Past

The QDWH algorithm can compute the polar decomposition of a matrix in a stable and efficient way. We generalize this method in order to compute generalized polar decompositions with respect to signature matrices. Here, the role of the QR decomposition is played by the hyperbolic QR decomposition. However, it doesn't show the same favorable properties concerning stability as its orthogonal counterpart. Remedies are found by exploiting connections to the LDL^T factorization and by employing well-conditioned permuted graph bases. The computed polar decomposition is used to formulate a structure-preserving spectral divide-and-conquer method for pseudosymmetric matrices. Applications of this method are found in computational quantum physics, where eigenvalues and eigenvectors describe optical properties of condensed matter or molecules. Additional properties guarantee fast convergence and a reduction to symmetric definite eigenvalue problems after just one step of spectral divide-and-conquer.

Numerous mathematical models have been proposed for modelling cancerous tumour invasion (Gatenby and Gawlinski 1996), angiogenesis (Owen et al 2008), growth kinetics (Wang et al 2009), response to irradiation (Gao et al 2013) and metastasis (Qiam and Akcay 2018). In this study, we attempt to model the qualitative behavior of growth, invasion, angiogenesis and fragmentation of tumours at the tissue level in an explicitly spatial and continuous manner in two dimensions. We simulate the effectiveness of radiation therapy on a growing tumour in comparison with immunotherapy and propose a novel framework based on vector fields for modelling the impact of interstitial flow on tumour morphology. The results of this model demonstrate the effectiveness of employing a system of partial differential equations along with vector fields for simulating tumour fragmentation and that immunotherapy, when applicable, is substantially more effective than radiation therapy.

A polynomial form is established for the off-shell CHY scattering equations proposed by Lam and Yao. Re-expressing this in terms of independent Mandelstam invariants provides a new expression for the polynomial scattering equations, immediately valid off shell, which makes it evident that they yield the off-shell amplitudes given by massless ϕ3 Feynman graphs. A CHY expression for individual Feynman graphs, valid even off shell, is established through a recurrence relation.

It is known that many real-world networks exhibit geometric properties.  Brain networks, social networks, and wireless communication networks are a few examples.  Storage and transmission of the information contained in the topologies and structures of these networks are important tasks, which, given their scale, is often nontrivial.  Although some (but not much) work has been done to characterize and develop compression limits and algorithms for nonspatial graphs, little is known for the spatial case.  In this talk, we will discuss an information theoretic formalism for studying compression limits for a fairly broad class of random geometric graphs.  We will then discuss entropy bounds for these graphs and, time permitting, local (pairwise) connection rules that yield maximum entropy properties in the induced graph distribution.

Given a number field K, it is natural to ask whether it has a finite extension with ideal class number one. This question can be translated into a fundamental question in class field theory, namely the class field tower problem. In this talk, we are going to discuss this problem as well as its solution due to Golod and Shafarevich using methods of group cohomology.
 

 The Steklov eigenvalue problem is an eigenvalue problem whose spectral parameters appear in the boundary condition. On a Riemannian surface with smooth boundary, Steklov eigenvalues have a very sharp asymptotic expansion. Also, a number of interesting sharp bounds for the $k$th Steklov eigenvalues have been known. We extend these results on orbisurfaces and discuss how the structure of orbifold singularities comes into play. This is joint work with Arias-Marco, Dryden, Gordon, Ray and Stanhope.

A broad challenge in the theory of finitely generated groups is to understand their subgroups. A group is commensurably coHopfian if its finite index subgroups are distinct from its infinite index subgroups (that is to say not abstractly isomorphic). We will focus primarily on hyperbolic groups, and give the first examples of one-ended hyperbolic groups that are not commensurably coHopfian.
This is joint work with Emily Stark.
 

"We consider a spatial Lambda-Fleming-Viot process, a model in mathematical biology, with a randomly chosen (rough) selection field. We study the scaling limit of this process in different regimes. This leads to the analysis of semi-discrete approximations of singular SPDEs, in particular the Parabolic Anderson Model and allows to extend previous results to weakly nonlinear cases. The subject presented is based on joint works with Aleksander Klimek and Nicolas Perkowski."

