University of Southampton

The analytic structure of scattering amplitudes exhibit striking
properties that are not at all evident from the first principles of
Quantum Field Theory. These are often rich and powerful enough to be
considered as their defining features, and this makes the problem of
finding a set of universal rules a compelling one. I will review the
recently mounting evidence for the relevance of tropical Grassmannians
in this respect, including implications on symbol alphabets and
adjacency conditions

In this talk I will introduce the study of lattices in locally compact groups through their actions CAT(0) spaces. This is an extremely rich class of groups including S-arithmetic groups acting on products of symmetric spaces and buildings, right angled Artin and Coxeter groups acting on polyhedral complexes, Burger-Mozes simple groups acting on products of trees, and the recent CAT(0) but non biautomatic groups of Leary and Minasyan. If time permits I will discuss some of my recent work related to the Leary-Minasyan groups.

We will discuss a generalisation of hyperbolic groups, from the group actions point of view. By studying torsion, we will see how this can help to answer questions about ordinary hyperbolic groups.

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.
 

Let $X$ be a countable discrete metric space, and think of operators on $\ell^2(X)$ in terms of their $X$-by-$X$ matrix. Band operators are ones whose matrix is supported on a "band" along the main diagonal; all norm-limits of these form a C*-algebra, called uniform Roe algebra of $X$. This algebra "encodes" the large-scale (a.k.a. coarse) structure of $X$. Quasi-locality, coined by John Roe in '88, is a property of an operator on $\ell^2(X)$, designed as a condition to check whether the operator belongs to the uniform Roe algebra (without producing band operators nearby). The talk is about our attempt to make this work, and an expander-ish condition on graphs that came out of trying to find a counterexample. (Joint with: A. Tikuisis, J. Zhang, K. Li and P. Nowak.)
 

The goal of topological data analysis is to apply tools form algebraic topology to reveal geometric structures hidden within high dimensional data. Mapper is among its most widely and successfully applied tools providing, a framework for the geometric analysis of point cloud data. Given a number of input parameters, the Mapper algorithm constructs a graph, giving rise to a visual representation of the structure of the data.  The Mapper graph is a topological representation, where the placement of individual vertices and edges is not important, while geometric features such as loops and flares are revealed.

However, Mappers method is rather ad hoc, and would therefore benefit from a formal approach governing how to make the necessary choices. In this talk I will present joint work with Francisco Belchì, Jacek Brodzki, and Mahesan Niranjan. We study how sensitive to perturbations of the data the graph returned by the Mapper algorithm is given a particular tuning of parameters and how this depend on the choice of those parameters. Treating Mapper as a clustering generalisation, we develop a notion of instability of Mapper and study how it is affected by the choices. In particular, we obtain concrete reasons for high values of Mapper instability and experimentally demonstrate how Mapper instability can be used to determine good Mapper outputs.

Our approach tackles directly the inherent instability of the choice of clustering procedure and requires very few assumption on the specifics of the data or chosen Mapper construction, making it applicable to any Mapper-type algorithm.

The concept of an acylindrically hyperbolic group, introduced by D. Osin, generalizes hyperbolic and relatively hyperbolic groups, and includes many other groups of interest: Out(F_n), n>1, most mapping class groups, directly indecomposable non-cyclic right angled Artin groups, most graph products, groups of deficiency at least 2, etc. Roughly speaking, a group G is acylindrically hyperbolic if there is a (possibly infinite) generating set X of G such that the Cayley graph \Gamma(G,X) is hyperbolic and the action of G on it is "sufficiently nice". Many global properties of hyperbolic/relatively hyperbolic groups have been also proved for acylindrically hyperbolic groups. 
In the talk I will discuss a method which allows to construct a common acylindrically hyperbolic quotient for any countable family of countable acylindrically hyperbolic groups. This allows us to produce acylindrically hyperbolic groups with many unexpected properties.(The talk will be based on joint work with Denis Osin.)
 

When high-dimensional problems are concerned, not much algorithms can break the curse of dimensionality, and solve them efficiently and reliably. Among those, tensor product algorithms, which implement the idea of separation of variables for multi-index arrays (tensors), seem to be the most general and also very promising. They originated in quantum physics and chemistry and descent broadly from the density matrix renormalization group (DMRG) and matrix product states (MPS) formalisms. The same tensor formats were recently re-discovered in the numerical linear algebra (NLA) community as the tensor train (TT) format.

Algorithms developed in the quantum physics community are based on the optimisation in tensor formats, that is performed subsequently for all components of a tensor format (i.e. all sites or modes).
The DMRG/MPS schemes are very efficient but very difficult to analyse, and at the moment only local convergence results for the simplest algorithm are available. In the NLA community, a common approach is to use a classical iterative scheme (e.g. GMRES) and enforce the compression to a tensor format at every step. The formal analysis is quite straightforward, but tensor ranks of the vectors which span the Krylov subspace grow rapidly with iterations, and the methods are struggling in practice.

The first attempt to merge classical iterative algorithms and DMRG/MPS methods was made by White (2005), where the second Krylov vector is used to expand the search space on the optimisation step.
The idea proved to be useful, but the implementation was based on the fair amount of physical intuition, and the algorithm is not completely justified.

We have recently proposed the AMEn algorithm for linear systems, that also injects the gradient direction in the optimisation step, but in a way that allows to prove the global convergence of the resulted scheme. The scheme can be easily applied for the computation of the ground state --- the differences to the algorithm of S. White are emphasized in Dolgov and Savostyanov (2013).
The AMEn scheme is already acknowledged in the NLA community --- for example it was recently applied for the computation of extreme eigenstates by Kressner, Steinlechner and Uschmajew (2013), using the block-TT format proposed by in Dolgov, Khoromskij, Oseledets and Savostyanov (2014).

At the moment, AMEn algorithm was applied
 - to simulate the NMR spectra of large molecules (such as ubiquitin),
 - to solve the Fokker-Planck equation for the non-Newtonian polymeric flows,
 - to the chemical master equation describing the mesoscopic model of gene regulative networks,
 - to solve the Heisenberg model problem for a periodic spin chain.
We aim to extend this framework and the analysis to other problems of NLA: eigenproblems, time-dependent problems, high-dimensional interpolation, and matrix functions;  as well as to a wider list of high-dimensional problems.

This is a joint work with Sergey Dolgov the from Max-Planck Institute for Mathematics in the Sciences, Leipzig, Germany.

We propose a new algorithm for the approximate solution of large-scale high-dimensional tensor-structured linear systems. It can be applied to high-dimensional differential equations, which allow a low-parametric approximation of the multilevel matrix, right-hand side and solution in a tensor product format. We apply standard one-site tensor optimisation algorithm (ALS), but expand the tensor manifolds using the classical iterative schemes (e.g. steepest descent).  We obtain the rank--adaptive algorithm with the theoretical convergence estimate not worse than the one of the steepest descent, and fast practical convergence, comparable or even better than the convergence of more expensive two-site optimisation algorithm (DMRG).
The method is successfully applied for a high--dimensional problem of quantum chemistry, namely the NMR simulation of a large peptide.

This is a joint work with S.Dolgov (Max-Planck Institute, Leipzig, Germany), supported by RFBR and EPSRC grants.

Keywords: high--dimensional problems, tensor train format, ALS, DMRG, steepest descent, convergence rate, superfast algorithms, NMR.