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In a spectacular breakthrough, Yitang Zhang proved that there are infinitely many pairs of primes differing by at most 70 million. Due to further advances of Maynard and Tao and the collaborative Polymath Project, 70 million has been reduced to a few hundred. This course will introduce attendees to the mathematics surrounding these developments.
Venue: University of Oxford
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The research school will be held in the School of Mathematics and Statistics University of Glasgow. The school is aimed at PhD students and early postdocs wanting to learn about topics in algebraic Lie theory and representation theory that are currently of great interest. It will be held the week before a conference at the ICMS in Edinburgh on Algebraic Lie theory. Participants in the research school are encouraged to apply to attend the conference too.
Venue: University of Glasgow
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The aim of the research school will be to give a thorough introduction to G2 geometry, starting from fundamental material and progressing through to recent breakthroughs and current research in which the UK plays a leading role. The school will also introduce participants to topics of broader interest in algebra (e.g. representation theory), analysis (e.g. elliptic regularity), geometry (e.g. holonomy) and topology (e.g. characteristic classes). The course will also indicate some connections beyond mathematics to contemporary theoretical physics (M-theory).
Venue: Imperial College
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Automorphic forms are present in almost every area of modern number theory. In recent decades there has been a starburst of activity and progress in this broad area, leading to many new directions, applications, and links with other areas within mathematics and mathematical physics. The purpose of this meeting is to bring together researchers from the EU, US, and elsewhere to understanding the latest developments in the subject, and to provide an environment in which researchers can freely exchange ideas and form new connections.
Venue: Bristol University
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The course will give an overview of the existing methods for investigating integer and rational solutions to Diophantine equations. It will include both the algebraic, analytic and model theory aspects of the subject. The course will take the format of three 6-hour mini-courses, supported by exercise classes.
1. Rational Points
A. Rational points on curves (Michael Stoll, Bremen)
B. Higher-dimensional varieties (Alexei Skorobogatov, Imperial)
2. Integral Points
A. Basic methods and solubility (Jennifer Park, McGill)
B. Analytic methods, (Trevor Wooley, Bristol)
3. Elliptic and modular curves
A. Elliptic curves (Tim Dokchitser, Bristol and Vladimir Dokchitser, Warwick)
B. Modularity (Andrew Granville, Montreal/UCL)
Venue: Baskervillel Hall, Hay-on-Wye
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The aim of this research school is to provide training and preparation for young researchers, many of whom will be attending SPA2015 the following week. The subjects of the courses represent some of the most exciting areas of current research in probability and have been chosen to tie in with a subset of the plenary lectures for SPA. The three main lecture course topics are:
• Two-dimensional Liouville quantum gravity and the Gaussian free field (Nathanael Berestycki, Cambridge)
• Random matrices and PDEs (Paul Bourgade, Courant)
• Integrable probability and the KPZ universality classes (Ivan Corwin, CMI, Columbia, IHP)
Venue: Mathematical Institute, University of Oxford
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This school will cover three interlinked discrete mathematics topics with compter science applications, which all saw exciting developments in the last few years: the Regularity Method, Limits of Combinatorial Structures, and Property Testing. The three main lecture course topics are:
• Regularity of Methods (David Colon, Oxford)
• Limits of Combinatorial Structures (Christian Borgs and Henry Cohn, Microsoft)
• Property Testing (Asaf Shapira, Tel Aviv)
The school will also include three special, more generally focussed, talks given by Noga Alon (Tel Aviv), Christian Borgs (Microsoft) and Ben Green (Oxford).
This school is colocated with the 25th British Combinatorial Conference, which immediately follows the school.
