## Previous Pure Mathematics Colloquia - 2012 to 2013

Previous Pure Mathematics Colloquia from: 2013/14, 2012/13, 2011/12, 2010/11, 2009/10, 2008/09, 2007/08### Thursday, 30th of May 2013, 4pm, Theatre C

Andrew Ferguson
*(University of Bristol)
***Dimension of the projections of dynamically defined sets
**

In 1954 Marstrand proved several theorems that concern the almost sure dimension of a projection and slice of a planar Borel set. Later, Kaufman gave bounds on the set of directions for which the projection is exceptional. In this talk I will discuss the situation when one considers certain classes of dynamically defined sets, showing that often one may say quite a bit more.

### Monday, 22nd of April 2013, 5pm, Theatre C

Susan Hermiller
*(University of Nebraska)
***Algorithms and topology of Cayley graphs for groups
**

A finitely generated group is called autostackable if it admits a certain topological/combinatorial property of the Cayley graph, namely a notion of ordering, or ``flow'', on edges outside of a maximal tree, together with a method for performing multiplication by generators using a finite automaton. This property implies a solution to the word problem. In turn this is a useful tool for computing asymptotic (``filling'') invariants for autostackable groups. In this talk I'll explain the connections between autostackability and other geometric group theory properties, and discuss many examples.

This is joint work with Mark Brittenham.

### Thursday, 18th of April 2013, 4pm, Theatre C

Peter Cameron
*(University of St Andrews)
***Derangements
**

A derangement is a permutation with no fixed points. One of the best-known results in enumerative combinatorics is that the proportion of derangements in a finite symmetric group is close to 1/e. Also Jordan showed that every transitive group of degree greater than 1 contains a derangement (though a simple algorithm to find one was only given this year). There are connections between derangements and a variety of topics in number theory, topology, game theory, combinatorial enumeration, and other areas, and they also raise interesting questions, many of them unsolved.

### Thursday, 11th of April 2013, 4pm, Theatre C

Jack Button
*(University of Cambridge)
***Tripling of finite sets in infinite non-abelian groups
**

Given a finite subset A of a group G, there has been much recent interest in comparing the size of A with that of the double product A.A and (in the case where G is non abelian) the triple product A.A.A. Following work of Helfgott and others on families of finite simple groups, we give examples of infinite groups G where the triple product of any generating set for G exhibits rapid growth, as well as new examples of infinite groups where this phenomenon does not occur.

### Thursday, 14th of March 2013, 4pm, Theatre C

Tom Leinster
*(University of Edinburgh)
***The Convex Magnitude Conjecture
**

(Joint work with Simon Willerton)

Magnitude is a real-valued invariant of metric spaces, springing from a category-theoretic study of size. Unlike most invariants of metric spaces, it changes unpredictably as the space is scaled up or down. It therefore assigns to each space a real-valued function: magnitude as a function of the scale factor. Roughly, the Convex Magnitude Conjecture states that for convex subsets of \(\mathbb{R}^n\), this function is a polynomial encoding all the most important quantities associated with convex sets: dimension, volume, surface area, perimeter, and so on.

I will explain where magnitude comes from, how it is defined, and what makes the conjecture interesting. I will also explain the conjecture's unusual status: while there is compelling evidence in its favour, not a single nontrivial example is known.

### Thursday, 28th of February 2013, 4pm, Theatre C

Pablo Shmerkin
*(University of Surrey)
***On sets containing circles/squares centered at every point
**

Let A be a subset of the plane with the property that, for each point of the plane, A contains a circle centered at that point. A deep classical result due independently to Marstrand and to Bourgain says that A must have positive Lebesgue measure.

I will review the history of the circle problem and tell about some recent developments on the similar problem where circles are replaced by (boundaries of) squares. Unlike the circle case, it was known that a set containing a square centered at every point may have zero Lebesgue measure. The question we study is: how small can A be? As we will see, the answer surprisingly depends on the notion of ``smallness'' used. This is joint work with T. Keleti and D. Nagy.

### Thursday, 21st of February 2013, 4pm, Theatre C

Armando Martino
*(University of Southhampton)
***The conjugacy problem for Houghton's groups
**

One of the fundamental problems in geometric group theory is to solve the conjugacy problem. That is, given a group presentation, find an algorithm which will decide whether two elements of the group are conjugate. We give a solution to this problem for Houghton's group, which is an infinite permutation group with many interesting properties, and for which standard techniques, namely those arising from the theory of hyperbolic and bi-automatic groups, are not applicable.

