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  1. Program Complementary Program 2019-20

    The Complementary Program has a limited number of memberships that are open to mathematicians whose interests are not closely related to the core programs; special consideration is given to mathematicians who are partners of an invited member of a core program. 

    Updated on Nov 27, 2018 12:28 PM PST
  2. Program Quantum Symmetries

    Organizers: Vaughan Jones (Vanderbilt University), LEAD Scott Morrison (Australian National University), Victor Ostrik (University of Oregon), Emily Peters (Loyola University), Eric Rowell (Texas A & M University), LEAD Noah Snyder (Indiana University), Chelsea Walton (University of Illinois at Urbana-Champaign)
    Program picture
    The study of tensor categories involves the interplay of representation theory, combinatorics, number theory, and low dimensional topology (from a string diagram calculation, describing the 3-dimensional bordism 2-category [arXiv:1411.0945]).

    Symmetry, as formalized by group theory, is ubiquitous across mathematics and science. Classical examples include point groups in crystallography, Noether's theorem relating differentiable symmetries and conserved quantities, and the classification of fundamental particles according to irreducible representations of the Poincaré group and the internal symmetry groups of the standard model. However, in some quantum settings, the notion of a group is no longer enough to capture all symmetries. Important motivating examples include Galois-like symmetries of von Neumann algebras, anyonic particles in condensed matter physics, and deformations of universal enveloping algebras. The language of tensor categories provides a unified framework to discuss these notions of quantum symmetry.

    Updated on Jan 14, 2020 02:21 PM PST
  3. Program Higher Categories and Categorification

    Organizers: David Ayala (Montana State University), Clark Barwick (University of Edinburgh), 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)
    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.

    Updated on Jan 10, 2020 03:55 PM PST
  4. Workshop Introductory Workshop: Quantum Symmetries

    Organizers: Vaughan Jones (Vanderbilt University), Victor Ostrik (University of Oregon), Emily Peters (Loyola University), LEAD Noah Snyder (Indiana University)
    Jellyfish floating to the surface, as in the evaluation algorithm for certain planar algebras.

    This workshop will consist of introductory minicourses on key topics in Quantum Symmetry: fusion categories, modular tensor categories, Hopf algebras, subfactors and planar algebras, topological field theories, conformal nets, and topological phases of matter.  These minicourses will be introductory and are aimed at giving semester participants exposure to the main ideas of subfields other than their own.

    Updated on Apr 09, 2018 02:20 PM PDT