The speakers for the plenary talks will be:
University of Newcastle
Research Interests: Algebra, representation theory, quantum groups, quantum affine algebras, Kac-Moody algebras, double affine Hecke algebras and their degenerations, quantum symmetric pairs.
Talk title: Algebras, categories, braids and knots
Abstract: I will expose a cunning plan of representation theorists to replace some difficult computations with some simple pictures, and along the way, use quantum groups to solve a problem from topology.
University of Warwick
Research Interests: Functional Analysis, High-dimensional and Discrete Geometry, Information Theory.
Talk title: The Second Law of Probability.
Abstract: The talk explains how a geometric principle gave rise to a new
variational description of information-theoretic entropy and how this
led to the solution of a problem dating back to the 50’s: whether the
the central limit theorem is driven by an analogue of the second law
University of Kent
Research Interests: Discrete variational methods, with applications to geometric integration, Noether’s Theorem in all its manifestations, Moving frames, Discrete moving frames, Multispace methods, Symbolic analysis for nonlinear differential and difference equations.
Talk title: Introduction to Moving Frames and its Applications
Abstract: I will give a gentle introduction to moving frames as they are now understood. Although the term is associated with Élie Cartan, the ideas underlying moving frames have a long history in differential geometry. The modern definition encompasses those ideas but can be used in modern applications such as numerical schemes for solving differential equations, to image processing and even to questions in computational algebraic geometry.After showing the basic ideas and the key results, I will show some applications in the smooth and discrete calculus of variations, in joint work with Tania Goncalves and Ana Rojo-Echeburúa.
The speaker for the public talk will be Paul Sutcliffe from University of Durham.
Research Interests: Topological solitons, skyrmions, monopoles, knot solitons.
Talk title: Tying Tornadoes in Knots
Abstract: We are all familiar with a tornado as a swirling vortex of air. In the mid-nineteenth century Lord Kelvin proposed the possibility of tying a knot in a tornado. Kelvin’s tornado was not in air but rather in the aether that was thought to fill all space at that time. He proposed that different atoms are simply different types of knots and that atoms are stable because tornado knots cannot be untied in a perfect aether fluid. Today we know that there is no aether, but Kelvin’s idea of tying tornadoes in knots is currently being studied in a variety of other materials. Examples will be presented from mathematical biology, chemistry and physics, and the fate of these tornado knots discussed.
The speakers for the keynote talks will be:
University of Warwick
Research Interests: Structure of rings of integers, of their units, of Mordell-Weil groups of elliptic curves, of higher K-groups of rings of integers as Galois modules, Cohen-Lenstra heuristics, Regulator constants and other invariants of Galois modules, Arithmetic of elliptic curves: growth of Mordell-Weil groups and of Selmer groups in extensions of number fields, Integral representations of finite groups, The Burnside ring and the rational representation ring of a finite group.
Talk title: The Cohen-Lenstra principle in mathematics; or: what does a random group look like?
Abstract: A common theme that has emerged in the last few decades across many branches of pure mathematics is that the behaviour of algebraic objects in families can be successfully predicted by probabilistic models. This was first observed in the case of ideal class groups of number fields by Cohen and Lenstra in the early 1980s, and many more instances have been found since. In this talk, I will explain the shape that these models take, and sketch some examples of the Cohen–Lenstra principle in nature, coming from number theory, combinatorics, and arithmetic geometry. Lots of open problems will be mentioned along the way.
University of Queen Mary, London
Research Interests: General relativity, higher dimensional black holes and numerical general relativity.
Talk title: Higher dimensional black holes: breaking Einstein’s gravity.
Abstract: In this talk I will give an overview of black holes in higher dimensions, and how they challenge our understanding of general relativity, Einstein’s theory of gravity. As I shall explain, in higher dimensions black holes can have different shapes, other than the spherical one. More interestingly, higher dimensional black holes can be unstable under small perturbation. Recent progress has shown that the evolution of these instabilities can lead to the formation of naked singularities, where Einstein’s theory of gravity breaks down.
University of Swansea
Research Interests: K-Theory, Hopf algebras, Miller-Morita-Mumford characteristic numbers of fibre bundles.
Talk title: What does topological K theory see?
Abstract: Algebraic topology uses different algebraic tools for studying the topological world, such as homology, homotopy, cobordism, or K theory, which is built on the idea of vector bundles. It is useful to know how effective a tool is, and how much information it can reveal. Bousfield gave a precise description of what p-local complex K theory sees, but in a rather complicated form. Clarke, Whitehouse and I identified this more simply as a category of “discrete” modules over a certain ring. In this talk I will present work with Neeran Khafaja about related, simpler categories that shed some light on the Bousfield category.
