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Geometric Problems in Linear and Nonlinear PDEs

Geometric analysis has intrigued mathematicians for centuries and remains a vibrant area of modern mathematics. Central to this field are the formulation of both linear and nonlinear partial differential equations (PDEs) on manifolds and the minimization of energy functionals in infinite-dimensional spaces. Our workshop will focus on four closely interconnected areas within geometric analysis: characterizing solutions to PDEs, Euler equations, minimal surface theory, and spectral geometry. The event aims to bring together leading experts and early-career researchers from around the world to discuss the latest advancements, exchange ideas, and explore major challenges and open problems in the field. 

Photo: Wikimedia Commons

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Venue: Sabhal Mor Ostaig, the Isle of Skye

Geometric Moduli Spaces – Rigidity, Genericity, Stability

The workshop’s goal is to explore connections across mathematical communities that study different geometric moduli spaces. Our workshop will bring together researchers from positive scalar curvature geometry, minimal surface theory, harmonic mapping theory, and special geometries. The study of moduli spaces naturally draws on a confluence of ideas from neighbouring fields, and in each of these areas there is a tradition of exploiting rigidity, genericity, and stability phenomena.
 
Despite many parallel developments, there has been surprisingly limited interaction and cross-collaboration across these communities. This workshop aims to foster new connections and to elucidate future research directions.

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Venue: ICMS Edinburgh

Arithmetic, Geometry, Space and Time: a Workshop on the Occasion of Minhyong Kim’s 61st Birthday

The aim of this workshop is to survey recent developments in reciprocity laws in arithmetic algebraic geometry, and in the nascent field of arithmetic quantum field theory. The event is also in celebration of the work of Minhyong Kim, who has made pioneering contributions to both subjects. This presents a chance to bring together world class researchers working on reciprocity laws in Diophantine geometry, p-adic Hodge theory, anabelian geometry, Iwasawa theory and the arithmetic and geometric Langlands programmes.

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Venue: ICMS Edinburgh

The William Rowan Hamilton Geometry and Topology Workshop: Celebrating Martin Bridson’s 60th Birthday

Geometric Group Theory began in the 1980s and 90s, formed from a rich synthesis of ideas from areas including geometry, topology, algebra, analysis and dynamics. One of its central themes is non-positive curvature – a fundamental concept in the geometry of manifolds for at least 200 years – and how this can manifest in a vast range of geometric spaces and their groups of symmetries. This was the topic of Martin Bridson’s highly influential book with André Haefliger, Metric Spaces of Non-Positive Curvature. Geometric Group Theory has led to spectacular advances in the study of 3-manifolds, algorithmic properties of groups, automorphisms of groups, and many other areas.

The workshop will be a major event in the field, presenting an unrivaled opportunity for researchers around the globe to interact and learn about some of the area’s best cutting-edge research through the workshop’s program of talks.

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Venue: Trinity College Dublin

Women in Numbers Europe 5

This workshop aims to foster and support new collaborations for women in number theory. Prior to the conference, the project leaders will design projects based on open research problems in number theory and provide background reading and references for their groups. Each participant will be assigned to a working group according to their research interests and background. The workshop will begin with presentations of projects and end with presentations of initial findings of each project by the youngest member of each group.

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Venue: University of Split

UK Workshop on Spectral Theory

Spectral Theory in the UK has a longstanding history, encompassing the influential works of scientists such as Titchmarsh, Born and Dirac. It naturally sits at the crossroads of numerous branches of mathematics, from the very pure to the more applied side of the subject, including but not limited to operator theory, analysis of PDEs, harmonic analysis, differential geometry, numerical analysis, number theory and mathematical physics.

Research in Spectral Theory is currently expanding rapidly on multiple fronts and experiencing worldwide growth. By bringing together leading experts in the field from all around the world, this workshop will provide an effective environment for exchanging ideas, developing new collaborations, and inspiring future generations in the field, building upon the rich and diverse research landscape in Spectral Theory in the UK.

