Seminars
In the academic year 2025-26, Semester 2, the Algebra and Combinatorics Seminar will run on Thursdays of Weeks 3, 5, 7, 9 and 12, at 1-2pm. The venue is Maths Lecture Theatre C.
Our Semester 1 programme is as follows.
Week 3 (Thursday 12th February) – Jeff Hicks (St Andrews)
Title: Topology in (non)-commutative algebra
Abstract: The polynomial ring in k-variables is one of the best understood rings. One of the fundamental results in the area is Hilbert’s Syzygy theorem (1890), which in the modern formulation states that a finitely generated module over this ring admits a free resolution of length at most k. In the 1990s Bayer and Sturmfels observed that many of these free resolutions had “underlying structure” given by a topological space. From this intuition, they introduced a framework (cellular resolutions) that provided a method for constructing new resolutions of a module from the data of a topological space. Unsurprisingly, many properties of these modules were encoded in the topology of the associated space.
If you, for some reason, think of the polynomial ring as the monoid ring of ℕk, then you might ask if there is a similar story for non-commutative monoids. A partial answer — which predates Bayer and Sturmfels — is in the foundational work of Eilenberg and Mac Lane (~1950) on classifying spaces and group cohomology. In this talk, I will mostly give an exposition of Bayer-Sturmfels’ work and its relationship to Eilenberg-Mac Lane spaces. Time permitting , I will discuss some work in progress with Lauren Cranton Heller, Mahrud Sayrafi, and Jay Yang on the general case monoid rings and applications.
Week 5 (Thursday 26th February) – Joseph Edwards/Struan McCartney (St Andrews)
Title (JE): Irreducible diagrams in the Jones Monoid
Abstract (JE): Over the past decade, significant advancements have been made in the theory of diagram monoids. For example, one particular project in the 2010s sought to classify and enumerate the idempotents of various families of diagram monoids. In this talk, we aim to expand on this work by providing further insight into the idempotent structure of the Jones Monoid. In particular, after defining the Jones Monoid, I will introduce the notation of irreducibility in the context of planar diagrams, and explore the structure of irreducible idempotent elements. No prior knowledge of diagram monoids will be assumed, and there will be lots of pictures throughout!
Title (SM): New External Difference Families and Related Constructions from Graph Valuations (POSTPONED TO CIRCA WEEK 8)
Abstract (SM): Digraph-defined external difference families were recently introduced as a natural generalization of several well-studied combinatorial objects motivated by information security. I will show various types of vertex-labelling for graphs and digraphs along with a blow up technique that can be used to create digraph-defined external difference families. Using these methods I will give a new infinite family of circular external difference families.
Week 7 (Thursday 19th March) – Callum Barber/Yayi Zhu (St Andrews)
Title (CB): Congruences, Diagonal Subsemigroups and the DSC Coefficient
Abstract: (CB) A diagonal subsemigroup of a semigroup $S$ is a subsemigroup of $S \times S$ that contains the diagonal $\Delta = \{(x,x) : x \in S \}$. When $S$ is finite the DSC coefficient of $S$, $\chi(S)$, is defined to be the ratio of the number of diagonal subsemigroups to the number of congruences. It is easy to show that $\chi(S) = 1$ if and only if $S$ is a group. It is natural to wonder what other values the DSC coefficient can take. It turns out that for every rational $\alpha$ in $(0,1]$ there is a finite semigroup $S$ with $\chi(S) = \alpha$. We prove this by utilizing the Rees matrix construction and the description of the congruences on such semigroups. We will also consider the DSC coefficient of some other semigroup constructions and introduce some open problems.
Title (YZ): Presentations for the Brauer monoid and its ideals
Abstract (YZ): In recent years, there has been a substantial amount of work on presentations for diagram monoids, including the partition monoid, the Brauer monoid, the Temperley–Lieb monoid, and their singular ideals. The J-classes of these semigroups form a chain, and the defining relations in such presentations lie in some specific J-classes. Similar structures also occur in transformation and matrix semigroups. We are interested in the minimal J-class whose elements appear in any presentation for the semigroup. In this talk, I will give results on presentations for the ideals of the Brauer monoid, with particular emphasis on identifying the minimal J-class.
Week 9 (Thursday 2nd April) – Dorte Behrens/Pierre Zhou (St Andrews)
Title (DB): MB-homogeneous, but not IA-homogeneous hypergraphs
Abstract (DB): In this talk I will give a brief introduction to properties ensuring the MB-homogeneity of a hypergraph and cover a construction that ensures these properties are fulfilled. I will then focus on a sketch of the proof that hypergraphs constructed in this way are not IA-homogeneous.
Title (PZ): Bounding the Lengths of Minimal B-Saturated Words over a Transitive Alphabet
Abstract (PZ): Let A be a finite set of letters, and I \subseteq A \times A an independence relation. We say that (A,I) is a transitive alphabet if the independence relation I is transitive. Let (A,I) be a transitive alphabet, and B \subseteq A* a (finite) antichain. A word w \in A* is said to be B-saturated if for all v \in [w], there exists some \beta \in B, such that \beta \leq v. By Higman’s Lemma, the set W(B) of all minimal B-saturated words over A is finite. The aim of this talk is to demonstrate that there exists a computable bound for the lengths of minimal B-saturated words over a transitive alphabet (A,I) whose graph realisation is connected. This means that there exists an algorithm that, given a transitive alphabet (A,I) of this kind, computes the set W(B). I will also briefly discuss the case in which the transitive alphabet (A,I) becomes disconnected, and explain the motivations behind these problems.
Week 12 (Thursday 23rd April) – Theodor Thorbjornsen/Joseph Ward
(St Andrews)
Title: TBC
Abstract: TBC