Calum Ross: Topology in Physics – Some Recent Applications

7th September, 2020

An LMS online lecture course in solitons.

The lectures will highlight some recent work on solvable models of topological solitons. The first involves generalisations of the U(1) Abelian-Higgs model whose integrability is intimately related to the geometry of constant curvature Riemann surfaces. The second piece of work is a study of magnetic skyrmions in chiral magnets. Recently a family of soluble models for magnetic skyrmions in chiral magnets was introduced. The energy functional for these models is bounded below by the topological charge, configurations which attain this bound solve first-order equations. The explicit solutions of these first-order equations are given in terms of arbitrary holomorphic functions. Finally I will explain how this model can be interpreted as a gauged non-linear sigma model.

Lecture 1: A primer on solitons. I will introduce the concept of a topological soliton through two prototypical examples, the φ4 and Sine-Gordon models in 1+1 dimensions. Next, we will meet Derrick's theorem and learn why solitons are hard to construct in higher dimensions. Finally, we will meet some examples of higher-dimensional models possessing soliton solutions.

Lecture 2: Solitons in chiral magnets. We will meet a specific model of 2-dimensional chiral magnetic systems which admits soliton solutions. For a special potential term exact, degree 1, skyrmion solutions can be constructed. This leads up to meeting a critically coupled version of the model where there is a whole zoo of analyitic skyrmion solutions.

Stefan Prohazka: Introduction to nonlorentzian geometry based on homogeneous spaces

7th September, 2020

An LMS online lecture course in nonlorentzian geometry.

The topics to be covered are:

   •  Motivation, definition and properties of homogeneous spaces;
   •  Examples to build intuition;
   •  Applications (in physics);
   •  Introduction to spacetimes;
   •  Sketch of classification of maximally symmetric spacetimes;
   •  Discussion of properties and invariants;
   •  Relevance in physics.

Liana Heuberger: Constructing Fano varieties via mirror symmetry

7th September, 2020

An LMS online lecture course in mirror symmetry.

Mirror symmetry conjecturally associates to a Fano orbifold a (very special type of) Laurent polynomial. Laurent inversion is a method for reversing this process, obtaining a Fano variety from a candidate Laurent polynomial. We apply this to construct new Fano 3-folds with terminal Gorenstein quotient singularities.

In this series of talks, I will go through some of the basics of toric geometry to showcase how one can use combinatorial data to systematically build geometric objects. We will restrict our attention to the well-studied Fano case, for which there is concrete evidence that the mirror theorem holds in many cases.

Sergiu Klainerman: On the mathematical theory of black holes

16th October, 2017

The gravitational waves detected recently by LIGO were produced in the final faze of the inward spiraling of two black holes before they collided to produce a more massive black hole. The experiment is entirely consistent with the so called Final State Conjecture of General Relativity according to which, generically, solutions of the initial value problem of the Einstein vacuum equations approach asymptotically, in any compact region, a Kerr black hole. Though the conjecture is so very easy to formulate and happens to be consistent with astrophysical observations as well as numerical experiments, its proof is far beyond our current mathematical understanding, let alone available techniques techniques. In fact even the far simpler and fundamental question of the stability of the Kerr black hole remains wide open.

In my lectures I will address the issue of stability as well as other aspects the mathematical theory of black holes such as rigidity and the problem of collapse. The rigidity conjecture asserts that all stationary solutions the Einstein vacuum equations must be Kerr black holes while the problem of collapse addresses the issue of how black holes form in the first place from regular initial conditions. Recent advances on all these problems were made possible by a remarkable combination of new geometric and analytic techniques which I will try to outline in my lectures.

Daniel Ueltschi: Quantum spin systems and phase transitions

2nd September, 2013

These lectures will be an introduction to the quantum Heisenberg model and other related systems. We will review the Hilbert space, the spin operators, the Hamiltonian, and the free energy. We will restrict ourselves to equilibrium systems. The main questions deal with the nature of equilibrium states and the phase transitions. We will review some of the main results such as the Mermin-Wagner theorem and the method of reflection positivity, that allows to prove the existence of phase transitions. Finally, we will discuss certain probabilistic representations and their consequences.