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Venue: ICMS Edinburgh

Self-interacting Process

This workshop will focus on the analysis of processes which interact locally with their own past. These self-interacting processes arise in a variety of models used in the physical sciences, e.g., describing experimentally observed behaviours in biology (ants preferentially following previous pheromone trails), or physical processes (polymers cannot intersect themselves). They can also appear as mathematical objects, for instance as a description of the effective interaction of a stochastic process in an annealed random environment, or as representations of correlations in random fields (random walk representations of spin systems).  We will concentrate both on specific memory or learning mechanisms that arise in concrete problems, as well as on general theory and methods for studying the effects of that memory.

The areas of interest of the workshop include, but are not limited to, (i) random walks arising in the study of statistical mechanics, both quantum and classical, (ii) stochastic processes in random environments, and (iii) the effects of long-range correlations.

This workshop will be held during the extended programme, Stochastic Systems for Anomalous Diffusion. Professor Gordon Slade will deliver the Clay Lecture

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Venue: Isaac Newton Institute

Structure and Randomness – a celebration of the mathematics of Timothy Gowers

This workshop is a celebration of the mathematics of Timothy Gowers, on the occasion of his 60th birthday. There will be speakers from across the spectrum of Gowers’s mathematical interests, from Banach Space Theory to Combinatorics and Additive Number Theory, Theoretical Computer Science and Pedagogy. 

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Venue: Isaac Newton Institute

K-stability and Moment Maps

The goal of this workshop is to bring together researchers in complex algebraic and differential geometry, working on notions of stability for algebraic varieties such as K-stability (and in particular the construction of moduli spaces of algebraic varieties), the existence of canonical metrics in Kähler geometry and the many nearby topics that interact with these areas. The relationship between K-stability and the existence of canonical Kähler metrics goes by the name of the Yau-Tian-Donaldson conjecture, and is motivated by a more classical correspondence between stability in the sense of geometric invariant theory and properties of moment maps as used in symplectic geometry—two of the overarching themes of the wider programme. This conference will emphasise the deep links between Kähler geometry, stability conditions and various related problems in equivariant algebraic and differential geometry.

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Venue: Isaac Newton Institute

Women in Commutative Algebra II

Commutative algebra is a branch of algebra with deep roots in number theory and algebraic geometry and with exciting connections to algebraic geometry, number theory, topology and combinatorics. In recent years commutative algebra has thrived: four different decades-old conjectures in commutative algebra were resolved, opening up new avenues for investigation and making a collaborative workshop particularly timely.

The major objectives of the workshop are to advance commutative algebra through cutting-edge research; promote the research of women in commutative algebra; facilitate collaborative interaction between junior and senior mathematicians and provide mentoring for early career mathematicians; cultivate a research network among women commutative algebraists; advance the academic careers of women algebraists through research publications and professional connections.

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Venue: CIRM, Trento, Italy

Polyhedral Products: a Path between Homotopy Theory and Geometric Group Theory

The overall aim of the workshop is to generate research opportunities at the intersection of the homotopy theory and geometric group theory associated to polyhedral products. These two areas have substantial common ground but to date have had only limited interaction. The workshop will significantly amplify the existing connections and investigate novel interactions between these two areas. 

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Venue: ICMS, Edinburgh

Women in Operator Algebras

Operator algebras, a major branch of functional analysis, deals mainly with C∗C∗-algebras and von Neumann algebras. These are ∗∗-subalgebras of the algebra of bounded linear operators on a Hilbert space that are closed in an appropriate topology. The field of operator algebras originated with von Neumann in the 1920s as a mathematical framework for modelling phenomena in quantum mechanics. It has since become a fundamental part of modern theoretical physics, as the natural setting for quantum information theory and quantum computing. In addition, operator algebras has connections to several areas of mathematics such as knot theory, dynamical systems, ergodic theory, and foundations.

The main goal of the workshop is to provide an environment for female operator algebraists to collaborate on pioneering research projects. The workshop also aims to develop the network of women in operator algebras and foster collaboration, support, and mentoring between its members. The participants will work in small groups of 4–6 members each led by one or two leaders.

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Venue: Banff International Research Station