Events

Title - Diagrammatic Algebra and p-Kazhdan—Lusztig theory

Speaker - Dr Chris Bowman-Scargill (University of York)

We introduce p-Kazhdan—Lusztig theory as a machine for encoding structures from representation theory in terms of polynomials.  We take a gentle tour of some of the state of the art concerning these objects, and discuss what is combinatorially knowable and unknowable in this field.

Title - Instabilities in spatially-periodic, and in slightly random, media

Speaker - Professor Jonathan Healey (Keele University)

We start with a brief introduction to fluid mechanics for the uninitiated, leading into the subject of 'hydrodynamic stability theory' (HST), which can describe how waves added to a steady fluid flow travel and grow, possibly leading to turbulence. The appearance of turbulence has countless practical applications (e.g. in physiological, atmospheric, oceanographic, astrophysical, aeronautical flows), and is often called the outstanding problem of classical mechanics. It has attracted the attention of an impressive list of mathematicians and scientists, including Leonardo da Vinci, who carried out experiments, making beautiful, detailed, sketches of his observations, Newton, who explained how friction can act in fluids (viscosity), Euler, who derived the equations of motion of inviscid (frictionless) fluid flow. The Navier-Stokes equations for viscous flow were finally derived mid-19th century and one of the millenium prize problems in mathematics is concerned with the Navier-Stokes equations.

The first HST was derived by Kelvin in 1871 for a flow proposed by Helmholtz in 1868 for two parallel streams moving at different speeds, resulting in a 'mixing layer' between the two streams (google 'billow clouds' for dramatic images and videos). While modern studies of unstable flows are often based on direct numerical simulations of the Navier-Stokes equations, many results and insights continue to be obtained using theoretical approaches, complementing and informing numerical and physical experiments. In this talk we review a line of development of the Kelvin-Helmholtz instability theory focused particularly on the interplay between the propagation and instability properties of waves in mixing layers, and the unexpectedly strong effects that can be produced when the basic flow varies by a small amount in the direction of the flow. The effect of spatially-periodic variation at the boundary of a flow is considered first (described by Floquet theory), which leads, surprisingly, to solutions for small random spatial variations at a boundary (described by stochastic differential equations) that differ greatly from the usual mathematically convenient, but physically unrealistic, model of perfectly flat boundaries. This work is in collaboration with Dr Matt Turner from the University of Surrey, another member of the wonderful MAGIC consortium!

Title - Exploring the landscape of Fano varieties: geometry, computation, and machine learning

Speaker - Alexander Kasprzyk, Associate Professor in Geometry (University of Nottingham) 

Algebraic geometry is one of the gems of pure mathematics. Essentially concerned with the study of shapes described by systems of polynomial equations, it builds a fundamental bridge between algebra and geometry, and informs many areas of modern mathematics and theoretical physics.

One of the fundamental open problems in algebraic geometry is the classification of Fano varieties: shapes with positive curvature which form the "atomic pieces" of geometry. Although known to be finite in each dimension (thanks in part to work by UK Fields Medalist Caucher Birkar), even the classification in dimension three is largely mysterious.

Recent ideas taken from mirror symmetry help translate this classification problem into a combinatorial question which can be tackled by computers. Many centuries of runtime on High Performance Computing clusters is providing the first conjectural glimpse of the landscape of Fano varieties. This has transported an abstract theoretical challenge into the world of Big Data. Much still remains to be understood, however tools from data science -- including machine learning -- are finally allowing us to begin making sense of the rich mathematical structure hidden within this classification.

Title: Coproducts, knots and quantum groups
Speaker: Martina Balagovic (Newcastle University)

Quantum groups are certain quasitriangular bialgebras defined in the 1980s as deformations of enveloping algebras of Lie algebras, with motivation coming from quantum physics.

I will explain what that means, and explain two ideas which lead to one of their applications:

1. 1. How reversing arrows in a definition of an algebra (a structure with a product and a unit) produces a coalgebra (a structure with a coproduct and a counit), and how a compatible algebra and coagebra structure provides a setting in which it is possible to pose a certain "universal" version of the Yang Baxter equation, which describes scattering in statistical mechanics.
2. 2. How quantum groups provide a "universal" solution to the Yang Baxter equation, and how such solutions can be used to generate invariants of knots.

Despite some long words from physics and topology, this will be a talk about algebra and representation theory, and assume only a good knowledge of linear algebra (including some understanding of tensor products).

At the end, to provide a link with some recent developments in the field, I will briefly discuss how adding the boundary on the physics side of this story changes the algebra and the topology. 

