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Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Mathematically, the characterization of localization and delocalization is a rich and delicate problem. Many disciplines, including probability, the analysis of partial differential equations, and mathematical physics, offer tools to tackle localization questions. This workshop will convene researchers from across these fields to share new techniques, explore common questions, and initiate novel lines of research. Moreover, the workshop is designed to nurture a community of early-career researchers who are comfortable communicating and collaborating across disciplines.

Applied Mathematics (in general),    Calculus, Differential Equations and Integration

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Hyperbolic spaces and, in particular, a generalization called asymptotically hyperbolic spaces, have in recent decades been the focus of intense mathematical interest. They have surprising applications in the physics of string theory and are important for the study of isolated systems in the theory of general relativity. The closely related de Sitter and anti-de Sitter spaces also arise in the study of cosmology. The purpose of this workshop is to bring together established experts and emerging new talents in the field who study diverse aspects of these fascinating objects. We will discuss some of the remarkable recent developments in the geometric analysis of these spaces, including developments in inverse problems, the various meanings of mass-energy in this setting, the interaction between the mathematical developments and physics, and new ideas to extend the theory beyond the asymptotically hyperbolic setting.

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Baryonic matter in the Universe is almost exclusively (99,9 \%) in the plasma state, which is one of the four common states (with solid, liquid and gas), having the specificity of being strongly affected by electrical and magnetic fields. Already three generations of plasma physicists have worked at understanding more precisely the dynamics of plasmas, unravelling their secrets and bringing order to what seems by nature chaotic. Still, the plasmas continue to preserve a part of their mystery. First, our project aims to gain more insight into the microscopic reactions induced by the presence of radiation. Secondly, it is to mathematically elucidate how inertial effects coming from the mixing of charged species (like electrons and ions) is able to stabilize the plasma dynamics at very small scales. Incidently, this would bring innovative insight about some specificities of (magnetohydrodynamics).

Applied Mathematics (in general),    Number Theory, Arithmetic

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

The goal of this workshop is to bring together experts from mathematics, quantum field theory, string theory and cosmology to make progress on an ambitious research program to obtain emergent space-time and early universe cosmology from a well defined quantum mechanical matrix model. The aim of the research program is nothing less than to develop a unified quantum theory of space, time and matter based on a well-defined mathematical foundation.

Astronomy, Astrophysics and Cosmology,    High Energy Physics, Particles and Fields

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

Particle physics, astrophysics, and cosmology cover the study of the universe from its smallest to largest scales. We study them in order to understand both the fundamental interactions that govern the universe and its large-scale structure and history. Advanced detectors at collider facilities that accelerate particles to near the speed of light and telescopes that monitor the night's sky looking back to the time just after the Big Bang are used to collect vast amounts of data in complicated datasets. Analyzing these data requires modern tools, making use of advanced machine learning and high performance computing infrastructure. This workshop brings together physicists, statisticians, and machine learning experts in order to make the most use of this data and learn as much from it as possible by discussing how to address the complexities of these large and detailed datasets, searching for 1 in a trillion events, and understanding correlations between thousands of quantities.

Neural Networks and Artificial Intelligence, Machine Learning,    Probability and Statistics, Game Theory

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

A team of three leading experts in algebraic and quantum geometry will convene at the Banff International Research Station (BIRS) to tackle a longstanding conjecture on the classification of vector bundles on the quantum projective line. Their collaboration aims to bridge the gap between classical algebraic geometry and emerging structures in quantum geometry, advancing our understanding of noncommutative spaces. The classification of vector bundles has been a cornerstone of modern algebraic geometry, with the celebrated Grothendieck’s Theorem providing a complete classification of vector bundles over the classical projective line. However, the quantum analog remains an open problem, representing a major challenge in the study of noncommutative algebraic geometry. A positive solution to the conjecture under investigation would significantly advance the field, providing new insights into the structure of quantum projective spaces and their applications in both mathematics and theoretical physics.

Geometry and Topology,    Quantum Mechanics and Quantum Technology

Banff International Research Station for Mathematical Innovation and Discovery (BIRS)

This event will facilitate a broad overview of the state of the art in quantum invariants, including fast computations, universal invariants and their relationships to famous equations arising from quantum physics and algebra, and the interplay between topological and geometric properties of spaces. The participant list features many of the leading world experts in this area, and provides opportunities for young researchers to become involved through an early career showcase, problem sessions, mentoring, and informal interactions.