Fourier analysis on planar algebras., at Jilin University, Monday, July 17, 2017


Abstract: Planar algebras are a 2D picture language first introduced by Jones to study subfactors. In recent joint work with Chunlan Jiang and Jinsong Wu, we initiated the study of Fourier analysis on planar algebras. It is surprising that many results in Fourier analysis can be worked out on pictures. These analytic results also provide new insights to understand the algebraic and topological structure of planar algebras.

Synergy on quon language, at Issac Newton Institute, Cambridge, UK, Tuesday, June 13, 2017

Operator algebras: subfactors and their applications

Abstract: We will talk about the discovery of the quon language, which was inspired by ideas in different areas: subfactors, quantum information, CFT, TQFT. New proofs and results were carried out by synergy on the quon language. The Fourier analysis on quons turns out to be powerful to study modular tensor categories.

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Picture language program Friday, May 19, 2017


Abstract: We discuss the appearance of pictures in mathematics. We emphasize a bi-directional process between picture language and mathematical concepts: abstraction and simulation. This provides a program to understand different subjects, using virtual and real mathematical concepts simulated by pictures.

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Pictures in Mathematics, at Tsinghua Univeristy, Thursday, May 11, 2017

Short courses:

Course 1: From algebras to planar algebras

Course 2: General constructions: Free product and quotient

Course 3: Quantum groups and quantum subgroups

Course 4: Pictorial Fourier analysis

Course 5: Picture language in quantum information

Course 6: Quon language: Fourier duality

The string Fourier transform on quons, at Indiana University, Bloomington, Saturday, April 1, 2017

AMS section meeting: special section on fusion categories and applications

Abstract: In recent joint work with Arthur Jaffe and Alex Wozniakowski, we introduced quon 3D language for quantum information associated with a unitary modular tensor category (MTC). In joint work with Feng Xu, we proved that the string Fourier transform (SFT) on quons is the S matrix of the MTC. I will talk about string quons defined by strings in 3-manifolds. String quons include the Greenberger-Horne-Zeilinger (GHZ), Max, and 6j-symbol states. We obtain an algebraic identity for the MTC from...

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Title: Fourier cousins and new planar algebras, at Shanghai Center for Mathematical Sciences & School of Mathematical Sciences, Fudan University , Monday, February 27, 2017

2017 Young Mathematician Forum


Abstract: We will talk about and a new one-parameter family of planar algebras and answer a widely-discussed open question. When Jones introduced planar algebras, people expected that one could construct new examples by skein theory. But they encountered three fundamental problems: evaluation, consistency, and positivity. Here we solve the three problems. The new parameterized planar algebra is a Fourier cousin of the Birman-Wenzl-Murakami planar algebra. We give irreducible representations of the new planar algebra...

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