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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.

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)

A tensor category is a category with a tensor product that satisfies properties similar to the tensor product of vector spaces. A typical example is given by the representation category of a group with the usual tensor product. While the most important examples of tensor categories arise from representation theory, the theory has outgrown its origins and has emerged into a vast and complex theory on its own, providing a unified language for many phenomena in different fields. Tensor categories are now ubiquitous in areas such as representation theory, invariants of links and 3-manifolds, algebraic geometry, higher category theory, quantum computing and mathematical physics. All in all the abstract theory of tensor categories along with its applications has seen a rapid growth over the last 10 - 20 years.

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)

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.

Graph Theory and Combinatorics,    Geometry and Topology

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

Free noncommutative (nc) function theory is an analogue of classical function theory where one replaces functions between two complex vector spaces by functions between square matrices of all sizes over these vector spaces that respect matrix size, direct sums, and similarities. It has emerged in the last decade and a half as a vibrant research area with results that are sometimes similar to and sometimes strikingly different from the classical commutative setting, and with interconnections and applications ranging from free rings and free skew fields to free probability and random matrix theory to dimension independent linear matrix inequalities in systems and control. The purpose of this research in teams is to develop new relations between free probability and nc function theory by launching a systematic investigation and usage of asymptotic integration over classical compact matrix groups and their homogeneous spaces. We expect this to open the road for an asymptotic study of Hardy and Bergmann spaces of nc functions and exhibit completely new phenomena regarding Shilov boundaries and uniqueness sets in the nc setting.

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

This workshop aims at bringing together an interdisciplinary scientific community to share some of the state-of-the-art results in the theory of Herglotz-Nevanlinna functions, and to solve common open problems lying at the frontier of pure mathematics, applied mathematics, physics, and engineering. Further development of this theory would lead to significant progress not only in the area of pure mathematics, but also in its applications to antennas, metamaterials, and composite materials.

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

Higher-rank graphs generalize path categories of directed graphs and are intricately connected to various mathematical disciplines. This workshop aims to illuminate the rich interplay between higher-rank graph C*-algebras and fields such as algebra, theoretical physics, and geometry. By examining how these algebras both influence and are influenced by these diverse areas, the event will provide insights into their broader implications and applications. The workshop will bring together experts from different fields and career stages, each offering a unique perspective on higher-rank graphs. and interactions.

Courses and Events for Math Students and Early Career Researchers,    Graph Theory and Combinatorics