University of Warwick

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. During the talk I will discuss how and why it is connected to the Alexander polynomial of the underlying manifold.  

There is a wide range of problems in continuum mechanics that involve transformation fronts, which are non-stationary interfaces between two different phases in a phase-transforming or a chemically-transforming material. From the mathematical point of view, the considered problems are represented by systems of non-linear PDEs with discontinuities across non-stationary interfaces, kinetics of which depend on the solution of the PDEs. Such problems have a significant industrial relevance – an example of a transformation front is the localised stress-affected chemical reaction in Li-ion batteries with Si-based anodes. Since the kinetics of the transformation fronts depends on the continuum fields, the transformation front propagation can be decelerated and even blocked by the mechanical stresses. This talk will focus on three topics: (1) the stability of the transformation fronts in the vicinity of the equilibrium position for the chemo-mechanical problem, (2) a fictitious-domain finite-element method (CutFEM) for solving non-linear PDEs with transformation fronts and (3) an applied problem of Si lithiation.

Most Ricci flow theory takes the short-time existence of solutions as a starting point and ends up concerned with understanding the long-time limiting behaviour and the structure of any finite-time singularities that may develop along the way. In this talk I will look at what you can think of as singularities at time zero. I will describe some of the situations in which one would like to start a  Ricci flow with a space that is rougher than a smooth bounded curvature Riemannian manifold, and some of the situations in which one considers smooth initial data that is only achieved in a non-smooth way. A particularly interesting and useful case is the problem of starting a Ricci flow on a Riemann surface equipped with a measure. I will not be assuming expertise in Ricci flow theory. Parts of the talk are joint with either Hao Yin (USTC) or ManChun Lee (CUHK).

For almost calibrated Lagrangian mean curvature flow it is known that all singularities are of Type II. To understand the finer structure of the singularities forming, it is thus necessary to understand the structure of general ancient solutions arising as potential limit flows at such singularities. We will discuss recent progress showing that ancient solutions with a blow-down a pair of static planes meeting along a 1-dimensional line are translators. This is joint work with J. Lotay and G. Szekelyhidi.

iii) identify a special class of correlated compressors based on the idea of random permutations, for which we coin the term PermK. The use of this technique results in the strict improvement on the previous MARINA rate. In the low Hessian variance regime, the improvement can be as large as √n, when d > n, and 1 + √d/n, when n<=d, where n is the number of workers and d is the number of parameters describing the model we are learning.

Trophic coherence, a measure of a graph’s hierarchical organisation, has been shown to be linked to a graph’s structural and dynamical aspects such as cyclicity, stability and normality. Trophic levels of vertices can reveal their functional properties, partition and rank the vertices accordingly. Trophic levels and hence trophic coherence can only be defined on graphs with basal vertices, i.e. vertices with zero in-degree. Consequently, trophic analysis of graphs had been restricted until now. In this talk I will introduce a novel  framework which can be defined on any simple graph. Within this general framework, I'll illustrate several new metrics: hierarchical levels, a generalisation of the notion of trophic levels, influence centrality, a measure of a vertex’s ability to influence dynamics, and democracy coefficient, a measure of overall feedback in the system. I will then discuss what new insights are illuminated on the topological and dynamical aspects of graphs. Finally, I will show how the hierarchical structure of a network relates to the incidence rate in a SIS epidemic model and the economic insights we can gain through it.

Article link: https://www.nature.com/articles/s41598-021-93161-4

We construct smooth flows with surgery that approximate weak mean curvature flows with only spherical and neck-pinch singularities. This is achieved by combining the recent work of Choi-Haslhofer-Hershkovits, and Choi-Haslhofer-Hershkovits-White, establishing canonical neighbourhoods of such singularities, with suitable barriers to flows with surgery. A limiting argument is then used to control these approximating flows. We demonstrate an application of this surgery flow by improving the entropy bound on the low-entropy Schoenflies conjecture.

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.