University of Glasgow
Research Interests: Discrete and ultra discrete integrable systems, quasideterminants.
Talk title: Box and Ball Systems and the Ultra-discrete KdV equation
Abstract: The Ultra discrete KdV equation is a discrete version of the celebrated KdV equation. It is discrete in the independent variables (time and space). The dependent variable $u_i^t$ conventionally takes the binary values 0 or 1 and then soliton solutions take the form of blocks of 1s moving to the right, the larger blocks moving at a faster speed. In this talk we will look at how to represent this equation in terms of a box and ball system. We shall look, in particular, at the case where the possible values of $u_i^t$ are extended to the reals. If time, we will also look at the connection between these systems and max plus algebras.
University of Kent
Research Interests: Topological solitons in mathematical physics, in particular the classical and quantum behaviour of Skyrmions. See: Topological Soliton Group.
Talk title: Topological Solitons
Abstract: Solitons are solutions of differential equations which look and behave like extended particles. “Topological solitons” have the additional property that the number of solitons cannot be changed by smooth deformations. In this talk, I will give examples of solitons and describe some interesting properties. For example, in some models, the static solutions with N solitons form a smooth manifold M whose dimension is proportional to N. Then, the dynamics of solitons can often be approximated as geodesic motion on M. I will end by describing a current research project.
Talk title: The Arakelov class group
Abstract: The Arakelov class group of an algebraic number field is an abelian group that can be viewed as a combination of the class group of the field and the unit group of its ring of integers, and that is better behaved than both. In the lecture we give several definitions of the group, and we discuss the role it plays in formulating conjectures about the distribution of class groups of random algebraic number fields.
Talk title: Localised planar patterns with applications to crime hotspots and magnetic fluids
Abstract: Understanding the formation of localised patterns in simple nonlinear partial differential equations has many applications to long-standing questions in nature as well as more recent ones in sociology. In this talk, I will present the theory for the emergence and behaviour of various types of localised planar patterns in a prototypical PDE and show how this theory can be applied to models of crime hotspots and magnetic fluids. Finally, I will present the many open problems in this exciting field.
Talk title: What kind of Mathematicians for Tomorrow’s Big Human Challenges
Abstract: Mathematics for human challenges of the future: languages rather than techniques. Why should mathematics not just be memorised or viewed as techniques, but be learned and understood as living and evolving languages? Just as there is not a “single” mathematics, but on the contrary such a great number of mathematical domains that it is unthinkable to imagine one mathematician could seek to practice them all, let alone master them all…nor is there just one way of doing mathematical research, or even using mathematics and, hence, of tackling a complex problem and striving to solve it. There is a well-trodden path taught and familiar to students – the “academic method” – and there are ‘the other’ methods. The talk aims to be an introduction to the idea that alternative rigorous approaches exist and to the impact of those new mechanisms of reflection in this “extraordinary quest”, which is to try to understand Reality in order to find solutions to the numerous challenges it poses for mankind.
University of Kent
Research Interests: Stable homotopy theory, in particular model categories and chromatic homotopy theory and homological algebra and A-infinity algebras.
Talk title: Stable homotopy groups of spheres.
Abstract: Stability is a phenomenon that adds a lot of structure to algebraic topology, particularly to homotopy groups. I will explain and motivate the construction of stable homotopy groups and give a brief overview of examples, known results and open challenges. This talk does not require any prerequisites in topology.
Rubén Sánchez García
University of Southampton
Research Interests: Interface between pure and applied mathematics within complexity theory, particularly mathematical aspects of complex networks, and topological data analysis.
Talk title: Network functions in the presence of symmetry
Abstract: Network models inherit all the redundancies of the system they represent, and they manifest themselves as symmetries of the underlying graph. The presence of such symmetries, a remarkable feature of real-world networks, have profound consequences on network structure and eigenvalues, and, crucially, symmetries are inherited by any function, or measurement, on the network. In this talk, I will explain the theoretical framework to study network symmetry in the context of arbitrary network functions, and the practical consequences of redundancies for compression, computational reduction, and spectral decomposition. The talk will start with an overview of previous and current work in applied graph theory and topology, including spectral clustering for power transmission networks, and the discrete Laplacian applied to ranking in horse racing. I will conclude with a reflection on working in the interface between pure and applied mathematics.