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

Computations with matrices, such as their multiplication, are fundamental operations that are executed on millions of devices, from smartphones to supercomputers, every day. But what is the minimum number of operations it takes to multiply two matrices? Surprisingly, the answer is currently unknown. The problem of determining the complexity, or the number of operations required to perform a certain computation, goes back to the very origins of computing. In the age of large-scale scientific computing, big data, and machine learning, understanding the complexity of computations is crucial for designing algorithms that are efficient in terms of both time and energy. There are two distinct research communities in computer science and mathematics working to answer these questions. Each community brings different perspectives and different approaches, but they rarely interact with each other to the detriment of both. To truly advance the state-of-the-art and tackle huge computational problems, cooperation and collaboration is needed. This workshop will bring together a diverse group of researchers in theoretical computer science and numerical analysis to shape the future of research on the complexity of matrix computations.

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

This workshop will bring together mathematicians with expertise spanning the theoretical, numerical, discrete and computational geometry and partial differential equations (PDEs) and computer scientists with related interests to identify and discuss some of the most promising areas for advances in the understanding and application in geometric data analysis.

Calculus, Differential Equations and Integration,    Computational Physics and Numerical Simulation

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

Most of the many discrete optimization problems arising in the sciences, engineering, and mathematics are NP-hard, that is, there exist no efficient algorithms to solve them to optimality, assuming the conjecture that P does not equal NP. The area of approximation algorithms focuses on the design and analysis of efficient algorithms that find solutions that are within a guaranteed factor of the optimal one. Loosely speaking, in the context of studying algorithmic problems, an approximation guarantee captures the quality of an algorithm -- for every possible set of input data for the problem, the algorithm finds a solution whose cost is within this factor of the optimal cost. A hardness threshold indicates the difficulty of the algorithmic problem -- no efficient algorithm can achieve an approximation guarantee better than the hardness threshold assuming that P does not equal NP. Over the last two decades, there have been major advances on the design and analysis of approximation algorithms, and on the complementary topic of the hardness of approximation. The goal of the workshop is to focus on a few key topics that could lead to deep new results in the areas of approximation algorithms, combinatorial optimization, hardness of approximation, and proof complexity.

Mathematical Physics,    Computational Physics and Numerical Simulation