University of Warwick

The torus T of projective space also acts on the Hilbert
scheme of subschemes of projective space, and the T-graph of the
Hilbert scheme has vertices the fixed points of this action, and edges
the closures of one-dimensional orbits. In general this graph depends
on the underlying field. I will discuss joint work with Rob
Silversmith, in which we construct of a subgraph, which we call the
spine, of the T-graph of Hilb^N(A^2) that is independent of the choice
of field. The key technique is an understanding of the tropical ideal,
in the sense of tropical scheme theory, of the ideal of the universal
family of an edge in the spine.

Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. It is a generalisation of an older invariant, the Teichmüller polynomial, defined by McMullen in 2002.

The aim of my talk is to demonstrate that veering triangulations provide a convenient setup for computations. More precisely, I will use fairly easy arguments to obtain a fairly strong statement which generalises the results of McMullen relating the Teichmüller polynomial to the Alexander polynomial.

I will not assume any prior knowledge on the Alexander polynomial, the Teichmüller polynomial or veering triangulations.

Right-angled Artin groups (RAAGs) were first introduced in the 70s by Baudisch and further developed in the 80s by Droms.
They have attracted much attention in Geometric Group Theory. One of the many reasons is that it has been shown that all hyperbolic 3-manifold groups are virtually finitely presented subgroups of RAAGs.
In the first part of the talk, I will discuss some of their interesting properties. I will explain some of their relations with manifold groups and their importance in finiteness conditions for groups.
In the second part, I will focus on my PhD project concerning subgroups of direct products of RAAGs.

If asked, we all say we aim to to good research and make sensible decisions. In mathematics, the choice of criteria to optimise is often explicit, and we know there is no complete ordering in more than one dimension.

Statisticians involved in multi-disciplinary research need to reflect on how their understanding of uncertainty and statistical methods can contribute to reliable and reproducible research. The ISI Declaration of Professional Ethics provides a framework for statisticians.  Judging what is "normal" and what is "best" requires an appreciation of the assumptions and guidelines of other disciplines.

I will briefly discuss the requirements for design and analysis in medical research, and relate this to debates on reproducible research and p-values in social science research. Issues arising from informed and uninformed consent will be outlined.

Examples might include medical research in developing countries, toxic tort or wrongful birth claims, big data and use of routine administrative or commercial data.

I will describe some new-ish results on the average and maximum size of the Riemann zeta function in a "typical" interval of length 1 on the critical line. A (hopefully) interesting feature of the proofs is that they reduce the problem for the zeta function to an analogous problem for a random model, which can then be solved using various probabilistic techniques.

The flow of a thin film down an inclined plane is an important physical phenomenon appearing in many industrial applications, such as coating (where it is desirable to maintain the fluid interface flat) or heat transfer (where a larger interfacial area is beneficial). These applications lead to the need of reliably manipulating the flow in order to obtain a desired interfacial shape. The interface of such thin films can be described by a number of models, each of them exhibiting instabilities for certain parameter regimes. In this talk, I will propose a feedback control methodology based on same-fluid blowing and suction. I use the Kuramoto–Sivashinsky (KS) equation to model interface perturbations and to derive the controls. I will show that one can use a finite number of point-actuated controls based on observations of the interface to stabilise both the flat solution and any chosen nontrivial solution of the KS equation. Furthermore, I will investigate the robustness of the designed controls to uncertain observations and parameter values, and study the effect of the controls across a hierarchy of models for the interface, which include the KS equation, (nonlinear) long-wave models and the full Navier–Stokes equations.

The Navier-Stokes paradigm does not capture thermal fluctuations that drive familiar effects such as Brownian motion and are seen to be key to understanding counter-intuitive phenomena in nanoscale interfacial flows.  On the other hand, molecular simulations naturally account for these fluctuations but are limited to exceptionally short time scales. A framework that incorporates thermal noise is provided by fluctuating hydrodynamics, based on the so-called Landau-Lifshitz-Navier-Stokes equations, and in this talk we shall exploit these equations to gain insight into nanoscale free surface flows.  Particular attention will be given to flows with topological changes, such as the coalescence of drops, breakup of jets and rupture of thin liquid films for which both analytic linear stability results and numerical simulations will be presented and compared to the results of molecular dynamics.

It is known that random matrix distributions such as those that describe the largest eignevalue of the Gaussian Orthogonal and Symplectic ensembles (GOE, GSE) admit two types of representations: one in terms of a Fredholm Pfaffian and one in terms of a Fredholm determinant. The equality of the two sets of expressions has so far been established via involved computations of linear algebraic nature. We provide a structural explanation of this duality via links (old and new) between the model of last passage percolation and the irreducible characters of classical groups, in particular the general linear, symplectic and orthogonal groups, and by studying, combinatorially, how their representations decompose when restricted to certain subgroups. Based on joint work with Elia Bisi.

We will introduce the Witt vectors of a ring with coefficients in a bimodule and use them to calculate the components of the Hill-Hopkins-Ravenel norm for cyclic p-groups. This algebraic construction generalizes Hesselholt's Witt vectors for non-commutative rings and Kaledin's polynomial Witt vectors over perfect fields. We will discuss applications to the characteristic polynomial over non-commutative rings and to the Dieudonné determinant. This is all joint work with Krause, Nikolaus and Patchkoria.