Many of our seminars are Pure Mathematics Colloquia, so why not check here?

## Previous CIRCA & Algebra Semigroups - 2010 to 2011

Previous Pure Mathematics Colloquia from: 2016/17, 2015/16, 2014/15, 2013/14, 2012/13, 2011/12, 2010/11, 2008/09, 2007/08**Friday, 11th of February 2011, 10am, Theatre A**

Markus Pfeiffer
*(University of St Andrews)
***A trilogy of Algebraic Automata Theory, Episode 1.
**

Continuing into the theory of Semirings and Semimodules we will look at one of the other important ingredients: Limits. In particular we will see how we can define Limits axiomatically and how the idea of arbitrary finite iteration and fixed points is used in this theory. Hopefully by the end of the talk we define regular languages algebraically and see how we can extend the concept to infinite state systems.

**Friday, 4th of February 2011, 10am, Theatre C**

Markus Pfeiffer
*(University of St Andrews)
***A trilogy of Algebraic Automata Theory, Episode 6.
**

Continuing into the theory of Semirings and Semimodules we will look at one of the other important ingredients: Limits. In particular we will see how we can define Limits axiomatically and how the idea of arbitrary finite iteration and fixed points is used in this theory. Hopefully by the end of the talk we define regular languages algebraically and see how we can extend the concept to infinite state systems.

**Thursday, 20th of January 2011, 4pm, Theatre C**

Markus Pfeiffer
*(University of St Andrews)
***A trilogy of Algebraic Automata Theory, Episode 5.
**

Continuing into the theory of Semirings and Semimodules we will look at one of the other important ingredients: Limits. In particular we will see how we can define Limits axiomatically and how the idea of arbitrary finite iteration and fixed points is used in this theory. Hopefully by the end of the talk we define regular languages algebraically and see how we can extend the concept to infinite state systems.

**Tuesday, 14th of December 2010, 2pm, Theatre D**

Markus Pfeiffer
*(University of St Andrews)
***A trilogy of Algebraic Automata Theory, Episode 4.
**

I will introduce a relatively little known approach to the concept of formal languages and regular languages in particular. This approach employs methods and notions from linear algebra. In this session I hope I will be able to introduce the basic concepts involved, namely semirings and formal powerseries, and how these tie into the theory of automata and formal languages. I will also hint at possible generalisations and extensions of this approach.

**Wednesday, 27th of October 2010, 2pm, Theatre A**

Yann Peresse
*(University of St Andrews)
***Sierpinski, surjections and a quadratic function in infinities
**

Let X be an infinite set. A classical theorem by Sierpinski states that every countable set of functions from X to itself can be obtained as a composition of just two such functions. In other words, if Self(X) is the semigroup of all functions from X to X, then any countable subset of Self(X) is contained in a 2-generated subsemigroup. Analogue properties are known to hold for many other semigroups. For instance, Galvin showed that every countable subset of the symmetric group Sym(X) of all bijections from X to X is contained in a 2-generated subgroup. Given a semigroup S we now say that S has Sierpinski rank n if every countable subset of S is contained in an n-generated subsemigroup and n is the least such number. For example, Self(X) and Sym(X) have Sierpinski rank 2. In this talk we will consider the semigroups Inj(X) and Surj(X) of all functions from X to itself that are injective respectively surjective. The results we will obtain are somewhat different from the aforementioned ones: the Sierpinski ranks of Inj(X) and Surj(X) depend on the cardinality of X. More precisely, if the size of X is aleph_n (i.e. the n+1st infinite cardinal), then the Sierpinski rank of Inj(X) is n+4 and the Sierpinski rank of Surj(X) is (n^2+9n)/2+7.

**Wednesday, 20th of October 2010, 4pm, Theatre C**

Collin Bleak
*(University of St Andrews)
***Using dynamics to analyze element centralizers in R. Thompson's group V
**

R. Thompson's group V is a finitely presented infinite simple group introduced in the 1970's. Amongst many of V's interesting properties are the facts that it contains infinitely many copies of any finite group, of any finitely generated free group, of Z^n for any positive integer n, of Q/Z, and of H*G, the free product of H and G, for many (but not all) subgroups H and G of V.

In this talk, for any given g in V, we use an analysis of the dynamics of g's action on the Cantor Set to produce an algebraic description of the structure of C_V(g), the centralizer of g in V.

This talk features joint work with Hannah Bowman, Alison Gordon, Garrett Graham, Jacob Hughes, Francesco Matucci, and Eugenia Sapir.

**Wednesday, 13th of October 2010, 4pm, Theatre C**

Ahmad Khalaf
*(University of Al-Baath, Syria)
***Groups with the basis property
**

In this talk I will give a description of the finite groups with the basis property. A group G is called a group with the basis property, if for each subgroup H of G, any two minimal generating sets of H have the same number of elements. I proved that every finite group G with the basis property is a Frobenius group, such that the Frobenius kernel is a p-subgroup which is the Fitting subgroup of G. Furthermore, the Frobenius complement is a q-subgroup for a different prime q. Therefore it follows that the nilpotency class of the Fitting subgroup of a group with the basis property can be arbitrarily large.

**Wednesday, 6th of October 2010, 4pm, Theatre C**

Istvan Szollosi
*(University of Cluj-Napoca, Romania)
***On the Ringel-Hall product of preinjective Kronecker modules and the matrix subpencil problem
**

We give a description of the terms in the Ringel-Hall product of preinjective Kronecker modules. We characterize in this way all the short exact sequences of preinjective modules. As an application we also give an explicit solution to the column completion challenge for pencils with only minimal indices for columns (corresponding to preinjective modules).