Conférences - Géométrie et topologie - Canada

Birational geometry is a classical mathematic subject dating back to the late 1800s. It aims to classify geometric shapes by looking at the majority of their points and asks "if I remove some points from shape A and some other points from shape B, do they become the same shape?". This turns out to be a surprisingly difficult and fundamental mathematical question. Perhaps more surprising is that the classification of geometric shapes from birational geometry is related to the classification of abstract mathematical gadgets called derived categories, a formal language developed by prominent mathematicians in France in the 1960s. However, the relationship between derived categories and birational geometry remains unproven.

The possibility of using quantum mechanics to build computers that are both faster and more secure is an exciting technological leap. One of the most promising pathways to a robust architecture, referred to as topological quantum computing, has lead to fascinating research at the frontier of mathematics and physics. Properties of new, exotic states of matter can be understood - and hence implemented in practice - using tools that were developed by pure mathematicians in connection to surfaces and their curvature. This workshop is bringing together international experts to explore the intimate connections between such topological properties of quantum matter and the key resource of quantum computing: entanglement.

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

This workshop will have a special focus on control theory and control systems. Control theory deals with the problem of shaping the behavior of ``systems,'' where a system can be an engineered object, such as a robot or a train, or a natural one, such as a flock of birds or even the brain. The basic driving questions are how to actuate, and how to design, the system to get it to perform as desired. Due to the wide variety of physical systems that control theory handles, the use of more abstract mathematical tools is very well suited to this field, and has already proven useful many instances, such as the study of system stability or optimal design of control system.

Systèmes, contrôle et automatisation,    Mathématiques appliquées (en général)

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

The workshop will bring together some of the world's top experts and promising young researchers working at the interface of Stochastics and Geometry, broadly understood. Through expository talks and research presentations, the workshop will give a vibrant overview of the most recent developments in the interaction between analysis, geometry and stochastic analysis. The focus of the workshop will be different approaches to random processes in geometric settings, with emphasis on problems on heat kernel analysis on finite- and infinite-dimensional manifolds, sometimes equipped with degenerate geometries, as well as random matrices and non-commutative probability and geometric rough paths. Another objective of the workshop is to identify fundamental open problems in this area that are motivated by mathematical physics and other potential applications. We plan to maintain geographic diversity and also to invite some of the most promising junior mathematicians.

Cours et évènements pour étudiants en mathématiques,    Statistique, Théorie des jeux

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

Rich mathematical structures appear in supersymmetric quantum field theories. These provide new perspectives on deep mathematics while at the same time allow for a better understanding of the quantum field theories. This workshop brings together experts from different areas in mathematics and theoretical physics with the aim of deepening our understanding and to foster new interdisciplinary collaborations.

Cours et évènements pour étudiants en mathématiques,    Physique des hautes énergies, des particules et des champs

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

The aim of this workshop is to bring together experts on many developing perspectives on knot homology (physical, geometric and algebraic) to draw connections between them, and to explore applications.

Cours et évènements pour étudiants en mathématiques,    Algèbre

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

In arithmetic geometry, one studies solutions to polynomial equations defined with arithmetically interesting coefficients, such as integers or rational numbers. One way to study such objects, which has seen tremendous success in the last several decades, is by investigating their symmetries. Quite surprisingly, in several interesting situations, many of the geometric and arithmetic properties of the objects in question are actually controlled by the object’s symmetries.

Cours et évènements pour étudiants en mathématiques,    Théorie des groupes

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

There is an age-old relationship between arithmetic and geometry, going back at least to Euclid's Elements. Historically, it has usually been geometry that has been used to enrich our understanding of arithmetic, but the purpose of this workshop is to study the flow of information in the other direction. Namely, how can arithmetic enhance our understanding of geometry? This meeting will bring together researchers from both sides of the partnership, to explore ways to bind the two fields ever closer together.

Cours et évènements pour étudiants en mathématiques,    Théorie des nombres, arithmétique

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

One focus of modern number theory is to study symmetries of numbers that are roots of polynomial equations. Collections of such symmetries are called Galois groups, and they often encode interesting arithmetical information. The theory of Galois representations provides a way to understand these Galois groups and in particular, how they interact with other areas of mathematics. This workshop will investigate how these Galois representations can be put together into families, and search for new arithmetic applications of these families.

Cours et évènements pour étudiants en mathématiques,    Théorie des nombres, arithmétique

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

The focus of this meeting is the connections between lattices and number theory and geometry. Number theory, one of the oldest branches of pure mathematics, is devoted to the study properties of the integers and more sophisticated number systems. Lattices and number theory have many deep connections. For instance using number theory it was recently demonstrated that certain packings of balls in high dimensions are optimally efficient. Lattices also appear naturally when one studies certain spaces that play an important role in number theory; one of the main focuses of this meeting is to investigate computational and theoretical methods to understand such spaces and to expand the frontier of our algorithmic knowledge in working with them.

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

The cohomology of arithmetic groups is the study of the properties of ``holes'' in geometric spaces that contain information about number theory. The workshop will bring together mathematicians with expertise in number theory, topology, and geometric group theory to tackle these problems and explore recent developments.

Théorie des groupes,    Théorie des nombres, arithmétique