RHUL Mathematics Seminar

Andrew Tregown: Sum-free sets in the integers
A sum-free set is simply a set of integers which does not contain any solution to x + y = z. In this talk we discuss a range of recent developments concerning sum-free sets. We give a sharp count on the number of maximal sum-free subsets in [n], thereby answering a question of Cameron and Erdős. We also consider similar questions in the setting of solution-free sets (that is now we forbid solutions to some other fixed equation). Related complexity results will also be discussed. Underlying all this work is a connection to independent sets in graphs and hypergraphs.

The talk includes joint work with Jozsi Balogh, Hong Liu and Maryam Sharifzadeh; Robert Hancock; Kitty Meeks.

Peter Allen: Packing sparse graphs
Given a family of graphs G₁, …, G_t and a graph H, a packing of the G_i into H means a collection of embeddings of the graph G_i into H such that each edge of H is used in at most one embedding. Equivalently, this means a colouring of the edges of H with t + 1 colours, where the colour i edges form a graph isomorphic to G_i for each 1 ≤ i ≤ t and the colour t + 1 edges are left over.

The question of what families one can pack into a large complete graph, or into a typical large graph, has been quite actively studied recently (in particular motivated by conjectures of Gyarfas and Ringel from the 60s on packing families of trees). We now have quite a good understanding of when one can expect to find an almost-perfect packing (i.e. all but a tiny fraction of edges are used in the packing), and some idea of how to find perfect packings (when all edges are used in the packing). I will outline one approach to these kinds of problem, using a simple probabilistic process.

This is joint work with Julia Boettcher, Dennis Clemens, Jan Hladky, Diana Piguet and Anusch Taraz.

Jonathan Chapman: Partition regularity and multiplicatively syndetic sets
The study of partition regularity investigates properties of sets which are preserved under finite partitions. For example, if we are given an integer polynomial and a finite colouring of the positive integers, can we always find a monochromatic root? In this talk I will introduce multiplicatively syndetic sets, which are sets of integers with 'bounded multiplicative gaps', and explore their connections with partition regularity. In particular, I will show that a homogeneous integer polynomial has a monochromatic root under any finite colouring if and only if the polynomial has a root over any multiplicatively syndetic set. I will then show how this fact can be used to obtain quantitative bounds for Brauer's generalisation of van der Waerden's theorem. I will also mention some open problems concerning minimal multiplicatively syndetic sets.

Ehud Meir: Rings of invariant and the symmetric groups
Let V be a vector space of dimension n over a field K of characteristic zero, and let T : V → V be a linear endomorphism. By choosing a basis of V we can describe T as an n × n matrix (a_ij).

While the matrix entries depend on the particular choice of basis, certain polynomials in the variables a_ij are invariant to the choice of basis, and provide the most fundamental information about the linear transformation T. In fact, we know that K[a_ij]^GL(V) = K[c₁, … c_n] where the c_i are the coefficients of the characteristic polynomial of T.

In this talk I will describe the ring of invariants A := K[End(V ⊗ W]^{GL(V) × GL(W)} where V and W are two finite dimensional vector spaces. This invariant theory problem arises naturally in the study of finite dimensional Hopf algebras. The representation theory of the symmetric group comes into play here in a fundamental way, and the dimensions of the homogeneous components of A are given by a formula involving the Kronecker coefficients.

I will then concentrate on the case dim(V) = dim(W) = 2. In this case the Hilbert function of A was calculated explicitly by Dejan Govc using Mathematica. I will explain these calculations and how they are relate to Zelevinsky's algebra.

Kevin Buzzard: The future of mathematics?
Over the last few years, something (possibly a mid-life crisis) has made me become concerned about the reliability of modern mathematics, and about how the methods we mathematicians have traditionally used to prove theorems are scaling with the advent of the internet / ArXiv, and pressure on academics to get big results out there. I have started experimenting with a formal computer proof verification system called Lean, integrating it into my undergraduate teaching at Imperial and pushing it to see if it can handle modern mathematical definitions such as perfectoid spaces and the other ideas which got Peter Scholze a Fields Medal in 2018. I personally believe that Lean is part of what will become a paradigm shift in the way humans do mathematics, and that people who do not switch will ultimately be left behind. Am I right? Only time will tell. This talk will be be suitable for a general scientific audience -- mathematics undergraduates, computer scientists and philosophers will all find it comprehensible.

Cong Ling: Post-quantum cryptography based on division algebras
The Learning with Errors (LWE) problem is the fundamental backbone of modern lattice based cryptography. However, schemes based on LWE are often impractical, so Ring LWE was introduced as a form of 'structured' LWE. Another popular variant, Module LWE, generalizes this exchange by implementing a module structure over a Ring LWE instance. In this work, we introduce a novel variant of LWE over cyclic algebras (CLWE). The proposed construction is both more efficient than Module LWE and conjecturally more secure than Ring LWE, the best of both worlds.

