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.

As a consequence of the Littlewood-Richardson  commuters coincidence and the Kumar-Torres branching model  via  Kushwaha-Raghavan-Viswanath flagged hives, we have solved the Lecouvey-Lenart conjecture on the  bijections between the Kwon and Sundaram branching models for  the pair (GL_2n(C), Sp_2n(C)) consisting of the general linear group GL_2n(C) and  the symplectic group Sp_2n(C).

In the 1950’s Davenport, Mirsky, Newman and Rado proved that if the integers are partitioned by a finite set of arithmetic progressions, then the largest difference must appear more than once. In other words, if g1, . . ., gn and a1 ≤ a2 ≤ . . . ≤ an are integers such that {gi+aiℤ}n i=1 is a partition of ℤ then an−1 = an. This confirmed a conjecture of Erdös and opened a broad area of research (see Covering systems of Paul Erdös. Past, present and future, Paul Erdös and his mathematics, I (Budapest, 1999), Bolyai Soc. Math. Stud., vol. 11, pp. 581-627. János Bolyai Math.

We investigate finite and profinite groups with the Magnus property, where a group is said to have the Magnus property if whenever two elements generate the same normal subgroup then the elements are conjugate or inverse-conjugate. This is joint work with Martino Garonzi and Pavel Zalesskii.

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