In this talk I will introduce a continuous wetting model consisting of the law of a Brownian meander tilted by its local time at a positive level h, with h small. I will prove that this measure converges, as h tends to 0, to the same weak limit as for discrete critical wetting models. I will also discuss the corresponding gradient dynamics, which is expected to converge to a Bessel SPDE admitting the law of a reflecting Brownian motion as invariant measure. This is based on joint work with Jean-Dominique Deuschel and Tal Orenshtein.

McDuff and Schlenk determined when a four-dimensional symplectic ellipsoid can be symplectically embedded into a four-dimensional ball. They found that if the ellipsoid is close to round, the answer is given by an ``infinite staircase" determined by the odd index Fibonacci numbers, while if the ellipsoid is sufficiently stretched, all obstructions vanish except for the volume obstruction. Infinite staircases have also been found when embedding ellipsoids into polydisks (Frenkel - Muller, Usher) and into the ellipsoid E(2, 3) (Cristofaro-Gardiner - Kleinman). In this talk, we will see how the sharpness of ECH capacities for embedding of ellipsoids implies the existence of infinite staircases for these and three other target spaces.  We will then discuss the relationship with toric varieties, lattice point counting, and the Philadelphia subway system. This is joint work with Dan Cristofaro-Gardiner, Alessia Mandini,
and Ana Rita Pires.

In the process of studying the zeta-function for one parameter families of Calabi-Yau manifolds we have been led to a manifold, for which the quartic numerator of the zeta-function factorises into two quadrics remarkably often. Among these factorisations, we find persistent factorisations; these are determined by a parameter that satisfies an algebraic equation with coefficients in Q, so independent of any particular prime.  We note that these factorisations are due a splitting of Hodge structure and that these special values of the parameter are rank two attractor points in the sense of IIB supergravity. To our knowledge, these points provide the first explicit examples of non-singular, non-rigid rank two attractor points for Calabi-Yau manifolds of full SU(3) holonomy. Modular groups and modular forms arise in relation to these attractor points in a way that, to a physicist, is unexpected. This is a report on joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.

In this talk, we will provide an overview of results in the setting of granular crystals, consisting of spherical beads interacting through nonlinear elastic spring-like forces. These crystals are used in numerous engineering applications including, e.g., for the production of "sound bullets'' or the examination of bone quality. In one dimension we show that there exist three prototypical types of coherent nonlinear waveforms: shock waves, traveling solitary waves and discrete breathers. The latter are time-periodic, spatially localized structures. For each one, we will analyze the existence theory, presenting connections to prototypical models of nonlinear wave theory, such as the Burgers equation, the Korteweg-de Vries equation and the nonlinear Schrodinger (NLS) equation, respectively. We will also explore the stability of such structures, presenting some explicit stability criteria for traveling waves in lattices. Finally, for each one of these structures, we will complement the mathematical theory and numerical computations with state-of-the-art experiments, allowing their quantitative identification and visualization. Finally, time permitting, ongoing extensions of these themes will be briefly touched upon, most notably in higher dimensions, in heterogeneous or disordered chains and in the presence of damping and driving; associated open questions will also be outlined.

Molecules are dynamical systems that can adopt a variety of three dimensional conformations which, in general, differ in energy and physical properties. The identification of energetically favourable conformations is fundamental in molecular physics and computational chemistry, since it is closely related to important open problems such as the prediction of the folding of proteins and virtual screening for drug design.
In this talk I will present theoretical and data-driven approaches to the study of molecular conformational spaces and their associated energy landscapes. I will show that the topology of the internal molecular conformational space might change after taking its quotient by the group action of a discrete group of symmetries. I will also show that geometric and topological tools for data analysis such as procrustes analysis, local dimensionality reduction, persistent homology and discrete Morse theory provide with efficient methods to study the mathematical structures underlying the molecular conformational spaces and their energy landscapes.
 

Taught courses offer a range of distinctive learning opportunities from lectures to tutorials/supervisions through to individual study. Orchestration refers to the combining and sequencing of these opportunities for maximum effect. This raises a question about who does the orchestration. In school, there is a good case for suggesting that it is teachers who take responsibility for orchestration of students’ learning opportunities. Moving to university, do students take on more responsibility for orchestration?