### Thursday, 14th of February 2013, 4pm, Theatre C

Elizabeth Lewis
*(University of St Andrews)
***C-V Mourey's single science of algebra and geometry
**

In 1828 C-V Mourey published some of his results relating to the difficulties presented by the theory of algebra.* Seeking algebraic reform, Mourey set out to discover a new set of definitions and fundamental principles. To this end he developed a theory of directed lines, which constituted a single science of algebra and geometry, and as an application of the theory, gave a proof of the fundamental theorem of algebra. The purpose of the talk is to bring attention to Mourey's mathematics. I feel strongly that the details of his mathematics, which demonstrate in particular the extent to which he understood the subtleties of algebra, deserve to be thoroughly understood.

* C-V Mourey, "La vraie théorie des quantités négatives et des quantités prétendues imaginaires", Paris, 2nd edn., 1861; online edn., www.hathitrust.org [http://hdl.handle.net/2027/njp.32101028427001]

### Monday, 14th of January 2013, 4pm, Theatre C

Mark Kambites
*(University of Manchester)
***Monoids acting by isometric embeddings
**

### Thursday, 20th of December 2012, 4pm, Theatre C

Nik Ruskuc
*(University of St Andrews)
***John Howie's Work on Ranks of Semigroups
**

This is a talk in honour of John Howie, Regius Professor of Mathematics 1970-1997, who died on 26 December last year.

### Thursday, 13th of December 2012, 4pm, Theatre C

Peter Mayr
*(University of Linz)
***Computation in direct powers
**

For a fixed finite algebraic structure \(A\) (e.g., a group, a ring, \(\dots\)) we consider computational problems in substructures of direct powers of \(A\), for example, the following Subpower Membership Problem:

INPUT: tuples \(a_1,\ldots, a_k\), and \(b\) in \(A^n\)

PROBLEM: Is \(b\) in the subalgebra of \(A^n\) generated by \(a_1,\ldots,
a_k\)?

We are interested in the complexity in terms of \(k\) and \(n\). There exist algebraic structures for which deciding subpower membership is Exptime-complete (Kozik, 2007). However, if \(A\) is a group, then an adaptation of a permutation group algorithm works in polynomial time (Furst, Hopcroft, Luks, 1980). More generally, for rings, algebras over fields, Lie rings, \(\dots\) all finite groups with additional multilinear operations the Subpower Membership Problem is easily seen to be in P.

I will show polynomial algorithms for Subpower Membership, Subpower Intersection, ... for any finite \(p\)-group with arbitary additional operations. The proof uses the fact that on \(p\)-groups every function behaves almost like a homomorphism.

### Thursday, 6th of December 2012, 4pm, Theatre C

Iain Gordon
*(University of Edinburgh)
***Galois problems in Schubert calculus, and related topics
**

### Thursday, 25th of October 2012, 4pm, Theatre C

Stefanie Eminger
*(University of St Andrews)
***Behind the scenes of the first International Congress of Mathematicians
**

Georg Cantor voiced the need for opportunities facilitating international mathematical cooperation as early as in 1888. A decade and efforts by a number of mathematicians later, the first International Congress of Mathematicians marked the beginning of an era where personal relations between mathematicians were considered to be of great importance. Furthermore, it set the standards for future congresses. As well as talking about the pre-history and the organisation of the congress, I hope to put it into a wider historic context and conjecture on the reasons why it was held in Zurich and why such a great emphasis was placed on the social aspect of the congress. I will also talk about some of the Swiss organizers, including Carl Friedrich Geiser, Ferdinand Rudio, and J\'eri\^ome Franel.

### Thursday, 27th of September 2012, 4pm, Theatre C

Matt Nicol
*(University of Houston)
***Limit laws for the shrinking target problem
**

Suppose \(T: X \rightarrow X\) is an ergodic map of a measure and metric space \( (X,\mu) \). Suppose \(B_i:= B(p,r_i) \) are nested balls of radius \( r_i \) about a point \( p \) in a the dynamical system \( (T,X,\mu) \). The question of whether \(T^i x \in B_i \) infinitely often for \( \mu \)-a.e. \( x \) is often called the shrinking target problem. In many dynamical settings it has been shown that if \( E_n:=\sum_{i=1}^n \mu (B_i) \) diverges then there is a quantitative rate of entry and \( \lim_{n\to \infty} \frac{1}{E_n} \sum_{j=1}^{n} 1_{B_i} (T^i x) \rightarrow 1 \) for \( \mu \)-a.e. \( x\in X \). This is a self-norming type of strong law of large numbers, sometimes called the Strong Borel Cantelli property. We discuss recent results on the Strong Borel Cantelli property and also self-norming central limit theorems of the form \( \lim_{ n\rightarrow \infty} \frac{1}{a_n} \sum_{i=1}^{n} [1_{B_i} (T^i x)-\mu(B_i)] \to N(0,1) \) in distribution (i.e. the Normal distribution with zero mean and variance 1). These results apply to a variety of hyperbolic and non-uniformly hyperbolic dynamical systems.