Seminars

In the academic year 2024-25, the Algebra and Combinatorics Seminar will run on Thursdays of odd-numbered weeks, at 1-2pm.  Please note that (due to the university’s week-numbering system) these are Weeks 1,3,5,7,9,12 in Semester 2.  The venue is Maths Lecture Theatre A.

Our Semester 2 programme is as follows.

Week 1 – Collin Bleak (St Andrews)

Title: Embedding certain automatic groups into the rational group R.

Abstract: We introduce left-continuous automatic groups as a subclass of the well-known automatic groups.  The class of left-continuous automatic groups is a bit mysterious as a subclass of the automatic groups, but, we know at least that the class contains the CAT(0) Cubical Complex groups (CCC groups), which themselves represent a broad class of groups of topical interest.  Our main theorem states that all left-continuous automatic groups embed as subgroups of the rational group R, a group introduced in 2000 by Grigorchuk, Nekrashevych, and Suschanskii.  We will provide definitions and examples of these various groups along the way, and have some discussion as well of different forms of boundaries of groups.  We also discuss how similar embedding theorems have been of use in larger programs of discovery.  Joint work with Belk, Chatterji, Matucci and Perego.
 
 
 

Week 3 –  Coen Del Valle/Theodor Thorbjornsen (St Andrews)

  • Coen Del Valle: A character theoretic formula for base size
  • Abstract: Let Omega be a finite set and let G be a group of permutations of Omega. A base for G is a subset of Omega whose pointwise stabiliser is trivial; the size of a smallest base for G is denoted b(G). In this talk, we will explore a surprising connection between the invariant b(G) and character theory.
  • Theodor Thorbjornsen: The twisted partition monoid
  • Abstract: I am in the early stages of an investigation into the combinatorial and algebraic properties of twisted partition monoids. In this talk I will first introduce partition monoids, the concept of twisting and general twisted products. I will then introduce one natural way of twisting partition monoids, and prove a couple initial results about the twisted partition monoid $P_n^{\Phi}$

Week 5 – Pilar Duque Paez/ Pierre Zhou (St Andrews)

  • Pilar Duque Paez: Constructing groups with undecidable conjugacy problem.
  • Abstract: In 1973, C. F. Miller III asked whether there exists a finitely presented linear group, over the integers, with undecidable conjugacy problem. Motivated by this question, in 1981, Elizabeth Scott showed that if one can construct a finitely presented subgroup L of GL(n,Z), with undecidable orbit problem on Z^n, then the commutator subgroup of the group <V_m, Z^n⋊ L> will be a finitely presented simple group with undecidable conjugacy problem. At the time, it was unclear whether such an L exists. In 1984, Scott answered C. F. Miller’s question using an alternative approach. In this talk we will show how to construct such an L using a modification of a technique of Bogopolski, Martino, Šunic and Ventura, thus achieving Scott´s original vision.
  • Pierre Zhou: Immutable Subsequences in Words Saturated by Singleton Bases
  • Abstract. Let A be a finite set of letters and τ a trace relation over A. A basis B is a finite subset of the free monoid A*. A word w over A is said to be B-saturated if for all trace companions v of w, there exists some basis element β such that v involves β as a subword. In terms of avoidance classes, a word w is B-saturated if the intersection of the trace [w] and the class Av(B) is empty. The aim of this talk is to demonstrate that for all singleton bases B = {β}, a word w is B-saturated precisely when w involves β as an immutable subsequence. If time permits, I shall also discuss how combinatorics on words may help us determine the atomicity of certain permutation classes, and how the aforementioned result fits into a bigger picture.

Week 7 – Violeta Lopez Lopez /Murray Whyte (St Andrews)

  • Violeta Lopez Lopez: Combinatorics in Brill-Noether theory
  • Abstract: Tropical Geometry is, roughly speaking, a combinatorial approach to studying Algebraic Geometry. In this last field, Brill-Noether theory studies algebraic curves, i.e., Riemann surfaces. In this talk, I will show the types of combinatorics that appear in tropical Brill-Noether theory with explicit examples.
  • Murray Whyte: Presentations for subsemigroups of the symmetric inverse monoid which contain the symmetric group
  • Abstract: A (reasonably) natural thing to want to do is ‘extend’ a permutation group to a transformation semigroup by equipping it with an extra generator (for our purposes, a partial bijection). There is a general construction for presentations of the monoids that might result from doing this — so-called factorable inverse monoids — due to Easdown, East and FitzGerald in 2004. We use this to find large presentations for the subsemigroups of the symmetric inverse monoid which contain the symmetric group — generated by the symmetric group, together with a partial bijection of any rank we like. We then considerably reduce the number of relations involved, and discuss the extent to which the resulting presentations are minimum-sized.

Week 9 – Struan McCartney /Yayi Zhu (St Andrews)

  • Struan McCartney: New Circular External Difference Families and related constructions
  • Abstract: External difference families were introduced in 2004 by Ogata et al to create secret sharing schemes with desirable properties. Circular external difference families (CEDFs) were introduced more recently in 2023 by Stinson and Veitch as a new way to create non-malleable secret sharing schemes. I will present new constructions of infinite families and new parameters for CEDFs along with constructions for related combinatorial objects.
  • Yayi Zhu: Relational depth of transformation semigroups and their ideals
  • Abstract: Presentations for classical transformation semigroups have long been of interest, including the partial transformation semigroup, the symmetric inverse monoid, and the full transformation semigroup. Green’s J-relation partitions those semigroups into subsets consisting of (partial) transformations of a certain rank, and the J-classes form a chain. We are interested in the minimal J-class J whose elements are involved in presentations for the transformation semigroups and their ideals, and we use ‘depth’ to describe the position of J in the chain. In this talk, I will present results on the relational depth of these semigroups, and give an example for PT_9, which contains all partial transformations on [n].

Week 12 – Dorte Behrens /Jung Won Cho (St Andrews)

  • Dorte Behrens: Homogeneous oriented two-graphs and MB-homogeneous k-hypergraphs
  • Abstract: In 1954 Fraïssé wrote his theorem on amalgamation classes and the existence of a homogeneous structure that forms their limit. Since then classifications of the homogeneous graphs (Lachlan & Woodroow, 1980), digraphs (Cherlin, 1998) and others have followed. As has the introduction of the concept of homomorphism-homogeneity (Cameron & Nešetril, 2006), which was further explored by Lockett and Truss (2012 & 2014). In this talk we will be exploring both these directions in research by considering the classification of homogeneous oriented two-graphs, and MB-homogeneous k-hypergraphs as an example of one type of homomorphism-homogeneity.
  • Jung Won Cho: Howson property for monogenic inverse semigroups
  • Abstract: An algebra is said to have the Howson property if the intersection of any two finitely generated subalgebras is again finitely generated. In this talk, we will look at the Howson property for monogenic inverse semigroups regarded as ordinary semigroups. We will see that a monogenic inverse semigroup has the Howson property if and only if it is not free. This is joint work with Craig Miller and Nik Ruškuc.

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