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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?

Géométrie et topologie,    Statistique, Théorie des jeux

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

How hard is it to figure out if there is a way to make a set of logical statements true at the same time by choosing appropriate truth values for their variables? This satisfiability problem is of immense importance both theoretically and practically, and lies right at the heart of mathematics and computer science. On the one hand, today so-called Boolean satisfiability (SAT) solvers are routinely and successfully used to solve large-scale real-world formulas in a wide range of application areas (such as hardware and software verification, electronic design automation, artificial intelligence research, cryptography, bioinformatics, operations research, and sometimes even pure mathematics). On the other hand, this problem is believed to be intractable in general --- though proving that this is so is so is one of the famous million dollar Clay Millennium Problems (the P vs. NP problem) --- and there are tiny formulas for which even the very best SAT solvers today fail miserably.

Boolean satisfiability (SAT)

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.

Algèbre,    Géométrie et topologie

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)

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.

Théorie des nombres, arithmétique,    Statistique, Théorie des jeux

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

This workshop concerns cutting edge research in combinatorics and graph theory, an area central to understanding structures which appear in modern technology, such as the internet and communication devices. Problems that were previously intractable can now be approached due to advances in computing. This workshop will feature 4-6 working groups on research on topics including graph matchings and algorithms, combinatorial coding theory, graph invariants, and algebraic graph theory. Research results from the workshop will be incorporated into Sage, a free open-source computer algebra system, and Python.

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.

Cours et évènements pour étudiants en mathématiques,    Algèbre

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.