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The Fields Institute for Research in Mathematical Sciences

Homotopy theory is in the midst of a renaissance as its usefulness in other areas of mathematics is becoming increasingly recognised. From polyhedral products in geometric group theory, to combinatorial methods in group theory, to topological properties of manifolds and spaces of embeddings, to Morava K-theory in symplectic geometry, to simplicial methods in topological data analysis, to a host of applications in mathematical physics - and these name only a few areas of interaction - homotopy theory over the past ten to fifteen years has significantly extended its reach. This program will assemble a wide range of leading experts to discuss the state of the art in current research, both in terms of core fundamental problems and crossover to other areas, and explore exciting new directions.

The Fields Institute for Research in Mathematical Sciences

This intensive graduate school will cover topics central to the Workshop on Interactions between Group Theory and Homotopy Theory the following week. This school is designed to give graduate students and early career researchers in related fields a better understanding of each subject, thus enabling the participants to better engage in the subsequent workshop.

The Fields Institute for Research in Mathematical Sciences

This intensive graduate school will cover topics central to the Workshop on Interactions between Group Theory and Homotopy Theory the following week. This school is designed to give graduate students and early career researchers in related fields a better understanding of each subject, thus enabling the participants to better engage in the subsequent workshop.

Cours et évènements pour étudiants en mathématiques,    Théorie des groupes

The Fields Institute for Research in Mathematical Sciences

Group theory and topology have a long and rich history of interaction. This workshop will bring together experts working on different aspects of this interaction in order to share ideas and boost the development of new methods that can be applied to the topology of configuration spaces, braid groups, mapping class groups and manifolds, and spaces of commuting elements.

Théorie des groupes,    Cours et évènements pour étudiants en mathématiques

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

This workshop will bring together researchers working on two fields seemingly at odds, and incompatible: combinatorial geometry and uncertainty. Combinatorial geometry is a field of counting discrete objects, it is about how things are connected, or in general how discrete objects like points, lines, and circles intersect with each other. On the other hand, modeling uncertainty is about not allowing discrete events to happen precisely, or at least not with confidence. It replaces these notions with probabilistic concepts. How can one count geometric objects and their connectivity properties if there is uncertainty in their very existence?

Théorie des graphes et combinatoire,    Statistique, Théorie des jeux

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

Mathematicians have been studying notions of space and geometry for thousands of years. Over time our models for spaces have grown successively more abstract. A major conceptual leap in the 1960's was taken with the theory of algebraic stacks, which are a sort of space with "internal symmetries'' not visible in usual geometric objects. There has been a flurry of recent activity among mathematical researchers in understanding the structure of these spaces. This workshop will bring together researchers developing this state-of-the-art theory with mathematicians at various career stages who are interested in applying this theory in their work. The organizers hope that this week of focused discussion on this topic will help bring these methods to a wider audience and lead to significant new developments both in the foundations of the theory and in applications.

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

Cluster algebras, webs and canonical bases are three different methods that mathematicians have developed in order the understand the symmetries of certain mathematical structures. Each of these three ways arose from, and has applications to, different areas of mathematics. These areas include algebraic geometry, representation theory, topology, and combinatorics, as well as physics. The goal of this workshop is to bring together experts and up-and-coming researchers from each of these areas in order to better understand the still mysterious connections between cluster algebras, webs, and canonical bases. A better understanding of these connections could allow for ideas from one area to be applied to solve problems in other areas.

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)

Core objects of study in algebraic geometry and in commutative algebra are called varieties and rings, respectively. Varieties are geometric objects that consist of sets of solutions to polynomial equations. Each variety has a corresponding ring consisting of the polynomial functions that are defined on the variety.

Algèbre,    Théorie des graphes et combinatoire

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

The study of geometric properties of convex bodies based on information about sections or projections of these bodies belongs to the area of geometric tomography and has important applications to many areas of mathematics and science, in general. In recent years, there has been a significant increase in the interaction between harmonic analysis and convex geometry, leading to solutions for several longstanding open problems, the discovery of new phenomena, and many new intriguing open questions. The goal of this workshop is to bring together young and established researchers to discuss recent developments and advance applications of harmonic analysis to convex geometry and geometric tomography.

Analyse,    Calcul infinitésimal, équations différentielles et intégrales

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.

Physique mathématique,    Mécanique quantique et technologie quantique

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.

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

The goal is to bring together researchers of different experience levels, from advanced graduate students to established mathematicians, in order to jointly work on research projects on topics of common interest within Toric Topology and Polyhedral Products. The focus of the workshop will be on active collaboration in teams, and while the weeklong workshop will be the most condensed part of the collaboration, it is expected that a significant amount of work on the projects will be done before as well as after the workshop. It is expected that at least one research paper be produced from each collaboration which will be published in a workshop proceedings after obtaining referees' approval.