(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.
(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.
(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.
(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.