### Programme of the Pure Mathematics Colloquium

The colloquium takes place on Thursdays at 4pm in Theatre C of the Mathematical Institute (unless indicated otherwise).

The schedule for Semester 2 of 2017-18 is:
• 8th Feb, 2018: Tadahiro Oh (Edinburgh)
Title: On singular stochastic dispersive PDEs
Abstract: In this talk, I will go over some of the recent developments on nonlinear dispersive PDEs, such as the nonlinear Schroedinger equations (NLS) and nonlinear wave equations (NLW), with rough and random data and/or forcing. In particular, taking the stochastic NLS and the stochastic nonlinear heat equations as examples, I will describe the difference between the dispersive and parabolic problems from the viewpoint of critical regularities, etc. If time permits, I will also discuss some recent development for the stochastic NLW.
• 15th Feb, 2018: Tom Coleman (St Andrews)
Title: Permutation monoids and MB-homogeneous structures
Abstract: There is a widely studied connection between subgroups of the infinite symmetric group and automorphisms of first-order structures; in particular, homogeneous structures (as characterised in the celebrated theorem of Fraisse) provide examples of interesting infinite permutation groups. This connection can be generalised in a surprising fashion, as there exist submonoids of the infinite symmetric group that are not necessarily groups; these are called permutation monoids. Much like the group case, there is a correspondence between permutation monoids and bimorphism monoids (monoids of bijective endomorphisms) of first-order structures. In this talk, I will explore this connection, introduce the idea of an MB-homogeneous structure and characterise these by demonstrating a Fraisse-like theorem. I will then go on to examine MB-homogeneous graphs in more detail, leading to some surprising results. The material in this talk almost wholly consists of work done during my PhD, and is joint with David Evans (Imperial) and Bob Gray (UEA).
• 22nd Feb, 2018: Dane Flannery (Galway)
Title: Algebraic design theory
Abstract: Algebraic perspectives and techniques have become increasingly prevalent in combinatorial design theory. The designs of interest can be viewed as square matrices whose rows (or columns) taken pairwise obey a fixed constraint depending on the matrix size and coefficient ring. Hadamard matrices and their generalisations are important examples; the constraint in these cases is orthogonality. The literature on Hadamard matrices is immense, covering numerous applications in areas such as signal processing, cryptography, and quantum computing.

The algebraic approach has been especially successful in solving existence and classification problems for `cocyclic' designs, which are defined via certain regular subgroups of their automorphism groups. The pairwise row/column constraint for cocyclic designs translates to a simpler balance condition (e.g., a cocyclic matrix is Hadamard if and only if all non-initial row sums are zero).

We present a survey of algebraic design theory, emphasizing some key results and open problems. In particular, we mention recent work extending the notion of cocyclic design when necessary restrictions on matrix size are modified (e.g., what is the analogue of cocyclic Hadamard matrix when the size is even but not divisible by 4?). This arose from considerations of the maximal determinant problem originally posed by Hadamard.
• 8th Mar, 2018: Maura Paterson (Birkbeck) - this talk was originally scheduled for 1st Mar, but moved due to snow
Title: Recent directions in Private Information Retrieval
Abstract: Private Information Retrieval (PIR) is a technique that allows a user to obtain a record from a database without the database owner learning which record the user wishes to access. One way to achieve this is for the user to download the entire database. However, it is clearly desirable to construct PIR schemes in which this is not necessary. In the unconditionally secure model (where we make no assumptions about the computational power of the database owners) interesting combinatorial problems arise naturally from the study of these schemes as we consider trade-offs between properties such as the size of the queries to the database, the number of bits downloaded from the database and the number of bits required for storing the database. In this talk we will provide a background survey on unconditionally secure PIR before discussing some newer directions in the study of these schemes. https://arxiv.org/abs/1609.07027
• 15th Mar, 2018: Colloquium cancelled
• 5th Apr, 2018: Jess Enright (Edinburgh)
Title: Changing times in temporal graphs
Abstract: Temporal graphs (in which edges are active only at specified time steps) are an increasingly important and popular model for a wide variety of natural and social phenomena. I'll talk a bit about what's been going on in the world of temporal graphs, and then go on to the idea of graph modification in a temporal setting.

Motivated by cows and rumours, I'll talk about a particular modification problem in which we assign times to edges so as to maximise or minimise reachability sets within a temporal graph. I'll mention an assortment of complexity results on these problems, showing that they are hard under a disappointingly large variety of restrictions. In particular, if edges can be grouped into classes that must be assigned the same time, then the problem is hard even on directed acyclic graphs when both the reachability target and the classes of edges are of constant size, as well as on an extremely restrictive class of trees. For fans of parameterised complexity, I'll note that one version of the problem is W[1]-hard when parameterised by the vertex cover number of the instance graph. The situation is slightly better if each edge is active at a unique timestep - in some very restricted cases the problem is solvable in polynomial time. (Joint work with Kitty Meeks.)

If you're not a lifelong fan of graphs or computational complexity, never fear! I will try to make this talk as enjoyable and accessible as possible, and will be sure to point out the important bits. There will probably be at least one picture of livestock.
• 12th Apr, 2018: John Mackay (Bristol)
Title: Poincaré inequalities and non-embeddings of groups and spaces
Abstract: A classical Poincaré inequality states that for a smooth function on a ball in Euclidean space, the L^p norm of the deviation of the function from its average is controlled by the L^p norm of the gradient of the function. In recent years analogous inequalities have been studied on general metric spaces, where they may or may not hold depending on the particular space.

In recent work with Hume and Tessera, we have introduced the "Poincaré profile" of a space which measures, on large scale, to what extent such inequalities hold. This idea generalises the "Separation Profile" of Benjamini, Schramm and Timar. In this talk I'll survey some of the history and motivations of these ideas, and some results and applications of our work, for example to show non-embedding results for groups.
• 19th Apr, 2018: Marianne Johnson (Manchester)
Title: Identities in upper triangular matrix tropical semigroups and the bicyclic monoid
Abstract: Izhakian and Margolis noted that the bicyclic monoid can be faithfully represented by $2 \times 2$ upper triangular tropical matrices, and furthermore that the semigroup of all such matrices satisfies Adjan's minimal length identity $ABBA AB ABBA = ABBA BA ABBA$ for the bicyclic monoid. In light of these results they posed the natural question of whether these two semigroups satisfy exactly the same semigroup identities.

In joint work with Laure Daviaud and Mark Kambites, we provided a necessary and sufficient condition for an identity to hold in the semigroup $UT_n$ of $n \times n$ upper triangular tropical matrices, and used this result in the case $n=2$ to give a positive answer to the above. In further joint work with Ngoc Tran, we provide geometric methods and algorithms to verify, generate and enumerate $UT_n$ identities of a fixed length over a two letter alphabet. This leads to some new observations in the case of the bicyclic monoid.
• 26th Apr, 2018: Fenny Smith (BSHM)
Title: So where did our numbers come from anyway?

• 29th May, 2018: Volodymyr Nekrashevych (Texas A&M)
Title: Simple torsion groups of subexponential growth
Abstract: I will describe a new class of torsion groups (groups of Burnside type) obtained by a simple procedure of "fragmenting" an action of the infinite dihedral group on a Cantor set. The class contains many interesting examples: a torsion group of piecewise isometries of a polygon, the first example of a simple infinite finitely generated group of sub-exponential growth, torsion groups associated with irrational rotations of the circle, etc.
Past colloquia can be found here.