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

The Fields Institute

This workshop will further explore interactions between set theory, Ramsey theory, and the geometry and dynamics of Polish groups. An emphasis will be placed on the large scale geometry and dynamics of homeomorphism groups, mapping class groups of spaces of infinite type, and of automorphism groups of countable structures. It will also include work on automatic continuity in the context of transformation groups.

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

We aim to unite the study of structure and dynamics by fostering conversations between researchers who use the HHD in areas including economics, game theory, statistical physics, and machine learning. We also aim to promote collaboration between industrial and academic teams using the HHD.

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

The central problem in algebraic geometry is to classify so-called algebraic varieties: geometric shapes cut out by algebraic equations. Algebraic varieties are parametrized by certain moduli spaces (parameter spaces whose points correspond to these different varieties) and the geometry of these moduli spaces encodes the ways of continuously deforming these shapes. Furthermore, classification questions for algebraic varieties often boil down to understanding the geometry of these moduli spaces. In the past few years, powerful new tools have been developed in moduli theory but there are still many open questions, especially in higher dimensions. This workshop will bring together a diverse group of researchers in moduli theory in order to apply these new tools to explicit examples of the classification of higher dimensional algebraic varieties and their moduli spaces and push our understanding of higher dimensional algebraic geometry to new frontiers.

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

The researchers participating in this workshop come from several mathematical subdisciplines, including combinatorics, algebraic geometry, and theoretical computer science. Their work and common interest here is to investigate new interactions between discrete mathematics, algebra, algebraic geometry, tropical geometry, and combinatorial algorithms. Some participants have collaborated before in smaller groups, but this meeting will provide an opportunity to work together more intensively. Several recent mathematical breakthroughs will be highlighted in the workshop, including the recent resolution of some long-standing open problems about graphs and matroids. These breakthroughs used a surprising combination of interdisciplinary techniques which the organizers expect to have exciting further applications.

Théorie des graphes et combinatoire,    Théorie de l'information, fondations de l'informatique

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

The workshop will bring together some of the world's top experts and early career mathematicians 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, often 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 invite many promising early career mathematicians.

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

A shape is "convex" if for any two points in the shape, the line segment connecting them does not pass out of the shape. This fundamental notion is the beginning of a wide array of ideas in geometry and analysis, leading to much more general theories in which "line segment" is replaced by other more flexible objects, or where "convexity" is determined by properties only analogous to the classical ones. Such generalized ideas of convexity are particularly important in the mathematical fields of symplectic topology and complex geometry. While links between the symplectic and complex versions of complexity have been recognized for some time, recent years have begun to see the emergence of an even closer relationship in which techniques and tools from one field can shed light or inspire new directions in the other. This workshop will include specialists in complex analysis, symplectic geometry, contact topology, and smooth manifold theory, with the goal of introducing researchers in these fields to the techniques, ideas, and open problems in the others, united by the theme of convexity in various forms. Students and emerging researchers will be grouped with established scholars to establish new professional connections and research directions, and groups will work together to begin to bring their various tools together toward the solution of major open problems in the field.

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

High-quality randomness is a valuable commodity in computation, experimentation, and discovery. This workshop focuses on randomness as a resource within real and complex algebraic geometry. The program highlights three current research trends: understanding the statistical properties of fundamental geometric invariants, developing randomized algorithms for symbolic and numerical calculations in algebraic geometry, and exploring the synergies between algebraic geometry and data analysis. This workshop will, for the first time, bring together a wide range of mathematicians interested in random algebraic geometry.

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

Soon after Einstein postulated the General Theory of Relativity, he introduced what he called his "greatest mistake'', the so-called cosmological constant. Solutions of the Einstein equation with cosmological constant are now called Einstein manifolds, and despite Einstein's pessimistic view, play key roles both in mathematics and physics. Quasi-Einstein manifolds are generalizations of Einstein manifolds, which include not only Einstein manifolds but Ricci solitons, warped product Einstein manifolds, static Einstein manifolds, and other special manifolds. These are important in many areas of modern mathematical research, including Ricci flow, collapsed Gromov-Hausdorff limits, and general relativity. This workshop will examine progress in seemingly disparate fields which in fact share a common nexus, namely quasi-Einstein manifolds or the related notion of Bakry-Emery Ricci curvature. Problems to be discussed include the classification problem for near horizon geometries (of black holes), for Ricci solitons and for static Einstein metrics, as well as the fundamental gap problem for Laplace and Schroedinger operators, and the formulation and application of synthetic Bakry-Emery Ricci curvature.

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

The influence of networks in our daily life is swiftly increasing. Examples of physical networks are those that involve passenger transfers or simply the connections between different computers to send information. On the other hand, virtually, we find social networks, such as Facebook, Instagram and so on, which are widely used and known by a high percentage of the world's population. Currently, the construction of networks that optimize certain real hypotheses, is a vast area that provides mathematical problems whose answers solve real situations.

The Fields Institute

The study of Lefschetz properties for Artinian algebras was motivated by the Lefschetz theory for projective manifolds, begun by S. Lefschetz, and well established by the late 1950's. Many of the important Artinian graded algebras appear as cohomology rings of an algebraic variety or manifold, though recent important developments have demonstrated important cases of the Lefschetz properties beyond such geometric settings (such as Coxeter groups or matroids). This renews interest in understanding the Lefschetz property, and Artinian algebras that admit them, systematically.

Algèbre,    Théorie des graphes et combinatoire

The Fields Institute

This is the 17th in the sequence of annual workshops traditionally held in the month of May at Nipissing University. Among our participants, we expect to have a number of prominent mathematicians from Canada, USA and abroad. The purpose of these workshops/seminars has been mainly to provide opportunities to researchers to learn a new area as well as opportunity to collaborate. As a result of these workshops there have been numerous collaborations dealing with problems in dimension theory, geometric topology, C*-algebras, set-theoretic topology, continuum theory and complex dynamics.