All MAGIC students are invited to join a virtual question and answer session on Tuesday 31 May at 11am to ask any questions concerning MAGIC. The panel will consist of Prof Peter Ashwin – Director of MAGIC (University of Exeter) Dr Yuri Bazlov – Deputy Director of MAGIC (University of Manchester) Prof Tim Phillips – Chair of MAGIC Programme Committee (Cardiff University) The session will take place via Zoom and a link to join the session will be made available under the “This week” section of the website on the day of the session. Please ensure that you are logged in to your MAGIC Zoom account in order to attend. We look forward to seeing you!

Dear All,

You are invited to join this year’s MAGIC launch lecture on Thursday 30 September at 16:00.

Please do attend if you are able to. Staff, students and researchers are all welcome.

To join please go to “This week” and select the “Join lecture” button which will appear shortly before the start of the lecture.

Please note you will you need to have set up a MAGIC Zoom account to join this live lecture – instructions on how to do this can be found in the "How to join a  live lecture" thread on the MAGIC General Forum.

Should you be unable to join on the day the lecture will be recorded and available to view at a later time.

MAGIC Launch Lecture Title: Thermalisation in weakly nonlinear chains
Speaker: Davide Proment (University of East Anglia)
Date: Thursday 30 September 2021 at 16:00 – 17:00

Abstract: I will present some theoretical results on the seminal $\alpha$-Fermi-Pasta-Ulam-Tsingou (FPUT) problem, that is a one-dimensional system formed N=16,32 and 64 masses connected by nonlinear quadratic springs.

The theoretical approach is based on resonant wave-wave interaction theory and wave turbulence theory.

Assuming the weakly nonlinear regime, the one originally considered by Fermi to model heat transport in crystals, I will show that the long time dynamics of the system is ruled by exact discrete wave resonances.

After giving a general introduction to the 1955 Fermi-Pasta-Ulam-Tsingou model, a system that inspired areas in mathematics and physics like integrable systems, ergodic theory, nonlinear waves, and statistical mechanics, I will discuss the main idea of the wave-wave interaction theory and wave turbulence theory.

I will show that the first non-trivial wave resonances correspond to six-wave interactions as solutions of Diophantine equations.

These interactions are responsible for the thermalisation process in the FPUT system and explain its extremely long time scale.

Numerical simulations supporting out theoretical predictions will be briefly discussed (based on the joint work with Miguel Onorato, Lara Vozella and Yuri V. L'vov entitled "A route to thermalisation in the $\alpha$-Fermi-Pasta-Ulam(-Tsingou) system", PNAS 112, 4208-4213, 2005)

I hope you will be able to join us.

Kind regards,

Sam Blackburn
MAGIC Administrator

MAGIC launch lecture 2016: Topology, Machine Learning and Statistics: three faces of data. (J. Brodski, University of Southampton) In recent years, topology has emerged as a new approach to analysis of complex data, and has provided some very interesting ideas and methodologies that have led to interesting results in genomics, medicine, mathematical biology, and others. In this talk I shall describe how topology interacts with machine learning and statistics, and how this exciting synthesis may advance research in data science. All are welcome to join this general interest lecture that will be in the MAGICPUBLIC virtual room.

MAGIC Launch Lecture: Chaos, numbers and fractuals: the MAGIC of ergodic theory
Charles Walkden (University of Manchester)
Abstract: Dynamical systems is the study of iteration: given a map T acting on a space X and a point x ∈ X, what happens when one repeatedly applies the map T to x? Often, dynamical systems are chaotic: two points x,y ∈ X that start very close to each other can have very different behaviours under iteration. Instead of trying to understand the behaviour of every point under iteration, ergodic theory takes a more qualitative view: what does the long-term behaviour of a typical point look like? In addition, subsets of X that are invariant under the dynamics have a complicated structure and are commonly `fractal' in nature. We will see how these notions fit together and allow us to obtain some striking and surprising results in number theory.

MAGIC annual lecture 2014/15
Noether's theorem 100 years on
Professor Elizabeth Mansfield University of Kent

In 1918, Emmy Noether showed how to calculate conserved quantities for variational problems which have a Lie symmetry, such as invariance under translations and rotations in time and space. Noether obtained the complete result, including the case of invariance under pseudo group actions.
In this talk I will show how Noether's two theorems have been updated to finite difference and to numerical schemes. One outstanding motivation is the desire to incorporate a particular conservation law, potential vorticity, into numerical schemes, needed (I am told) for extreme weather prediction.

Abstract: I will give a gentle introduction to some deep and important quantum phenomena associated with topology. The first is the Aharonov-Bohm effect, and the second is quantum statistics for identical particles. No prior knowledge of quantum mechanics will be assumed. I will go on to discuss some new work on quantum statistics in discrete spaces associated with recent developments in characterizing their topology.

Dynamics of Social perceptions, Attitudes Norms and Applications to Digital Media and Security