The Simons Foundation is pleased to announce the establishment of the Simons Collaboration on Perfection in Algebra, Geometry and Topology, directed by Martin Olsson of the University of California, Berkeley.
The connection between geometry, algebra and arithmetic is at the heart of many parts of mathematics, such as arithmetic and algebraic geometry, number theory and representation theory. This interplay has been critical in some of the greatest achievements in these fields and has informed the development of key tools, including arithmetic cohomology theories and guiding visions in the Langlands program and the theory of motives. The last decade has seen remarkable new ideas and techniques emerging in mixed characteristic geometry, such as perfectoid spaces, prismatic cohomology and systematic use of derived algebraic geometry. The key new notions, unified by the theme of perfection in a very broad sense, have revolutionized a number of algebraic fields.
These new tools have strengthened existing connections between fields and established new opportunities. In particular, perfection techniques have shed new light on classical parts of number theory and arithmetic geometry, but also created new links with commutative algebra and topology. This collaboration brings together experts in a wide variety of algebraic disciplines to explore these new structures and their applications to major open problems across algebra, geometry, topology and number theory.