Previous CIRCA & Algebra Seminars - 2010 to 2011
Previous CIRCA & Algebra Seminars from: 2010/11, 2008/09, 2007/08.Thursday, 15th of March 2012, 4pm, Theatre C
Mikael Vejdemo-Johansson
(University of St Andrews)
Algorithms of computational and applied algebraic topology
There is a growing field of computational and applied algebraic topology right now, in which techniques from algebraic topology -- a field that translates qualitative geometric questions ("How many separate pieces are there?" "How many essentially different paths from A to B are there subject to certain constraints?" etc.) into, essentially, linear algebra.
At the core of these developments lies an algorithm for computing persistent homology originally described by Edelsbrunner, Letscher and Zomorodian in 2002, and subsequently refined and expanded by a number of researchers in mathematics, computational geometry and theory fields.
In this talk, we shall take a look at the concept of persistent homology, the algorithm proposed by ELZ and some of its immediate refinements, and finally discuss the first stabs at parallelism that were achieved by Lipsky, Skraba and the speaker in 2011.
Thursday, 23th of February 2012, 4pm, Theatre C
Ursula Martin
(Queen Mary University of London, SICSA Distinguished Visitor, University of Edinburgh)
1377 questions and counting - what can we learn from online math?
Online blogs, question answering systems and distributed proofs provide a rich new resource for understanding what mathematicians really do, and hence devising better tools for supporting mathematical advance.
In this talk we discuss the first steps in such a research programme, looking at two examples using the tools of qualitative sociology, to see what we can learn about mathematical practice, and whether the reality of mathematical practice supports the theories of researchers such as Polya and Lakatos.
Polymath provides structured way for a number of people to work on a proof simultaneously: we analyse a polymath proof of a math olympiad problem to see what kinds of techniques the participants use. Mathoverflow supports asking and answering research level mathematical questions: we look at a sample of questions about group theory, and provide a typology of the kinds of questions asked, and consider the features of the discussions and answers they generate.
Finally we outline a programme of further work, and consider what our results tell us about opportunities for further computational support for proof and question answering.
Thursday, 24th of November 2011, 2:30pm, Purdie (Chemistry Building) Theatre D
Primoz Skraba
(Jozef Stefan Institute, Ljubljana)
Computing Well Groups for Maps in Euclidean Space
Well groups are a recently introduced concept in computational topology, related to persistent homology. They were introduce to quantify the robustness of topological maps. In this talk, I will introduce the necessary background of well groups and of standard persistence theory. In general, we do not know how to compute these well groups, however, I will discuss the cases where we can compute it, concentrating in particular on Euclidean space: that is maps which go from R^N to R^N.
Wednesday, 5th of October 2011, 3pm, Theatre C
Geoff Whittle
(Victoria University of Wellington)
Matroid Representation over Infinite Fields
A canonical way to obtain a matroid is from a finite set of vectors in a vector space over a field F. A matroid that can be obtained in such a way is said to be representable over F. It is clear that when Whitney first defined matroids he had matroids representable over the reals as his standard model, but for a variety of reasons most attention has focussed on matroids representable over finite fields.
There is increasing evidence that the class of matroids representable over a fixed finite field is well behaved with strong general theorems holding. Essentially none of these theorems hold if F is infinite. Indeed matroids representable over the reals - the natural matroids for our geometric intuition - turn out to be a mysterious class indeed. In the talk I will discuss this striking contrast in behaviour.
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).