Meetings/Workshops on Graph Theory and Combinatorics in Canada

Dynamics is the study of systems that evolve over time. Combinatorics is the study of discrete structures such as binary strings, well-formed expressions in simple languages, data trees, colorings of graphs, etc. In Dynamical Algebraic Combinatorics, we study key parameters associated with these systems as they evolve, which helps us make sense of the underlying dynamics.

Two groups of mathematicians, who specialize either in topological combinatorics or in combinatorial commutative algebra, are organizing a workshop in order to identify and to work on problems in the intersection of both fields. Topological combinatorics uses tools from topology to study discrete structures, while combinatorial commutative algebra also studies discrete structures but uses algebraic tools. As a result, the two groups of mathematicians may be studying the exact same problem about discrete structures, but using completely different tools and mathematical language. As an example, questions about the enumeration of the faces of a topological object called a Cohen--Macaulay flag complex can be translated into questions about Hilbert functions of ideals. The goal of this workshop is to bring together 21 mathematicians in both fields to facilitate discussions and encourage new collaborations between these two groups. The organizers have identified five possible themes that will bring an additional focus to the direction of the workshop.

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

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.

Courses and Events for Math Students and Early Career Researchers,    Information Theory, Foundations of Computer Science

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

The systematic study of random graphs commenced with the seminal work of Erd" os and Renyi in the 1960s. Beyond the classical connections and synergies between random graphs and computer science, over the last few years a new exciting connection between random graphs and statistical inference has been discovered. These connections have led to important new insights on computational problems and statistical inference problems and to new methods for the analysis of random graphs or random discrete structures in general. In addition, since the late 1990s an insightful approach called the cavity method by physicists has been applied to put forward predictions on structural properties of random graphs and their phase transitions. More recently, it has inspired new algorithms for statistical inference problems such as message passing algorithms and novel spectral algorithms. With this workshop we hope to stimulate research and foster new exciting interdisciplinary collaborations between researchers with different backgrounds and expertise in random graphs, computer science, probability, statistics, statistical inference and information theory.

Courses and Events for Math Students and Early Career Researchers,    Probability and Statistics, Game Theory

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

Despite the enormous commercial potential that quantum computing presents, the existence of large-scale quantum computers also has the potential to destroy current security infrastructures. Post-quantum cryptography aims to develop new security protocols that will remain secure even after powerful quantum computers are built. This workshop focuses on isogeny-based cryptography, one of the most promising areas in post-quantum cryptography. In particular, we will examine the security, feasibility and development of new protocols in isogeny-based cryptography, as well as the intricate and beautiful pure mathematics of the related isogeny graphs and elliptic curve endomorphism rings. To address the goals of both training and research, the program will be comprised of keynote speakers and working group sessions.

Courses and Events for Math Students and Early Career Researchers,    Cryptography, Blockchain and Information Security

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

Advances in data collection and social media have led to network and graph data becoming available in many areas, including social sciences, biological sciences, and engineering. Understanding and modeling network structure, as well as conducting rigorous statistical inference to assess uncertainty, can provide crucial insights into the dynamics and interaction mechanisms of the system. Statistical network analysis to date has largely focused on the setting where a single network is observed as a noisy version of some underlying structure of interest. This setting in itself is challenging, requiring adaptation of existing statistical frameworks to networks and bridging the gap between theoretically optimal performance and computational feasibility. The workshop will start by covering recent advances in this setting.

Courses and Events for Math Students and Early Career Researchers,    Probability and Statistics, Game Theory

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

Graphs are the standard mathematical model of many real-world entities: computer networks, road and highway networks, drainage systems, river networks, electrical networks, and so on. In some cases, these highly-complicated graphs are contained in the product of two or more much simpler graphs. In a recent breakthrough, it was shown that this is the case for planar or near-planar graphs like those that model river, road, and highway networks; any such graph is contained in the product of a simple tree-like graph and a path. This product structure gives deep insight into these graphs and their properties, allowing a host of mathematical and algorithmic tools to be applied to these graphs. The goal of this workshop is to continue the search for product structure in more general classes of graphs as well as to find new methods to exploit such product structure mathematically and algorithmically, when it is present.