Sergey Kitaev
*(University of Strathclyde)
***On a hierarchy related to interval orders
**

A partially ordered set (poset) is an interval order if it is isomorphic to some set of intervals on the real line ordered by left-to-right precedence. Interval orders are important in mathematics, computer science, engineering and the social sciences. For example, complex manufacturing processes are often broken into a series of tasks, each with a specified starting and ending time. Some of the tasks are not time-overlapping, so at the completion of the first task, all resources associated with that task can be used for the following task. On the other hand, if two tasks have overlapping time periods, they compete for resources and thus can be viewed as conflicting tasks.

A poset is said to be (2+2)-free if no two disjoint 2-element chains have comparable elements. In 1970, Fishburn proved that (2+2)-free posets are precisely interval orders. Recently, Bousquet-Mélou, Claesson, Dukes, and Kitaev introduced ascent sequences, which not only allowed us to enumerate interval orders, but also to connect them to other combinatorial objects, namely to Stoimenow's matchings, to certain upper triangular matrices, and to certain pattern avoiding permutations (a very active area of research these days). A host of papers by various authors has followed this initial paper.

Juan Luis García Zapata
*(Universidad de Extremadura)
***A parallel method for polynomial root finding
**

The proposed method is based on the winding number of curves in the complex plane. By Cauchy's argument principle, the number of roots of \(f\) inside a curve \(\Gamma\) is the winding number (around origin) of the curve \(f(\Gamma)\). We can divide the plane region inside \(\Gamma\) in smaller regions, compute the winding number of its borders and, recursively, divide them in subregions if they contain some root, until the desired accuracy is reached.

With this method we avoid three drawbacks of traditional iterative root finding methods (of point estimations, or eigenvalue based): 1) They are not easy parallelizable, 2) can not restrict your search to a predetermined region, 3) show problems for certain types of polynomial, and therefore the complexity analysis is difficult.

The computation of the winding number is not done by numerical integration, but by discrete geometry procedures. This helps to give a bound for the complexity of the algorithm depending on the distance from the roots to the border of the region. This distance plays a role similar to the condition number, which is used in numerical linear algebra.

Nik Ruskuc
*(University of St Andrews)
***Free idempotent generated semigroups (Part II)
**

Let \(S\) be a semigroup, and let \(E\) be its set of idempotents. The structure of the set \(E\) can naturally be described as a bi-ordered set, a notion arising as a generalisation of the semilattice of idempotents in inverse semigroups. The free-est idempotent generated semigroup with this bi-order of idempotents is given by the presentation \[ \langle E \:|\: \:|\: e\cdot f= ef\ (e,f\in E,\ \{ e,f\}\cap\{ef,fe\}\neq\emptyset)\rangle. \] (Here \(e\cdot f\) is a product of two generating symbols, while \(ef\) stands for their product in \(S\), which is an idempotent as a consequence of the condition \(\{ e,f\}\cap\{ef,fe\}\neq\emptyset\).) Given the controlling position that idempotents have in a free idempotent generated semigroup, it is natural to ask after the maximal subgroups in this semigroup. For instance, it is known that if \(S\) is a completely \(0\)-simple semigroup, then all the maximal subgroups of the corresponding free idempotent generated semigroup are free (Pastijn). In the talk I will present recent work with Robert Gray, with the emphasis on the following aspects:

-- a new approach to computing maximal subgroups in free idempotent generated semigroups based on Reidemeister--Schreier rewriting and a certain combinatorics on rows, columns and cells in the Rees matrix representation of a \(\mathcal{D}\)-class;

-- construction of two families of examples, showing that every group arises as a maximal subgroup of a free idempotent generated semigroup, and that every finitely presented group arises as a maximal subgroup of a free idempotent generated semigroup over a finite semigroup;

-- discussion of how to employ our methodology to compute free idempotent generated semigroups over some natural semigroups, such as full transformation monoids for example;

-- discussion of open problems and possible paths for further investigation in this area.

Alexander Konovalov
*(University of St Andrews)
***Prime graphs of normalised unit groups of integral group rings of sporadic simple groups
**

For an arbitrary group \(G\), the Gruenberg-Kegel of \(G\) (also called the prime graph \(G\)) is the graph whose vertices are labeled by primes \(p\) for which there exists an element of order \(p\) in \(G\) and with an edge from \(p\) to a distinct \(q\) if and only if \(G\) has an element of order \(pq\). I will report on the current state of the project to determine prime graphs of normalised unit groups of integral group rings of sporadic simple groups, in which we already proved that they coincide with prime graphs of underlying groups for the following thirteen sporadic simple groups: Mathieu groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\), \(M_{24}\); Janko groups \(J_1\), \(J_2\), \(J_3\); Held, Higman-Sims, McLaughlin, Rudvalis and Suzuki groups. This is a joint work with Victor Bovdi, Eric Jespers, Steve Linton, Salvatore Siciliano et al. In this project we use the GAP system together with the two constraint solvers, Minion and ECLiPSe.

Nik Ruskuc
*(University of St Andrews)
***Free idempotent generated semigroups
**

Let \(S\) be a semigroup, and let \(E\) be its set of idempotents. The structure of the set \(E\) can naturally be described as a bi-ordered set, a notion arising as a generalisation of the semilattice of idempotents in inverse semigroups. The free-est idempotent generated semigroup with this bi-order of idempotents is given by the presentation \[ \langle E \:|\: \:|\: e\cdot f= ef\ (e,f\in E,\ \{ e,f\}\cap\{ef,fe\}\neq\emptyset)\rangle. \] (Here \(e\cdot f\) is a product of two generating symbols, while \(ef\) stands for their product in \(S\), which is an idempotent as a consequence of the condition \(\{ e,f\}\cap\{ef,fe\}\neq\emptyset\).) Given the controlling position that idempotents have in a free idempotent generated semigroup, it is natural to ask after the maximal subgroups in this semigroup. For instance, it is known that if \(S\) is a completely \(0\)-simple semigroup, then all the maximal subgroups of the corresponding free idempotent generated semigroup are free (Pastijn). In the talk I will present recent work with Robert Gray, with the emphasis on the following aspects:

-- a new approach to computing maximal subgroups in free idempotent
generated semigroups based on Reidemeister--Schreier rewriting and a
certain combinatorics on rows, columns and cells in the Rees matrix
representation of a \(\mathcal{D}\)-class;

--construction of two families of examples, showing that every group
arises as a maximal subgroup of a free idempotent generated semigroup,
and that every finitely presented group arises as a maximal subgroup of
a free idempotent generated semigroup over a finite semigroup;

-- discussion of how to employ our methodology to compute free idempotent generated semigroups over some natural semigroups, such as full transformation monoids for example;

--discussion of open problems and possible paths for further investigation in this area.