Courses
Course 1
Tensor categoriesVictor Ostrik
University of Oregon
Abstract: This series of talks will be devoted to the theory of tensor categories which is a mathematical side of quantum symmetries. We will discuss basic definitions, examples and constructions of such categories, as well as some structure theory and classification results.
Course 2
Hopf Algebras and Their Generalizations from a Categorical Point of ViewGabriella Böhm
Wigner Research Centre for Physics of the Hungarian Academy of Sciences
Abstract: Folklore says that (Hopf) bialgebras are distinguished algebras whose representation category admits a (closed) monoidal structure. In the planned course we give a precise mathematical meaning to this folklore and show how the definitions of (Hopf) bialgebras and their various generalizations can be derived from this principle. The algebraic structures discussed in this uniform framework will include classical (Hopf) bialgebras over fields and, more generally, in braided monoidal categories; (Hopf) bialgebroids over an arbitrary base algebra, as well as their particular instances whose base algebra possesses a separable Frobenius structure, known as weak (Hopf) bialgebras; and (Hopf) bimonoids in so-called duoidal categories.
Course 3
On finite-dimensional Hopf algebras and their representationsSiu-Hung Ng
Louisiana State University
Abstract: Hopf algebras are generalizations of group algebras. However, finite-dimensional Hopf algebras over C are not necessarily semisimple and there could be infinitely many isomorphism classes for a given dimension. In these lectures, we will talk about some basic theorems on finite-dimensional Hopf algebras, which include the uniqueness of integrals, Radford formula of the fourth power of antipodes, and the Nichols-Zoeller Theorem. Applications of these theorems on the classification of Hopf algebras of small dimensions and some invariants of the tensor categories of their representations will be discussed.
Course 4
The Mathematics of Topological Quantum ComputingEric Rowell
Texas A&M University
Abstract: In topological quantum computing, information is encoded in "knotted" quantum states of topological phases of matter, thus being locked into topology to prevent decay. Topological precision has been confirmed in quantum Hall liquids by experiments to an accuracy of 10^(-10) and harnessed to stabilize quantum memory. In these lectures, we will discuss the conceptual development of this interdisciplinary field at the juncture of mathematics, physics and computer science. Our focus is on computing and physical motivations, basic mathematical notions and results, open problems and future directions related to and/or inspired by topological quantum computing.
Course 5
Topological Quantum Field TheoryNoah Snyder
Indiana University, Bloomington
Abstract: Topological quantum field theories are algebraic invariants of n-manifolds which can be computed via cutting-and-gluing. This can be formalized a symmetric monoidal functor of higher categories from a bordism category to an algebraic category. First we will introduce ordinary TQFTs and their classification in 1 and 2 dimensions. Next we will discuss extended TFTs and discuss 210 and 321 field theories and their relationship to modular tensor categories and Reshetikhin-Turaev field theories following Bartlett--Douglas--Schommer-Pries--Vicary. Finally, we will discuss the Baez--Dolan--Hopkins--Lurie cobordism hypothesis and its relationship to Turaev--Viro 3-2-1-0 topological field theories in joint work with Douglas--Schommer-Pries.
Course 6
Subfactors, fusion categories, and planar algebrasScott Morrison
Australian National University
Abstract: We'll learn about algebra objects in tensor categories, and in particular the equivalences between three different worlds:
1. (finite depth) subfactor standard invariants,
2. algebras in (unitary) fusion categories, and
3. (unitary) planar algebras.
Using these, we'll study how combinatorial, algebraic, and topological techniques can be used to classify and understand examples.