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Banff International Research Station (BIRS) / UBCO

The workshop will tackle four of the main directions in which Higgs bundles have recently taken a leading role within the mathematics and physics communities, following the breakthroughs of the last years. Bringing together leaders in the area both in mathematics and physics, as well as young researchers in the subfields of the workshop, the meeting will present the ideal setting for novel discoveries and understanding of the uses that Higgs bundles and Hitchin systems may have.

Banff International Research Station (BIRS) / UBCO

This workshop will study this question for systems of geometric origin. One example of this is a particle bouncing inside a region -- think of a ball bouncing around a billiard table, or a light beam bouncing around a room whose walls are mirrors. If the walls are flat then the behavior is fairly predictable, but if they are curved then one often observes random-looking behavior in the long run. Or imagine walking in a straight line along a surface; if the surface is flat then changing the starting point a little will lead to a predictable change in the trajectory. However, if the surface is curved like a saddle then different trajectories will spread out and your location after walking for a long time will eventually seem to be random. The task of giving precise descriptions of systems such as these requires a more complete development of the theory of thermodynamic formalism for systems of geometric origin, which is precisely the goal of this workshop.

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.

Computational Physics and Numerical Simulation,    Algorithms and Data Structures

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

The workshop is conceived in response to recent breakthroughs in physics and mathematics. One is the real quantization of Hitchin moduli spaces due to Gaiotto and Witten, and the other is a deformation quantization theory of Kontsevich and Soibelman that has achieved a vast mathematical foundation and generalization of topological recursion of Eynard and Orantin in random matrix theory. The vision of the organizers is that these two theories should be related through mirror symmetry. The workshop is organized to test this vision, and to explore its mathematical consequences.

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

Motivated by theoretical work that pointed to new opportunities, tremendous progress has been made over the past several years in using moiré superlattices formed from two-dimensional materials as a laboratory for the study of quantum materials. This development has opened up an exciting new scientific opening, one in which mathematics and theoretical physics have a very large role to play by identifying moiré systems that are likely to exhibit new or poorly understood electronic phenomena, by inventing mathematical methods that enable testable predictions of physical properties, and by unravelling the meaning of experimental observations.