Kevin Grace: Templates for representable matroids
The matroid structure theory of Geelen, Gerards, and Whittle has led to an announced result that a highly connected member of a minor-closed class of matroids representable over a finite field is a mild modification (known as a perturbation) of a frame matroid, the dual of a frame matroid, or a matroid representable over a proper subfield. They introduced the notion of a template to describe these perturbations in more detail. In this talk, we will define templates and discuss how templates are related to each other. We define a preorder on the set of frame templates over a finite field, and we determine the minimal nontrivial templates with respect to this preorder.

We use templates to obtain results about representability, extremal functions, and excluded minors for various minor-closed classes of matroids, subject to the announced result of Geelen, Gerards, and Whittle. These classes include the class of 1-flowing matroids and three closely related classes of quaternary matroids – the golden-mean matroids, the matroids representable over all fields of size at least 4, and the quaternary matroids representable over fields of all characteristics. This leads to a determination of the extremal functions for these classes, verifying a conjecture of Archer for matroids of sufficiently large rank.

This talk will include a brief introduction to matroid theory and is partially based on joint work with Stefan van Zwam.

Joni Teräväinen: Chowla's conjecture at almost all scales
An unsolved conjecture of Chowla states that the Möbius function should not correlate with its own shifts. This can be viewed as a conjecture about the randomness of the Möbius function.

In the last few years, there has been a lot of progress on Chowla's conjecture, which I will survey in the talk. Nearly all of the previously obtained results have concerned correlation sums that are weighted logarithmically, so one wonders whether it is possible to get rid of these weights. We show that one can indeed remove logarithmic weights from previously known results on Chowla's conjecture, provided that one restricts to almost all scales in a suitable sense.

This is joint work with Terry Tao.

Ilya Molchanov: The semigroup of metric measure spaces and its infinitely divisible probability measures
A metric measure space is a complete, separable metric space equipped with a probability measure that has full support. Two such spaces are equivalent if they are isometric as metric spaces via an isometry that maps the probability measure on the first space to the probability measure on the second. The resulting set of equivalence classes can be metrized with the Gromov–Prohorov metric. We consider the natural binary operation on this space that takes two metric measure spaces and forms their Cartesian product equipped with the sum of the two metrics and the product of the two probability measures. We show that the metric measure spaces equipped with this operation form a cancellative, commutative, Polish semigroup with a translation invariant metric. There is an explicit family of continuous semicharacters that is extremely useful for establishing that there are no infinitely divisible elements and that each element has a unique factorization into prime elements. We investigate the interaction between the semigroup structure and the natural action of the positive real numbers on this space that arises from scaling the metric. We establish that there is no analogue of the law of large numbers. We characterize the infinitely divisible and stable probability measures and the Lévy processes on this semigroup.

Susan Sierra: Enveloping algebras of infinite-dimensional Lie algebras: a survey
Given a Lie algebra g, one may form its universal enveloping algebra, U(g). This is a ring which has the same representation theory as g. Enveloping algebras of finite-dimensional Lie algebras are some of the most well-studied, and thus well-understood, examples in noncommutative ring theory; knowledge of their structure has been important in understanding the representation theory of finite-dimensional Lie algebras and other questions.

In contrast, enveloping algebras of infinite-dimenensional Lie algebras are much more mysterious: for example, it is not known if any such are (left and right) noetherian. We discuss our result (with Walton) that if W is the Witt algebra of vector fields on the complex affine line, then U(W) is not left or right noetherian: there are one-sided ideals which cannot be finitely generated. We further discuss recent joint work with Petukhov and Iyudu suggesting that all two-sided ideals of U(W) may be finitely generated. We also discuss what is known about enveloping algebras of other infinite-dimensional Lie algebras.

Joel Larsson: Minimum weight spanning structures in randomly weighted graphs
Let the complete graph K_n be equipped with i.i.d. edge weights. Given a class of spanning subgraphs, we can ask 'what is the minimum total weight of a subgraph in the class?' This question has received considerable attention for some classes of subgraphs, such as perfect matchings, spanning trees, and Hamilton cycles. In these cases, the minimum weight is known to converge in probability to ζ(2), ζ(3) and ≈ 2.04 respectively when the common edge weight distribution is uniform on [0, 1]. In this talk I will discuss some of the proof ideas, and how to attempt to generalize them to denser spanning subgraphs such as K_t-factors.

Eamonn O'Brien: Algorithms for matrix groups
How can we compute effectively with a matrix group defined over a finite field? We identify some inherent challenges, and outline a practical model which exploits randomness, geometry and detailed knowledge of the group structure.