Venue: University of Warwick
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The aim of the school is to provide PhD students with knowledge and training on basic tools needed to study statistical properties of dynamical systems. Students are not assumed to have any knowledge in the subject. In the first two days the lectures will aim at general background and tools needed in the area. More advanced topics related to current advances in the area will start to appear on the third day of the school. The three main lecture course topics are:
• Strong statistical properties of hyperbolic and partially hyperbolic dynamical systems (Carlangelo Liverani, Roma Tor Vergata)
• Fast-slow skew product systems and convergence to stochastic differential equations (Ian Melbourne, Warwick)
• Open systems: fundamental questions and emerging perspectives (Mark Demers, Fairfield)
Venue: Loughborough University
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This school will cover important recent developments in three central areas of mathematical analysis, namely calculus of variations, geometric nonlinear PDE and mathematical phyiscs. For each of these research areas there will be a course consisting of two mini-courses:
The school will also include guest lectures given by Robert Jensen (Loyola) and Juan Manfredi(Pittsburgh)
Venue: University of Reading
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Symplectic manifolds arise as phase spaces in the Hamiltonian formulation of classical dynamics. In this setting, fixed energy levels naturally give rise to contact structures on odd-dimensional manifolds. The global topology of symplectic and contact manifolds has been a central area of study since the groundbreaking work of Gromov, Donaldson, Floer and others in the 1980s and 1990s. Contact topology has steadily increased in importance as a subject in its own right, and has been an essential ingredient in some of the biggest results in low-dimensional topology this century, including Kronheimer-Mrowka’s proof of Property P and the deep links exhibited by Ozsváth-Szabó and others between Floer-theoretic invariants such as Heegaard Floer homology and the topology of 3-manifolds.
Several key themes have emerged in three dimensional contact topology. These include: Gromov’sh-principle, and the resulting flexible/rigid dichotomy; the importance of open book decompositionsof manifolds; fillability questions concerning symplectic manifolds bounded by a given contact manifold; and the intimate relationship between symplectic and contact topology and gauge-theoretic invariants of smooth manifolds.
A key result of Borman-Eliashberg-Murphy in 2014 gave a dramatic advance in our understanding of contact topology in higher dimensions: using an h-principle argument, they extended Eliashberg’s classification of flexible contact structures in three dimensions to arbitrary dimension. This result generated a great deal of activity in high-dimensional contact topology, leading to a rapidly growing understanding of which aspects of the theory in three dimensions can be generalised to higher dimensions, and of new phenomena which arise in high dimensions.
This LMS-CMI Research School will give students a comprehensive and accessible introduction to key aspects of contact topology in three dimensions and to the new frontier of high-dimensional contact topology. This is a unique opportunity for students and early-career researchers to get a hands-on guided tour of an exciting and fast-developing area of research from a world-leading team of experts.
Lecturers
Vincent Colin, Université de Nantes
Emmy Murphy, Massachusetts Institute of Technology
András Stipsicz, Alfréd Rényi Institute of Mathematics
Each of the above will give a minicourse of 5 lectures, aimed at graduate students with a fairly broad background. These will be supported by teaching assistants who will lead tutorials/problem sessions. Additional lectures will be given by Patrick Massot, Gordana Matić, and Ivan Smith.
Image taken, with permission, from Ken Baker’s blog Sketches of Topology
Venue: University of Glasgow
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The Research School will feature a set of minicourses, offering an introduction to a number of central topics in low-dimensional topology, geometry and dynamics. Each minicourse will be broadly directed towards one of the workshops in the Warwick-EPSRC symposium on Geometric Topology in Low Dimensions, and will present the basic material needed to understand the central objects, ideas, and techniques that will appear there.
There will also be tutorial support for each course, at which attendees will have the opportunity to review the lectures, solve problems and ask questions related to the material presented in the course.
Venue: University of Warwick
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Manifolds are at the centre of much of geometry and topology, and through the influence of axiomatic topological quantum field theory they have become an important organizing force in category and representation theory.
Classically, in the 1960s, algebraic topology was at the heart of their classification theory in form of characteristic classes and characteristic numbers, cobordism theory, surgery theory, and later Waldhausen’s K-theory of manifolds. By the 1980s the machinery got heavy with diminishing returns. We are now experiencing a renaissance of the field as well as a paradigm shift where manifolds not only are the objects of study but become the tools.
The school aims at inspiring the next generation with this exciting success story of interwoven ideas bouncing between different fields, and giving the participants the tools to contribute to this lively research area.
Venue: Mathematical Institute, University of Oxford