Speakers

Talks:

• Naomi Bazlov (Technion- Israel Institute of Technology), Polynomials with Restricted Digits
  • The ring of polynomials in one variable over a finite field $F_q$ is a fascinating algebraic object, both similar to and different from the ring of integers $Z$. One of the earliest results about distribution of degree n irreducible polynomials satisfying an additional constraint is an analogue of the Prime Number Theorem in Arithmetic Progressions for $F_q[T]$, proved in 1924 by Emil Artin. In the 20th century, Richard Hayes and Stephen Cohen, among others, studied the distribution of polynomials over $F_q$ in more depth from both combinatorial and asymptotic point of view.
    Recently, much progress has been achieved in the problem of primes (in $Z$) with restricted digits, in particular due to work of James Maynard. Function field analogues of this problem attract a lot of interest. Questions about polynomials in $F_q[T]$ with coefficients restricted to a subset of the field can be attacked using the circle method, as well as more modern probabilistic methods. I will present my very recent results about asymptotic behaviour of squarefree polynomials of degree n with coefficients restricted to a subset of $F_q$
• Gemma Crowe (University of Manchester), Introduction to twisted conjugacy in groups
  • Dehn’s famous decision problems for finitely presented groups have been studied for over a century by combinatorial and geometric group theorists. In recent years, a variant of one of these classical problems, namely the twisted conjugacy problem, has been studied. The motivation for this work comes from Bogopolski, Martino and Ventura who, in 2009, proved an equivalence between conjugacy in group extensions and twisted conjugacy.
    In this talk I will give a brief survey of this lesser-known decision problem, and discuss some of the latest results in this area.

• Kanwal Khalid (University of Essex), An Introduction to Nearly Complete Intersection Ideals
  • Nearly complete intersection ideals (NCI)  are a recent class of ideals and are defined by Adam Boocher and James Seiner in 2017. They used them as a tool to establish a bound on the sum of betti numbers of a monomial ideal I in a polynomial  ring R,  when I is not a complete intersection ideal. They reduced their proof to Nearly complete intersection ideals and showed that their potential bound  holds for the sum of betti numbers of a nearly complete intersection ideal. Later on, Miller and stone gave us an equivalent definition of nearly complete intersections ideals generated in degree two, calling them nearly complete intersection graphs and then classified them. In our talk, we will have a brief introduction to Nearly complete intersection ideals and then we will see their transition into graph theory. We will quickly look into the classification given by Miller and Stone and will have a look on challenges while using their classification. Lastly, we will discuss a new  classification of Nearly complete intersection graphs and will learn construction of Nearly complete intersection ideals using this new classification.
• Orla McGrath (University of Leeds), The Ascending Chain Condition for Difference Ideals
  • An open problem in the study of difference algebra is finding classes of difference ideals that satisfy the ascending chain condition. In this talk I will give a brief overview of the basics of difference algebra before discussing the problem of finding properties of difference ideals that imply finite difference generation. I will summarise some results found so far, including the result that any strictly ascending chain of partial difference Hopf ideals is finite.
• Fernando Mendez (University of Leeds), Introduction to Center-Symmetric Structures and Lie Algebras over Graphs
  • This talk begins with an overview of pre-Lie structures, illustrated by an example that naturally arises in particle physics. We then introduce a new class of pre-Lie structures—center-symmetric structures (CSAs)—which are closely related to left-symmetric algebras (LSAs). Finally, we highlight some of the key properties and distinctive features of these structures.
• David Nkansah (Aarhus University), Homological algebra without grading
  • Complexes are a fundamental object in homological algebra, and as such, grading is deeply ingrained in the subject. But what happens if we remove the grading from a complex? This leads us to the notion of a differential module. In this talk, we will explore how the finiteness of the injective dimension of a finitely generated module over a local commutative noetherian ring can be detected using this ungraded framework. As an immediate application, we will see that this perspective also detects when such a ring is Cohen–Macaulay.
• Charlotte Roelants (Vrjie Universiteit Brussel), Irreducibility of Killing forms on finite simple groups of Lie type
  • We study Killing forms on finite groups arising from the extension of the theory of Killing forms on Lie algebras to braided-Lie algebras. For certain braided structures associated with a conjugation-stable subset of a finite group, these forms admit an expression in terms of the character of the conjugation action on the group.
    Motivated by Cartan’s criterion and earlier work by López Peña, Majid, and Rietsch, we investigate the non-degeneracy and irreducibility of such Killing forms defined on conjugation-stable subsets of finite groups. We will particularly focus on conjugacy classes of involutions and unipotent elements in simple groups of Lie type of rank one. Our methods reveal interesting connections with counting formulas in character theory, generation properties, and commuting graphs.
    This talk is based on a recent preprint in collaboration with Kevin Piterman.
• Anja Sneperger (University of Leeds), Some determinants and relations in Heronian friezes
  • Motivated by computational geometry of point configurations on the Euclidean plane, and by the theory of cluster algebras of type A, Sergey Fomin and Linus Setiabrata have recently introduced Heronian friezes - the Euclidean analogues of Coxeter’s frieze patterns. I will explain the way these friezes arise from a polygon in the complex plane, and that a sufficiently generic Heronian frieze is uniquely determined by a small proportion of its entries. Then, I shall proceed to some of my results, that hold in the cyclic case, i.e. when the vertices of the polygon are placed on the circle. Namely, I will speak about vanishing of certain determinants in the cyclic case, as well as explain some interesting algebraic relations that hold between the entries of the cyclic Heronian frieze.
• Pavel Turek (University of Birmingham), Decomposition columns labelled by d-balanced partitions
  • A central open problem of the modular representation theory of symmetric groups (and Hecke algebras) is the computation of decomposition numbers, which describe decompositions of Specht modules in prime characteristic. These numbers are organised in a decomposition matrix, the structure of which remains a mystery. In this talk, we present new results about columns of this matrix labelled by a novel family of partitions called d-balanced partitions, describing many of these columns explicitly. It turns out that these columns have a rich combinatorial structure, which will be explored during the talk and may be of independent interest. This is joint work with David Hemmer and with Bim Gustavsson and Stacey Law.

Posters:

• Laura Olivia Felder (University of Hertfordshire), Higher Chiral Algebras in a Polysimplicial Model
• Mathison Knight (University of Hertfordshire), Majority Polymorphisms in Cactus Graphs
• David Nkansah (Aarhus University), Homological Algebra Without Grading
• Iacopo Nonis (University of Leeds), $\tau$-exceptional Sequences for Representations of Quivers over Local Algebras
• Ibraheem Sajid (University of Leeds), Motion Groups