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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?

Geometry and Topology,    Graph Theory and Combinatorics

CRM / Université de Montréal

CRM

Number Theory, Arithmetic,    Courses and Events for Math Students and Early Career Researchers

CRM / Université de Montréal

Summer school “Universal Statistics in Number Theory” (May 4 - 15) will introduce the key topics from both sides of the program, including lecture series by probabilists as well as number theorists.

part of Montreal program 2026: Universal Statistics in number theory

Courses and Events for Math Students and Early Career Researchers,    Number Theory, Arithmetic

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

Particle physics, astrophysics, and cosmology cover the study of the universe from its smallest to largest scales. We study them in order to understand both the fundamental interactions that govern the universe and its large-scale structure and history. Advanced detectors at collider facilities that accelerate particles to near the speed of light and telescopes that monitor the night's sky looking back to the time just after the Big Bang are used to collect vast amounts of data in complicated datasets. Analyzing these data requires modern tools, making use of advanced machine learning and high performance computing infrastructure. This workshop brings together physicists, statisticians, and machine learning experts in order to make the most use of this data and learn as much from it as possible by discussing how to address the complexities of these large and detailed datasets, searching for 1 in a trillion events, and understanding correlations between thousands of quantities.

Mathematical Physics,    Neural Networks and Artificial Intelligence, Machine Learning

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

In recent years some of the most exciting breakthroughs in Combinatorics on longstanding conjectures have resulted from innovative applications of established techniques to areas where they have not necessarily been used before. We would like to harness the power of collaboration and bring together open-minded participants with different areas of expertise to produce novel research in a number of globally studied areas including. We aspire to create new productive long-term bonds between members of the global community.

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

Catastrophic events, even though they happen rarely, have a significant impact when they occur. Disastrous climate, financial, insurance or complex network failure events can have devastating social and environmental consequences. A complete risk analysis for modelling and prediction purposes requires understanding how these extreme, rare events occur, and what are the main drivers causing them. Machine learning methods open the road for methodological developments to forecast these extreme events and discover their complex, possibly high-dimensional nature. The aim of this workshop is to bring together researchers contributing to closely related, but culturally disconnected research communities: extreme value theory and machine learning. The goal is to discuss new directions and open mathematical problems, and foster further collaboration. The leading experts will introduce young researchers, postdocs and graduate students to the state-of-the-art in the field.

Applied Mathematics (in general),    Neural Networks and Artificial Intelligence, Machine Learning

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

The proposed workshop focuses on recent breakthroughs in arithmetic functions, L-functions, and pseudorandomness in Number Theory, as well as new connections to other areas (via both employed methods and applications), such as additive combinatorics, arithmetic statistics, and dynamical systems. Such breakthroughs include the latest advances in the study of various character and exponential sums (e.g., Kloosterman sums), which are powerful and versatile tools for understanding the anatomy of integers; moments, and other averages of L -functions (over number fields and function fields); integers having a prescribed arithmetic structure (e.g., prime, smooth, square-free, sum of two squares), along with their distribution in arithmetic progressions and short intervals, as well as function field analogs of these problems; distribution of elements of multiplicative subgroups; pseudorandomness of arithmetical functions. An essential aspect of these developments is the vast arsenal of tools and techniques that have been conceived, developed, and implemented, ranging from analysis to additive combinatorics, algebraic geometry, and random matrix theory. The proposed workshop will concentrate on the emerging methods that underlie such advances and on possible future directions.

Number Theory, Arithmetic,    Graph Theory and Combinatorics

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

The fields of kernel approximation and Gaussian processes have evolved, mostly independently, over the past few decades to become mature subjects in numerical analysis and statistics/applied probability, respectively. The two fields have natural affinities, both by relying on a common mathematical machinery of reproducing kernel Hilbert spaces and positive definite functions, and by treating a diverse assortment of scientific applications. A hallmark problem for kernels is the modeling of irregularly sampled, deterministic data, but kernels are also heavily employed in the simulation of a range of phenomena from atmospheric flows to financial forecasting.; a key feature of kernel approximation is the ability to deliver solutions to computational problems without requiring costly underlying geometric structures like meshing. Gaussian processes have traditionally focused on regression, classification, and learning of the underlying probability distribution from random samples, although recent activity has focused on successful undertakings in interpretable and explainable machine learning, inverse problems and partial differential equations with stochastic parameters. Despite their different languages and different objects of interest, the fields share a similar collection of important, challenging theoretical and computational problems.

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

Understanding the resilience of ecosystems will be critical for our future considering the multiple crises that we face. Current understanding of resilience relies largely on examining the consequences of disturbances in a deterministic framework. However, disturbances will be arguably stochastic in nature. Disturbances will be multifaceted, often repeated, and probably highly variable over time. Furthermore there is uncertainty and randomness in the processes that respond to the disturbances. Consequently, resilience is most naturally explored in the context of stochasticity: randomness in disturbances must be expected and prepared for. In this workshop, we aim to bring together stochastic and deterministic researchers studying the resilience of ecosystems and the resilience of human economic-ecosystem interactions.

Applied Mathematics (in general),    Ecosystems, Environment and Sustainable Development

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

Evolving random structures are discrete mathematical models that evolve randomly `step-by-step’ over time. Such evolving structures are powerful tools, and the topic has been receiving considerable attention from combinatorics, probability, computer science and related communities. For example, in combinatorics some of the best-known extremal constructions arise from natural evolving random hypergraph models, and in probability many interesting dynamic network models arise from carefully crafted evolving random graph models. Interestingly, in combinatorics and probability slightly different sets of ideas have been proposed for analyzing such random evolving structures, which in turn have also found applications in the analysis of randomized algorithms.