Graduate Student Seminar

Seminars in this series are for students to talk about something of  personal interest, or a professor to describe and explore their area of research.

The talks are at a very accessible level.

Free pizzas are a regular part  of this seminar series.

Further information on seminars can be found at the following link: http://www.math.toronto.edu/mgsa/seminar

Details

Kitaev's quantum double model for topological quantum computation -- representations, higher categories and extended TQFTs

by Lucy Zhang | University of Toronto
Time: 13:10  (Wednesday, Apr. 06, 2011)
Location: BA6183, Bahen Center, 40 St. George St.
Abstract:
I will introduce one of the major mathematical models of 'topological quantum computation', the Kitaev's quantum double model. As will be argued by Alexander Kirillov on Thursday, this family of models defined on planar graphs is related to another family, called the Levin-Wen models. Both families can be viewd as extended 3-2-1(-0) TQFTs (top dimension is 3 because of 2 spatial + 1 temporal dimensions) arising from the Turaev-Viro construction on the tensor category labeling the underlying 'string-nets'; alternatively, the same extended TQFTs may also be obtained by applying the Reshetikhin-Turaev construction on the tensor category of 'excitations'. Without further ado, I will dive into the Kiteav's description of these models, where the representation theory of quantum doubles (of certain Hopf algebras) plays an important role. A cool thing about these models for topological quantum computation is that (if implemented,) they provide physical realizations of recently invented/popularized mathematical notions of extended TQFTs and higher categories. We can even apply the trendy concept of categorification in this context and produce extended TQFTs for higher dimensional manifolds (e.g. a 4-3-2-1-0 theory might describe our space-time manifold). This talk should serve as a good preparation and complementation for Alexander Kirillov's talk on Thursday.

Dates in this series

· Thursday, Jan. 21, 2010: The Calderon Problem - From the Past to the Present (Leo Tzou)
· Wednesday, Feb. 03, 2010: Indecomposable characters on the group S(2) (Artem Dudko)
· Wednesday, Feb. 10, 2010: Self-referencing Recursions and Labeled Binary Trees (Abraham Isgur)
· Wednesday, Feb. 17, 2010: Introduction to the Well-posedness Theory of (Dispersive) PDEs (Hiro Oh)
· Wednesday, Mar. 03, 2010: Renormalization and universality in one-dimensional dynamics: critical circle maps (Michael Yampolsky)
· Wednesday, Mar. 10, 2010: The Dimer Model on a Planar Domain: From random domino tilings to limit shapes (Alex Bloemendal)
· Wednesday, Mar. 17, 2010: The Planar Dimer Model, continued (Alex Bloemendal)
· Thursday, Sep. 16, 2010: The calculus of Euler characteristic and some of its application (Dmitry Kerner)
· Thursday, Oct. 07, 2010: Equivalences of the Axiom of Choice (Mike Pawliuk)
· Thursday, Oct. 21, 2010: Covering arrays with row limit four (Nevena Francetic)
· Thursday, Oct. 28, 2010: Which groups are Lie groups? (Yevgeniy Likuomovich)
· Thursday, Nov. 04, 2010: All started with Weyl (Victor Ivrii)
· Thursday, Nov. 11, 2010: Sparse structures in Euclidean space (Dominic Dotterrer)
· Thursday, Nov. 18, 2010: A short introduction to curve singularities (Dmitry Kerner)
· Thursday, Nov. 25, 2010: Newton diagrams and applications (Dmitry Kerner)
· Thursday, Dec. 02, 2010: Convergence and smoothing properties of Steiner symmetrizations and other simple rearrangements (Almut Burchard)
· Tuesday, Jan. 11, 2011: TeXFreak (Victor Ivrii)
· Wednesday, Jan. 26, 2011: Two-sided chain condition in Leavitt path algebras (Pinar Colak)
· Tuesday, Feb. 01, 2011: SPECIAL GRADUATE SEMINAR -- How to classify algebraic varieties (Chenyang Xu)
· Wednesday, Feb. 02, 2011: Enrico Bombieri and the Prime Number Theorem (John Friedlander)
· Wednesday, Feb. 16, 2011: Basic Financial Derivatives and the Black-Scholes PDE (Ryan Donnelly)
· Wednesday, Mar. 09, 2011: Classical mechanics and symplectic geometry (Jonathan Fisher)
· Thursday, Mar. 17, 2011: On Computability of Julia sets (Artem Dudko)
· Wednesday, Mar. 23, 2011: Variational optimization in economics (Robert McCann)
· Wednesday, Mar. 30, 2011: Exact formulas in random growth and directed polymers (Jeremy Quastel)
· Wednesday, Apr. 06, 2011: Kitaev's quantum double model for topological quantum computation -- representations, higher categories and extended TQFTs (Lucy Zhang)
· Wednesday, Apr. 13, 2011: Tropicalization and Connection to Secondary Fans (Jihyeon Jessie Yang)
· Thursday, Sep. 29, 2011: Probabilistic Cauchy Theory and Invariant Measures for Hamiltonian PDEs (Geordie Richards)
· Thursday, Oct. 13, 2011: The cohomology ring of the GKM graph of a flag manifold (Yukiko Fukukawa)
· Friday, Nov. 04, 2011: Representations of the full group of a Bratteli diagram (Artem Dudko)
· Thursday, Nov. 10, 2011: How to Solve a Cubic Equation? (Askold Khovanskii)
· Thursday, Nov. 24, 2011: Rate of escape of random walks on groups (Balint Virag)
· Thursday, Dec. 01, 2011: Some aspects of geometric measure theory (Robert Jerrard)
· Thursday, Jan. 19, 2012: Bounded generation, congruence subgroups and the splitting of primes (Kumar Murty)
· Thursday, Jan. 26, 2012: How to solve a quintic polynomial? (Yuri Burda)
· Thursday, Feb. 02, 2012: Dynamics on the circle (Elio Mazzeo)
· Thursday, Feb. 09, 2012: How to keep a can from sliding down an incline (and make it roll instead) (David Li-Bland)
· Thursday, Mar. 15, 2012: Arithmetic graphs and the sum-product phenomenon (Brandon Hanson)
· Thursday, Mar. 22, 2012: Two Inverse Problems in Medical Imaging (Nicholas Hoell)
· Thursday, Mar. 29, 2012: Enveloping algebras of nonassociative structures (Marina Tvalavadze)
· Thursday, Nov. 08, 2012: Airy's rainbow and Stokes' mist: asymptotics through the prism of geometry (Brent Pym)
· Thursday, Dec. 13, 2012: C*-algebras and the Classification Program (Danny Hay)
· Thursday, Jan. 17, 2013: Why is the Riemann Hypothesis a big deal? (Asif Zaman)