CRM / Université de Montréal

Understanding the geometry of the finitely generated subgroups of a group informs our understanding of that group significantly. For example, some hyperbolic groups enjoy the property that every finitely generated subgroup is quasi-convex. Groups which are incoherent and groups which are algebraically fibered do not have this property. This conference aims to explore some of the many directions in this field, including quasi-convexity in other contexts.

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

There exists a mysterious algebraic structure in 8-dimensional space that does not behave in the same way as the ordinary arithmetic that we all learn as children. Somewhat surprisingly, this strange algebra, called the octonions, shows up naturally in the study of certain higher dimensional geometric spaces, called manifolds, that are of importance in physics and in the study of the equations introduced by Einstein in general relativity. Geometers have been studying such spaces for a number of years, mostly using tools from calculus and differential equations. The proposed workshops intends to introduce these researchers to experts in the special octonion algebra, as well as in related special algebraic structures called spinors, to facilitate a cross-pollination of ideas, methods, and techniques. Some of these groups have never before been brought together in this way.

CRM / Université de Montréal

This workshop focuses on several themes related to the fundamental groups of 3-manifolds. The themes include: the theory character varieties, Thurston norm polytopes and fibrations, orderability, profinite rigidity, and variations on the Cannon conjecture. The first week of the workshop will feature mini-courses by Ian Biringer (Boston College), Michelle Chu (Minnesota), Nathan Dunfield (Illinois-Urbana), and Daniel Groves (Illinois-Chicago). The second week will be devoted to research talks.

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

This workshop will focus on geometry, topology and their applications to control systems. Control theory deals with the problem of shaping the behavior of "systems", which can mean an engineered object, such as a robotic arm used for remote surgery, or a natural one, such as a flock of birds or even the brain. The basic driving questions are how to control or design a system which performs as desired. Geometric and topological ideas have led to technological breakthroughs throughout the years, but geometric thinking in engineering has not yet permeated the mainstream. The goal of this workshop is to bring together a diverse group of engineers and scientists working in geometry, topology and their intersection, in order to open new frontiers of research and applications.

CRM / Université de Montréal

The mini-courses and research talks will address several themes, including: nonpositively-curved cube complexes, median spaces, (coarsely) Helly spaces and groups, and various combinatorial notions of nonpositive curvature. The first week of the workshop involves mini-courses by Elia Fioravanti (MPI Bonn), Thomas Haettel (Université de Montpellier), Nima Hoda (ENS Paris), Nir Lazarovich (Technion). The second week will be devoted to research talks.

Théorie des groupes,    Théorie des graphes et combinatoire

Banff International Research Station (BIRS) / UBCO

Geometry is all around us. The circles and triangles of the Ancient Greeks have never left us. Film, art, and architecture ha ve all been touched by mathematics and geometry. 3D printing, robotics, and origami are a new wave of geometry. At Geometry: Education, Art, and Research, leading researchers pair up with educations and artists to push the boundaries of contemporary geometry.

Histoire et philosophie de la science,    Conférences publiques et sensibilisation

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

The workshop is concerned with geometric objects arising from polynomial equations known as algebraic curves. Such curves have been studied extensively since the 19th century, but recent developments in combinatorics have led to a plethora of new techniques and solutions to major problems in the field. Bringing together experts from classical algebraic geometry and the adjacent combinatorial areas, the workshop aims to bridge the different points of views, identify key open problems in the field, and to examine how progress can be achieved via a combination of old and new tools.

Calcul infinitésimal, équations différentielles et intégrales,    Théorie des graphes et combinatoire

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

The Banff International Research Station will host a diverse, international group of mathematicians in 2023 to study geometric approaches to fluids and optimal transport. The ideas of geodesic settings for fluids have been circulating since the late 1960s, but received a strong surge lately, in the last 5-10 years, with the development of new techniques in infinite-dimensional geometry and group theory. The original idea is that the particles in an ideal liquid or gas move in a curved infinite-dimensional space, in the same way as an airplane flying around the earth: along the shortest path. Properties of the fluid, such as how well it can be predicted in the future, how well it mixes together, and how extreme the speed can be, can all be related to how curved this infinite-dimensional space is, which helps us understand features of the weather and climate geometrically. More recently similar ideas have been used to understand solids as well, for example when a pile of rocks are transported from one place to another at the optimal cost.

Physique de la matière condensée et des matériaux,    Thermodynamique, dynamique des fluides et physique statistique

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

Many important questions in convex geometry are related to the study of sections and projections of convex bodies. For example, can convex bodies be uniquely determined by the size of their sections or projections? What are the largest and smallest sections of the cube? Do convex bodies with smaller sections have smaller volume? A common feature of all these problems is that they are solved with the help of harmonic analysis. The aim of the proposed workshop is to bring together leading experts and young researchers to discuss recent progress in applications of harmonic analysis to convex geometry.

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

The concept of manifold in mathematics corresponds approximately to the everyday idea of a smooth surface --- think of soap bubbles or sheets of fabric --- except that it can have any number of dimensions. A question that lies at the origin of topology, the mathematics of shape, is: can we write down a list of all manifolds? At first, such a classification was attempted through numerical invariants. Over time, the focus of algebraic topology has shifted from using numerical invariants to the algebraic objects underlying them, and then ascended further to studying moduli spaces. This workshop concerns recent breakthroughs in this philosophy for studying of manifolds, combining higher-algebraic methods with novel geometric techniques.

The Fields Institute

Scheduled as part of Thematic Program on Operator Algebras and Applications