Conferences  >  Mathematics  >  Algebra  >  United Kingdom

Isaac Newton Institute for Mathematical Sciences, Cambridge

The study of the geometric properties of eigenvalues is closely linked to geometric analysis, applied mathematics and mathematical physics. A striking interplay with the theory of minimal surfaces and recently discovered connections to homogenisation theory have led to new sharp isoperimetric inequalities for Laplace and Steklov eigenvalues on Riemannian manifolds. The latest progress on Neumann and Robin eigenvalues, as well as emerging research on the spectra of other physically important operators, such as Dirac and curl, further highlight the expanding scope of the field and its applications. The workshop aims to bring together a diverse group of experts working on these and related topics. By fostering the exchange of ideas and methodologies, the meeting seeks to inspire innovative approaches to open problems and to advance the broader landscape of spectral geometry.

Isaac Newton Institute for Mathematical Sciences, Cambridge

From spectral shape optimization to inverse problems and computational graphics, applications of spectral problems are ubiquitous in science and engineering, and their approximation presents fascinating mathematical challenges. There are three important areas of mathematical activity which examine spectral problems from distinct perspectives: applications, computation and geometry, with important overlapping interests. In this workshop researchers will present the state-of-the-art at the intersection of these areas, and help shape an ambitious research agenda pushing the frontiers of this intersection

Isaac Newton Institute for Mathematical Sciences, Cambridge

The purpose of the workshop is to bring together specialists from several communities: spectral geometers, numerical analysts, and researchers working in AI geometric data processing and PDEs. The development of AI based spectral analysis requires a deep interaction between them. Our primary objective is to exchange perspectives and gain a deeper understanding of the potential role that artificial intelligence and geometric data processing may play in addressing research questions in spectral geometry, and ultimately to foster the development of new collaborations. We aim to explore how AI might provide new methods, insights, or tools that could help us tackle some of numerical problems related either to the computation of the spectrum or the optimization of the geometry in relationship with spectral functionals.

Geometry and Topology,    Neural Networks and Artificial Intelligence, Machine Learning

The International Centre for Mathematical Sciences (ICMS)

Frieze patterns are celebrated for their ubiquity in mathematics, featuring in cluster algebras, continued fraction theory, difference equations, hyperbolic geometry, and moduli space theory. This workshop brings together researchers from frieze patterns and the cognate fields of continued fractions and discrete geometry to promote cross-disciplinary interactions in these fertile subjects. The event is framed around the legacy of the English geologist John Farey (1766–1826), who inspired the development of Farey sequences and the Farey tessellation, which are common themes of the workshop. There will be a mathematical art gallery accompanying the workshop to share the outputs of our research.

Isaac Newton Institute for Mathematical Sciences, Cambridge

There are several widely open problems about the geometry and the spectral theory of the Laplacian, many of them coming from physics, in particular from "quantum chaos'', including the statistics of the number of eigenvalues in appropriately sized intervals, which is related to the influential universality conjectures of Berry and Tabor and of Bohigas, Giannoni and Schmit; quantum unique ergodicity and Berry's random wave model; optimal lower bounds for the first eigenvalue; asymptotic estimates of Cheeger constants. Randomness plays an important role in the study of such problems. One incarnation is random wave theory, which is the study of random linear combinations of eigenfunctions. This and related models are used to understand the fine structure of nodal sets.

The International Centre for Mathematical Sciences (ICMS)

This workshop centers on the representation theory of finite groups and closely related structures. Some areas of the field are well-established and classical, while others include subjects born only recently. All are being rapidly evolving and actively explored by diverse research communities. These communities, often located in different regions around the world, offer distinct perspectives and approaches to the subject. A central aim of the workshop is to bring together experts and early-career researchers from these various groups, providing a forum to share progress, exchange ideas, and ask each other questions. The workshop also offers an opportunity to celebrate Dave Benson’s 70th birthday, which falls in December 2025. Dave’s work has had a profound influence on many of the themes represented at this meeting, and we look forward to honoring his contributions during the event.

Cohomology and modular representation theory of finite groups; Hochschild cohomology; Algebraic topology; Tensor-triangular geometry; Finite tensor categories

Group Theory,    Courses and Events for Math Students and Early Career Researchers

The International Centre for Mathematical Sciences (ICMS)

The UK has always been at the forefront of, and has a reputation in, integrable systems research. It is an area that glues together topics in mathematical physics, analysis, algebra, geometry, applied mathematics and probability. Research in non-commutative integrable hierarchies has had a recent resurgence – they have many applications, including (eg.) in non-commutative nonlinear optics, and have strong links to string theory. The main themes outlined align with, and will be led, by recent research by the invitees. We propose a different than usual workshop format, along the lines of those held at Oberwolfach and the American Institute of Mathematics. Nominated experts give feeder presentations in their areas on successive mornings, outlining the current frontier of research and open problems. In the afternoons, these are followed up by discussion sessions and working groups who report back to the workshop as a whole at the end of the day. The workshop structure develops dynamically throughout the week. The organisers and the Scientific Advisory Group meet daily to set the agenda ahead, using feedback from the end-of-day reports and participants.

(i) Regularity and solutions; (ii) Non-commutativity, both in terms of non-Abelian (eg.\/ matrix) versions of classical integrable systems and in terms of non-commutative independent variables; and (iii) The connections of such systems to random matrix theory.

The International Centre for Mathematical Sciences (ICMS)

Recent research has pointed to fundamental underlying relationships between each of three sub-fields of combinatorics: matroid theory, structural graph theory, and topological graph theory. While the connection between matroids and graphs is long-established, newer foundational links with topological graph theory are creating fresh opportunities for cross-disciplinary advances. The purpose of this workshop is to provide the first dedicated forum to bring together experts in all these three fields to share techniques and approaches, and seed ongoing cross-disciplinary collaboration and advances.

Graph Theory and Combinatorics,    Geometry and Topology