Pure Mathematics Colloquia
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Previous Pure Mathematics Colloquia - 2013 to 2014

Previous Pure Mathematics Colloquia from: 2013/14, 2012/13, 2011/12, 2010/11, 2009/10, 2008/09, 2007/08

Thursday, 6th December 2013, 4pm, Theatre C

Nik Ruskuc
(University of St Andrews)
Permutations and words

In this talk I will explore how concepts from theoretical computer science -- automata and languages -- can be utilised in a combinatorial context such as the theory of pattern avoidance classes of permutations.



Thursday, 28th November 2013, 4pm, Theatre C

Francesco Matucci
(Université Paris-Sud 11)
Measuring finiteness in groups

Celebrated theorems by Gromov and of Lubotzky-Mann-Segal have shown that growth functions in groups can give complete characterization of classes of groups or give useful invariants. Given a residually finite group, we analyze a growth function measuring the minimal index of a normal subgroup in a group which does not contain a given element. This growth (called residual finiteness growth) attempts to measure how "efficient" of a group is at being residually finite. We review known results about this growth, such as the existence of a Gromov-like theorem in a particular case, and explain how it naturally leads to the study of a second related growth (called intersection growth). Intersection growth measures asymptotic behavior of the index of the intersection of all subgroups of a group that have index at most n and gives a profinite invariant of analytical flavor. In this talk I will introduce these growths and give an overview of some cases and properties.

This is joint work with Ian Biringer, Khalid Bou-Rabee and Martin Kassabov.



Thursday, 21st of November 2013, 4pm, Theatre C

Sandro Vaienti
(Marseille/Toulon)
Loss of memory and extreme value theory in randomly perturbed dynamical systems

We present new results on relaxation to equilibrium and limit theorems for dynamical systems which are randomly perturbed. In the case of the extreme value issues, the perturbation with observational noise allows to recover the local dimensional properties of the invariant measure.



Thursday, 24th of October 2013, 4pm, Theatre C

Tuomas Orponen
(University of Helsinki, visiting Edinburgh)
On restricted families of projections

Let \(F\) be a family of 2-dimensional subspaces in \(\mathbb{R}^3\). We say that \(F\) has property (A), if the following holds:

(A) If \(K\) is a set (also in \(\mathbb{R}^3\)) with Hausdorff dimension at most 2, then there exist many subspaces \(V\) in \(F\) so that \( \dim \pi_V(K) = dim K \).

Here \(\pi_V(K)\) is the orthogonal projection of \(K\) into \(V\). A special case of the Marstrand-Mattila projection theorem states that \(F = G(3,2)\), the family of all 2-dimensional subspaces, has property (A). It is a challenging open problem to determine, whether (A) can be satisfied by significantly smaller families \(F\). The answers, both conjectured and known, depend not only on the size of \(F\), but also on its structural properties. In the talk, I will survey the results obtained so far.



Thursday, 10th of October 2013, 4pm, Theatre C

Lars Olsen
(University of St Andrews)
Why is \(1 + 2 + 3 + 4 +..... = -1/12\), or the fine art of counting primes (and other things).

Remark:

This is a ``rerun'' of a talk given to SUMS (St Andrews Undergraduate Mathematical Society). Unfortunately, the original presentation of the talk had to be abandoned halfway through due to technical problems. The talk will now be presented again; this time jointly between SUMS and the Pure Mathematics Colloquium.

The talk is elementary, non-technical and has a very clear historical outlook (it only requires that you remember what a complex number is -- and even this is not need to fully understand the talk). The talk is thus suitable for all types of mathematicians, including, pure mathematicians, applied mathematicians and statisticians.

Abstract:

What is the number of prime numbers less that 1000000? The answer to this question is 78498. More generally, what is the number of prime numbers less that x? Gauss knew already in 1792 (even though he could not prove it) that the number of prime numbers less than x is roughly \(x/\log(x)\). A proof of this was outlined (but not fully completed) by Riemann in 1859. Riemann's remarkably insight was that in order to count the prime numbers we must study the function \[\zeta(z) = 1^{-z} + 2^{-z} + 3^{-z} + 4^{-z} + \cdots \] for complex numbers z. Finally, in 1896, Riemann's proof was completed independently by Jacques Hadamard and Charles Jean de la Vallee-Poussin. Why does the study of \(\zeta(z)\) lead to information about the prime numbers? What is going on?



Wednesday, 9th of October 2013, 4pm, Theatre C

Jim Belk
(Bard College)
Turing Machines, Automata, and the Brin-Thomspon Group 2V

I will sketch a proof that the Brin-Thompson group 2V has unsolvable torsion problem. That is, there is no algorithm to decide whether a given element of 2V has finite order. As a consequence, I will show that the periodicity problem for asynchronous finite-state transducers is undecidable. This is joint work with Collin Bleak.



Thursday, 3th of October 2013, 4pm, Theatre C

Kathryn Hare
(University of Waterloo)
The size of orbits and the eigenvalues of sums of matrices

By the orbit of a (skew) Hermitian matrix we mean the set of all unitarily similar matrices. Orbits have interesting geometric properties. For example, although every non-zero orbit in \(su(2)\) is a 2-dimensional manifold, the sum of any two non-zero orbits contains a 3-dimensional ball.

Every orbit supports an orbital measure, a probability measure which is invariant under similarity transformations. We will discuss a geometric/smoothness dichotomy that orbits and orbital measures satisfy and apply this to study the spectrum of sums of Hermitian matrices.



Thursday, 26th of September 2013, 4pm, Theatre C

Tom Ramsey
(University of Hawaii)
Kronecker Constants of Sets of Integers

For subsets \(S\) of an arbitrary discrete abelian group \(G\) the Kronecker constant \(\kappa(S)\) is defined to be the minimum of E such that, for function \(f\) from \(S\) into the unit circle \(\mathbb T\) there is character \(x\) such that \(| f(\cdot) -x(\cdot) |_{\infty} <\)\(E\) . For sets of integers \(S\) Kathryn Hare and Tom Ramsey replaced this definition with an angular Kronecker constant \(\alpha(S)\) with the property that \[ \kappa(S)=\left| e^{2\pi i \alpha(S)} -1 \right| \]

This transformation greatly simplifies the calculation of \(\kappa(S)\). For example, Hare and Ramsey proved \(\alpha(S)\) is a rational number for finite sets of integers. Usefully, when \(S\) consists of small integers, \(\alpha(S)\) can be computed by an algorithm that exhausts a tree of linear programming problems.

The complexity of calculating \(\alpha(S)\) is unknown. Some recent results suggest that the complexity is less than that of the traveling salesman problem.



Thursday, 12th of September 2013, 4pm, Theatre C

Persi Diaconis
(Stanford University)
Adding numbers and shuffling cards