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The International Centre for Mathematical Sciences (ICMS)

The interplay between geometry and physics has been a driving force for new developments in both mathematics and physics for a very long time. In the last two decades, higher category theory has become a fine contributor in this dialogue, providing many invaluable ideas and very useful heuristics. Organised in partnership with the Clay Mathematics Institute, this workshop aims to present the latest results in the application of (higher) geometry to theoretical physics and to foster new collaborations between mathematicians and theoretical physicists.

Isaac Newton Institute for Mathematical Sciences

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.

Algebra,    Neuronale Netze und Künstliche Intelligenz, Maschinelles Lernen

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)

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.

Algebra,    Graphentheorie und Kombinatorik

Isaac Newton Institute for Mathematical Sciences, Cambridge

During the 20th century, interactions between geometers and theoretical physicists were revolutionary in both areas, leading to such unexpected developments as mirror symmetry and the impact of topological field theory on invariants of 4-manifolds. Another momentum is now developing around new connections between number theory and physics, mainly but not exclusively through the theory of automorphic forms. We have reached a time when contacts between number theorists and physicists have the potential for another revolution.

Zahlentheorie, Arithmetik,    Mathematische Physik

Isaac Newton Institute for Mathematical Sciences, Cambridge

This workshop will focus on a circle of problems lying at the interface between harmonic analysis and geometric measure theory, including the Kakeya and Restriction Conjectures, the Falconer distance problem and projection theory. Underlying all these questions are combinatorial incidence problems, concerning intersection patterns between geometric objects such as thin tubes and annuli. The workshop will bring together experts from a range of backgrounds to investigate these deep and important interdisciplinary problems.

Analysis,    Infinitesimalrechnung, Differentialgleichungen, Integralrechnung