This is an INI satellite programme is based at Gaelic College, Sabhal Mor Ostain, Isle of Skye.
Algebraic topology and homotopy theory have witnessed an impressive renaissance involving applications to geometric topology and number theory. The development of new technologies was central to the solution of long-standing problems in homotopy theory, for example, in the solution of the Kervaire invariant problem, which apart from solving a long standing conjecture, also had an impact on developments in equivariant stable homotopy theory, for example, the higher dimensional version of the Madsen–Weiss theorem. The interplay between key classes of groups arising in geometric and topological settings has proven to be extremely fruitful, leading on the geometric side to powerful results and solutions to long standing problems. In homotopy theory, work on fusion systems revived the interest in fusion systems in representation theory, while creating a new and active subject within homotopy theory and related subjects. At the same time, breakthroughs in manifold topology include the computations of homotopy types of cobordism categories, the immense success of gauge theory and ideas from physics in manifold topology, and the development of quantum invariants in the context of representation theory and quantum groups. Many of these developments are powered by the rapid developments in higher structures. Originally developed in their foundational aspects within higher category theory, they now find applications and are a unifying language for many diverse areas: from homotopy theory (higher groupoids, infinity categories) to mathematical physics (higher cobordism categories), algebraic geometry (higher stacks), logic and computer science (homotopy type theory).
The planned programme will bring together established researchers working in algebraic, combinatorial and geometric topology, as well as modern homotopy theory on the one hand and in representation theory of groups and algebras and category theory and higher structures on the other hand. The activities will be divided into four weeks. The first two weeks are designed for researchers at various career levels and expertise to interact, exchange ideas and knowledge. The third week will be a workshop with instructional mini-courses designed specifically for early career researchers. The activities will culminate in the fourth week with an international conference. The programme will take places in the Gaelic College Sabhal Mor Ostain on the Isle of Skye, which is an outstanding venue to host such activities. The location is quite remote but accessible and very attractive. The college offers the convenience of a fully equipped conference facility with plenty of break-room space, on site accommo- dation and dining facilities and, at the same time, a quiet absence of distractions for researchers to discuss future trends in this spectacularly stimulating atmosphere.
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