In this session there will be a chance to look back on the learning opportunities you experienced last term and to reflect on how (or even if) they were orchestrated. What could be different in the term ahead if you pay more attention to how distinctive learning opportunities are orchestrated?

Globally, almost 38 million people are living with HIV.  HPTN 071 (PopART) is the largest HIV prevention trial to date, taking place in 21 communities in Zambia and South Africa with a combined population of more than 1 million people.  As part of the trial an individual-based mathematical model was developed to help in planning the trial, to help interpret the results of the trial, and to make projections both into the future and to areas where the trial did not take place. In this talk I will outline the individual-based mathematical model used in the trial, the inference framework, and will discuss examples of how the results from the model have been used to help inform policy decisions.  

In this talk we present p-order methods for unconstrained minimization of convex functions that are p-times differentiable with Hölder continuous p-th derivatives. We establish worst-case complexity bounds for methods with and without acceleration. Some of these methods are "universal", that is, they do not require prior knowledge of the constants that define the smoothness level of the objective function. A lower complexity bound for this problem class is also obtained. This is a joint work with Yurii Nesterov (Université Catholique de Louvain).
 

We will outline the main features of Wooley's efficient congruencing method for the parabola. Then we will go on to prove new bounds for discrete restriction to the curve (x,x^3). The latter is joint work with Trevor Wooley (Purdue).

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.

Boundary integral equations are an elegant tool to model and simulate a range of physical phenomena in bounded and unbounded domains.

While mathematically well understood, the numerical implementation (e.g. via boundary element methods) still poses a number of computational challenges, from the efficient assembly of the underlying linear systems up to the fast preconditioned solution in complex applications. In this talk we provide an overview of some of these challenges and demonstrate the efficient implementation of boundary element methods on modern CPU and GPU architectures. As part of the talk we will present a number of practical examples using the Bempp-cl boundary element software, our next generation boundary element package, that has been developed in Python and supports modern vectorized CPU instruction sets and a number of GPU types.

Let's take a step back to understand what it means to use maths in society: Which maths, and whose society? I'll talk about some of the options I've come across, including time I spent at the US Census Bureau, and we will hear your ideas too. We might even crowdsource a document of maths in society opportunities together...

Do classical solutions of the compressible Navier-Stokes equations converge to an entropy solution of their inviscid counterparts, the Euler equations? In this talk we present a result which answers this question affirmatively, in the one-dimensional case, for a particular class of fluids. Specifically, we consider gases that exhibit approximately polytropic behaviour in the vicinity of the vacuum, and that are isothermal for larger values of the density (which we call approximately isothermal gases). Our approach makes use of methods from the theory of compensated compactness of Tartar and Murat, and is inspired by the earlier works of Chen and Perepelitsa, Lions, Perthame and Tadmor, and Lions, Perthame and Souganidis. This is joint work with Matthew Schrecker.

We take a look at difference fields with several commuting automorphisms. The theory of difference fields with one distinguished automorphism has a model companion known as ACFA, which Zoe Chatzidakis and Ehud Hrushovski have studied in depth. However, Hrushovski has proved that if you look at fields with two or more commuting automorphisms, then the existentially closed models of the theory do not form a first order model class. We introduce a non-elementary framework for studying them. We then discuss how to generalise a result of Kowalski and Pillay that every definable group (in ACFA) virtually embeds into an algebraic group. This is joint work in progress with Zoe Chatzidakis and Nick Ramsey.

Whitehead published two papers in 1936 on free groups. Both concerned decision problems for equivalence of (sets of) elements under automorphisms. The first focused on primitive elements (those that appear in some basis), the second looked at arbitrary sets of elements. While both of the resulting algorithms are combinatorial, Whitehead's proofs that these algorithms actually work involve some nice manipulation of surfaces in 3-manifolds. We will have a look at how this works for primitive elements. I'll outline some generalizations due to Culler-Vogtmann, Gertsen, and Stallings, and if we have time talk about how it fits in with some of my current work.