Ran Levi: Complexes of tournaments in directed networks
Clique graphs whose edges are oriented are referred to in the combinatorics literature as tournaments. We consider a family of semi-simplicial sets, that we refer to as "tournaplexes", whose simplices are tournaments. In particular, given a directed graph G, we associate with it a "flag tournaplex" which is a tournaplex containing the directed flag complex of G, but also the geometric realisation of cliques that are not directed. We define several types of filtration on tournaplexes, and exploting persistent homology, we observe that filtered flag tournaplexes provide finer means of distinguishing graph dynamics than the directed flag complex. We then demonstrate the power of those ideas by applying them to graph data arising from the Blue Brain Project's digital reconstruction of a rat's neorcortex. Time permitting we explore connection to another graph invariant.

Sasha Kleshchev: Schur algebras and their representations
We will describe Schur algebras as defined and studied first by Schur in his thesis in 1901 and which go back to the 19th century invariant theory. We will discuss some modern developments related to categorification and Broué’s conjecture.

Joshua Coyston: Limit points of Mahler measures coming from digraphs
The Mahler measure of a single variable polynomial f can be defined as the geometric mean of |f(z)| on the unit circle. The search to find the smallest Mahler measure, or a complete set of 'small' Mahler measures, has been ongoing for almost 100 years. I will introduce a recent concept — the Mahler measure of a directed graph (digraph) — and how some tools from graph theory have helped us find limit points of Mahler measures.

Jan-Christophe Schlage-Puchta: Graph colourings avoiding rainbow graphs
Let K_n = (G,E) be the complete graph with n vertices, χ : E → {1,…,k} an edge colouring of K_n. If G is another finite graph, we say that the colouring χ contains a rainbow G, if there is an embedding of G into K_n such that the edges of the image of G have pairwise different colours. Wagner classified all colourings which do not contain a rainbow P₆, that is, a path containing 6 vertices. Extending this result to more complex graphs would be extremely tedious. In fact, we showed that for slightly larger graphs G there exist extremely colourful colourings without rainbow G

Here we extend this result. We say that (α, β) ∈ [0, 1]² implies a graph G, if every colouring of a sufficiently large K_n, such that there are at least αn vertices which are adjacent to edges of at least βn different colours, contains a rainbow G. We show that the set of pairs implying G is either [0,1]² \ {(0,0)} or a triangle with vertices (1,0),(1,1),(β,1). We classify connected graphs such that (β,1), 0 ≤ β < 1 is on the 2 boundary. However, we cannot determine the boundary for the disjoint traingles. In fact, we show that this problem is equivalent to a special case of the Caccetta–Häggkvist conjecture.

Mark Wildon: The counter-intuitive behaviour of high-dimensional spaces.
Suppose a point is chosen uniformly at random from a high-dimensional sphere. Almost certainly, it is on the equator. I will discuss this and other counter-intuitive features of high-dimensional spaces, using examples from Euclidean geometry, coding theory, cryptography, and the theory of computable functions. In particular, we will see a striking difference between the adversarial (Hamming) and probabilistic (Shannon) models for coding theory that illuminates one sense in which F₂¹⁰⁰⁰ is a 'larger' space than R⁴. If time permits, I will make a connection with my research on representations of finite and infinite groups.

Jens Bolte: Many-particle quantum graphs
Quantum graphs are models of particles moving along the edges of a graph following the laws of quantum mechanics. I will explain the basic construction of quantum graph models and then introduce models that describe several particles with contact interactions. Time permitting, I will talk about Bose–Einstein condensation as well as about non-standard exchange statistics of identical particles.

Simon Blackburn: How many finite rings are there?
For a positive integer n, write f(n) for the number of isomorphism classes of rings of order n. What can we say about f(n)? Determining f(n) exactly for all n looks unrealistic, but in 1970, Kruse and Price (JLMS) stated an asymptotic result that gives the growth rate of f(n) as n tends to infinity. Sadly, there are problems with their proof. I will talk about recent joint work with K. Robin McLean (University of Liverpool) in which we fix the problems, and improve the error terms of the Kruse-Price result. (Joint work with Robin McLean, Liverpool).

Brita Nucinkis: An introduction to groups of piecewise linear homeomorphisms of the unit interval.
In this talk I will give an introduction to several ways of defining Thompson's groups F, T and V and their generalisations, including groups with irrational slopes. I will also give an overview about some of their most interesting properties such as finiteness conditions and simplicity. If time permits, I will talk about some connections to areas outside group theory.

Eoghan McDowell: Counting paths in lattices and a new symmetric polynomial identity
The Lindström–Gessel–Viennot lemma states that the number of tuples of paths in a given lattice is equal to the determinant of a certain matrix. In this talk I will explain the elegant combinatorial argument behind this result, and use it to obtain a new symmetric polynomial identity. This identity generalises both the binomial determinant duality of theorem of Gessel and Viennot and the symmetric function duality theorem of Aitken.