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Posted January 12, 2018

Last modified January 17, 2018

Christine Lee, University of Texas at Austin

Understanding quantum link invariants via surfaces in 3-manifolds

Abstract: Quantum link invariants lie at the intersection of hyperbolic geometry, 3-dimensional manifolds, quantum physics, and representation theory, where a central goal is to understand its connection to other invariants of links and 3-manifolds. In this talk, we will introduce the colored Jones polynomial, an important example of quantum link invariants. We will discuss how studying properly embedded surfaces in a 3-manifold provides insight into the topological and geometric content of the polynomial. In particular, we will describe how relating the definition of the polynomial to surfaces in the complement of a link shows that it determines boundary slopes and bounds the hyperbolic volume of many links, and we will explore the implication of this approach on these classical invariants.

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Posted January 19, 2018

Last modified January 21, 2018

Shawn X. Cui, Stanford, Institute for Theoretical Physics

Four Dimensional Topological Quantum Field Theories

Abstract: We give an introduction to topological quantum field theories (TQFTs), which have wide applications in low dimensional topology, representation theory, and topological quantum computing. In particular, TQFTs provide invariants of smooth manifolds. We give an explicit construction of a family of four dimensional TQFTs. The input to the construction is a class of tensor categories called $G$-crossed braided fusion categories where $G$ is any finite group. We show that our TQFTs generalize most known examples such as Yetter's TQFT and the Crane-Yetter TQFT. It remains to check if the resulting invariant of 4-manifolds is sensitive to smooth structures. It is expected that the most general four dimensional TQFTs should arise from spherical fusion 2-categories, the proper definition of which has not been universally agreed upon. Indeed, we prove that a $G$-crossed braided fusion category corresponds to a 2-category which does not satisfy the criteria to be a spherical fusion 2-category as defined by Mackaay. Thus the question of what axioms properly define a spherical fusion 2-category is open.

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Posted January 22, 2018

3:30 pm - 4:20 pm Lockett 241
Larry Rolen, Trinity College Dublin & Georgia Tech

tba

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Posted January 21, 2018

3:30 pm - 4:20 pm Lockett 241
Chun-Hung Liu, Princeton University

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Posted December 17, 2017

3:30 pm - 4:20 pm TBD
Habib Ouerdiane, University of Tunis El Manar

TBD

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Posted December 17, 2017

3:30 pm - 4:20 pm TBD
Guozhen Lu, University of Connecticut

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Posted December 26, 2017

3:30 pm - 4:20 pm TBD
Stefan Kolb, Newcastle University

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Posted January 12, 2018

3:30 pm - 4:20 pm TBD
Birge Huisgen-Zimmermann, University of California, Santa Barbara

TBD