School of Mathematics

This LMS Research School for PhD students and early stage researchers is devoted to quantum machine learning (QML) and quantum simulations, which constitute active and very promising areas of research. Both QML and Hamiltonian simulations encompass interesting mathematical structures and give a gateway into deep, open questions in computational complexity and quantum advantage. Both topics have generated considerable recent interest because of their applicability on current and near-term noisy quantum hardware.

The three main lecture courses at the school reflect these concerns and opportunities. Introduction to quantum information and algorithms  will introduce a mathematical audience to the basic concepts and key ideas that are used in constructing efficient quantum algorithms and measuring their effectiveness. The other two courses, Introduction to Quantum Machine Learning  and Hamiltonian Simulation  will look at the application of this framework in machine learning and the simulation of quantum systems and optimization.

Whilst the summer school is primarily designed to equip early career mathematicians with cutting-edge knowledge that will allow them to approach the current research landscape in QML, the courses and plenary talks will also address the wider industrial and societal impact of the research area.

The area of integrable systems is a large and active area of research with lots of fruitful connections with algebra, geometry, analysis and theoretical physics. The workshop is devoted to interactions between integrability and geometry, highlighting recent developments where there has been substantial progress both in terms of establishing new connections and new structures. 

The workshop will be centred around the following overlapping topics: Cherednik algebras and many-body problems; Poisson geometry, deformations and quantisation; Frobenius manifolds; discrete geometry and discrete dynamics.

The workshop will also be an occasion to celebrate the 70th birthday of Alexander Veselov who made foundational contributions to Integrable Systems and related areas. 

The aim of this workshop is to discuss the recent advances and new developments in the research area of anabelian geometry and the theory of arithmetic fundamental groups. This includes among others the newly discovered versions of the Grothendieck anabelian conjectures for hyperbolic curves and finitely generated fields, new results on the section conjecture, new developments in the theory of multizeta values, and the recent developments in combinatorial anabelian geometry  and Grothendieck-Teichmuller theory.

We aim to bring together experts in the field, as well as researchers who are interested in the theory of arithmetic fundamental groups and its applications, in order to discuss these developments and explore further research directions and possible applications.