Elements of a finitely generated group have a natural notion of length: namely the length of a shortest word over the generators that represents the element. This allows us to study the growth of such groups by considering the size of spheres with increasing radii. One current area of interest is the rationality or otherwise of the formal power series whose coefficients are the sphere sizes. I will describe a combinatorial way to study this series for the class of virtually abelian groups, introduced by Benson in the 1980s, and then outline its applications to other types of growth series.

Semidefinite convex optimization problems often have low-rank solutions that can be represented with O(p)-storage. However, semidefinite programming methods require us to store the matrix decision variable with size O(p^2), which prevents the application of virtually all convex methods at large scale.

Indeed, storage, not arithmetic computation, is now the obstacle that prevents us from solving large- scale optimization problems. A grand challenge in contemporary optimization is therefore to design storage-optimal algorithms that provably and reliably solve large-scale optimization problems in key scientific and engineering applications. An algorithm is called storage optimal if its working storage is within a constant factor of the memory required to specify a generic problem instance and its solution.

So far, convex methods have completely failed to satisfy storage optimality. As a result, the literature has largely focused on storage optimal non-convex methods to obtain numerical solutions. Unfortunately, these algorithms have been shown to be provably correct only under unverifiable and unrealistic statistical assumptions on the problem template. They can also sacrifice the key benefits of convexity, as they do not use key convex geometric properties in their cost functions.

To this end, my talk introduces a new convex optimization algebra to obtain numerical solutions to semidefinite programs with a low-rank matrix streaming model. This streaming model provides us an opportunity to integrate sketching as a new tool for developing storage optimal convex optimization methods that go beyond semidefinite programming to more general convex templates. The resulting algorithms are expected to achieve unparalleled results for scalable matrix optimization problems in signal processing, machine learning, and computer science.

Thompson's group F is a group of homeomorphisms of the unit interval which exhibits a strange mix of properties; on the one hand it has some self-similarity type properties one might expect of a really big group, but on the other hand it is finitely presented. I will give a proof of finite generation by expressing elements as pairs of binary trees.

Every topological space is metrisable once the symmetry axiom is abandoned and the codomain of the metric is allowed to take values in a suitable structure tailored to fit the topology (and every completely regular space is similarly metrisable while retaining symmetry). This result was popularised in 1988 by Kopperman, who used value semigroups as the codomain for the metric, and restated in 1997 by Flagg, using value quantales. In categorical terms, each of these constructions extends to an equivalence of categories between the category Top and a category of all L-valued metric spaces (where L ranges over either value semigroups or value quantales) and the classical \epsilon-\delta notion of continuous mappings. Thus, there are (at least) two metric formalisms for topology, raising the questions: 1) is any of the two actually useful for doing topology? and 2) are the two formalisms equally powerful for the purposes of topology? After reviewing Flagg's machinery I will attempt to answer the former affirmatively and the latter negatively. In more detail, the two approaches are equipotent when it comes to point-to-point topological consideration, but only Flagg's formalism captures 'higher order' topological aspects correctly, however at a price; there is no notion of product of value quantales. En route to establishing Flagg's formalism as convenient, it will be shown that both fine and coarse variants of homology and homotopy arise as left and right Kan extensions of genuinely metrically constructed functors, and a topologically relevant notion of tensor product of value quantales, a surrogate for the non-existent products, will be described. 


 

Nash-Williams showed that the collection of locally finite trees under the topological minor relation results in a BQO. Naturally, two interesting questions arise:

1.      What is the number \lambda of topological types of locally finite trees?

2.       What are the possible sizes of an equivalence class of locally finite trees?

 For (1), clearly, \omega_0 \leq \lambda \leq c and Matthiesen refined it to \omega_1 \leq \lambda \leq c. Thus, this question becomes non-trivial in the absence of the Continuum Hypothesis. In this paper we address both questions by showing - entirely within ZFC - that for a large collection of locally finite trees that includes those with countably many rays:

- \lambda = \omega_1, and

- the size of an equivalence class can only be either 1 or c.

With topics ranging from prime numbers to the lottery, from lemmings to bending balls like Beckham, Professor Marcus du Sautoy will provide an entertaining and, perhaps, unexpected approach to explain how mathematics can be used to predict the future. 

We are delighted to announce our first Oxford Mathematics Midlands Public Lecture to take place at Solihull School on 9th January 2019. 

Please email @email to register

Watch live:
https://facebook.com/OxfordMathematics
https://livestream.com/oxuni/du-Sautoy

We are very grateful to Solihull School for hosting this lecture.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.