The Pure Mathematics Colloquium takes place on Thursday afternoons at 4pm in Room 1A (unless otherwise indicated).
Abstract
McAlister's P-theorem is one of the most significant components of the structure theory for inverse semigroups: since its first appearance in 1974 several different proofs have been given and the theorem extended in application (e.g., by Gomes and Howie 1996, and Bulman-Fleming, Fountain and Gould 1997). We give a P-theorem for the class of ordered groupoids that covers all these variations in one go, and whose proof is based on ideas from the English Revolution circa 1647.
Abstract
In terms of the surviving written record, ancient China witnessed two so-called "golden ages", a millennium apart, in which problems concerning right triangles played a prominent part. This talk is an exercise in mathematics from history, rather than the history of mathematics, in that it takes the treatment of right triangles from the earlier period, some seventeen hundred years ago, and explores what can be done with it, including applying it to the problems from a millennium later. This simple mathematical expedient seems to shed fresh light on this history: even if there is little or no intervening record, could the Chinese mathematicians not have taken similar steps themselves?
Abstract
Fusion systems were introduced by L. Puig in order to systematically study p-fusion in finite groups and in p-blocks of finite groups. This axiomatic point of view has led to interesting developments in representation theory and in topology. I will attempt to give an overview of the subject and also explain how some classical theorems in group theory carry over to this setting.
Abstract
I'll explain how fruitful it is to study many classes of noncommutative algebras by looking at the extent to which their behaviour deviates from certain commutative rings to which they are associated.
Abstract
A detachment of a graph G is a graph which is obtained from G by splitting some or all of its vertices into 2 or more subvertices, with any edges incident with an original vertex being shared out arbitrarily among its subvertices. A well-known special case is when the detachment is a cycle; this is equivalent to an Eulerian circuit, which passes along each edge exactly once and returns to the starting point, and is possible if and only if G is connected and all of its vertices have even degree. The talk will consider the possibility of obtaining similar criteria in more complex cases.
Abstract
Given a finite presentation of a group G, proving properties of G can be difficult. Indeed, many questions about finitely presented groups are unsolvable in general. Algorithms exist for answering some questions while for other questions algorithms exist for verifying the truth of positive answers. An important tool in this regard is the Todd-Coxeter coset enumeration procedure. It is possible to extract formal proofs from the internal working of coset enumerations. We give examples of how this works, and show how the proofs produced can be mechanically verified and how they can be converted to alternative forms. We discuss these automatically produced proofs in terms of their size and the insights they offer. We compare them to hand proofs and to the simplest possible proofs. We point out that this technique has been used to help solve a longstanding conjecture about an infinite class of finitely presented groups.
Abstract
We extend the classical arithmetic defined over the set of natural numbers N to the set of all finite directed connected multigraphs having a pair of distinguished vertices. Specifically, we introduce a model F on the set of such graphs, and provide an interpretation of the language of arithmetic inside F. The resulting model exhibits the property that the standard model on N embeds in F as a submodel, with the directed path of length n playing the role of the standard integer n. We will compare the theory of the larger structure F with classical arithmetic statements that hold in N. For example, we explore the extent to which F enjoys properties like the associativity and commutativity of +, x, distributivity, divisibility, and order laws. We give a graph-theoretic characterization of + irreducibility, proving that left and right cancellation laws hold for addition in F. We use this to give a partial characterization of all pairs of graphs for which addition is commutative. This is achieved through the development of a theory of canonical + decompositions. Several open questions will be presented.
This is joint work with Delaram Kahrobaei (Univ. St Andrews) and Kiran Bhutani (Catholic Univ. of America).
Abstract
The talk is by way of a general introduction. I will try to tell you what they are and why I think they are interesting. I will also will try to indicate some of the directions in which the subject has been going in recent years.
Abstract
Let Fq denote the finite field of order q. For a polynomial f over Fq, let Vf = {f(a) | a in Fq} denote the value set of f. We will discuss a variety of problems and results concerning the cardinality of the value set of a polynomial and present some joint work with Pinaki Das which improves the best lower bound for the value set of a polynomial f over Fq. Polynomials whose value set has maximal cardinality q are called permutation polynomials. We will also discuss some results related to permutation polynomials over finite fields along with some of their applications.
Abstract
Euclid's algorithm for computing the greatest common divisor of 2 numbers is considered to be the oldest proper algorithm known. It has been well and truly studied and analyzed. Euclid's algorithm can be amplified naturally in various ways. The GCD problem for more than two numbers is interesting in its own right. We will consider how to compute multiple GCDs well in the worst case.
Also, we can do a constructive computation, the so-called extended GCD, which expresses the GCD as a linear combination of the input numbers. Extended GCD computation plays a basic role in various fundamental algorithms. It dates back to at least Euler. We will look at how hard it is to compute extended GCDs well.
Abstract
Results on positivity and boundedness of cosine and sine sums are easy to state and difficult to prove. We survey the basic results and sketch the proof of one of the most recent.