebraic groups and semi-simple Lie algebras. They possess nice homological properties, like always having finite global dimension.  They have inspired several generalisations, such as standardly and properly stratified algebras, which retain several homological features and stratification properties.

The Gruenberg-Kegel graph of a group is defined as the graph whose vertices are the primes that appear as orders of elements of the group, and there is an edge between two primes p and q if and only if pq is the order of an element of the group. For solvable groups with small fields of characters, like rational groups or inverse semirstional groups, the set of vertices of this graph is known to be bounded (e.g., for rational solvable groups, only the primes 2,3 and 5 can appear).

This has been an open question for more than 70 years. I'll review what is known, including some recent developments.

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There will be coffee and cake after the seminar in the common room.