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Mathematical Sciences Research Institute

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Program

Higher Categories and Categorification January 21, 2020 to May 29, 2020
Organizers David Ayala (Montana State University), Clark Barwick (Massachusetts Institute of Technology), David Nadler (University of California, Berkeley), LEAD Emily Riehl (Johns Hopkins University), Marcy Robertson (University of Melbourne), Peter Teichner (Max-Planck-Institut für Mathematik), Dominic Verity (Macquarie University)
Description
Higher adjunction axiom
swallowtail identity
Though many of the ideas in higher category theory find their origins in homotopy theory — for instance as expressed by Grothendieck’s “homotopy hypothesis” — the subject today interacts with a broad spectrum of areas of mathematical research. Unforeseen descent, or local-to-global formulas, for familiar objects can be articulated in terms of higher invertible morphisms. Compatible associative deformations of a sequence of maps of spaces, or derived schemes, can putatively be represented by higher categories, as Koszul duality for E_n-algebras suggests. Higher categories offer unforeseen characterizing universal properties for familiar constructions such as K-theory. Manifold theory is natively connected to higher category theory and adjunction data, a connection that is most famously articulated by the recently proven Cobordism Hypothesis. In parallel, the idea of "categorification'' is playing an increasing role in algebraic geometry, representation theory, mathematical physics, and manifold theory, and higher categorical structures also appear in the very foundations of mathematics in the form of univalent foundations and homotopy type theory. A central mission of this semester will be to mitigate the exorbitantly high "cost of admission'' for mathematicians in other areas of research who aim to apply higher categorical technology and to create opportunities for potent collaborations between mathematicians from these different fields and experts from within higher category theory.
Keywords and Mathematics Subject Classification (MSC)
Primary Mathematics Subject Classification No Primary AMS MSC
Secondary Mathematics Subject Classification No Secondary AMS MSC
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Programmatic Workshops
February 06, 2020 - February 07, 2020 Connections for Women: Higher Categories and Categorification
February 10, 2020 - February 14, 2020 Introductory Workshop: Higher Categories and Categorification
March 23, 2020 - March 27, 2020 (∞, n)-categories,factorization homology, and algebraic K-theory