The Fourier transform on subfactors, at HIT, Wednesday, August 10, 2016

Abstract: Subfactor theory is a natural framework to study quantum symmetries. I will talk recent work joint with Chunlan Jiang and Jinsong Wu about the analytic property of the Fourier transform on subfactors. This Fourier transform also has applications in mathematical physics and quantum information theory.

Planar anionic algebras, at Workshop on von Neumann algebras (HIM), Thursday, July 7, 2016

We recall some fundamental concepts, results and questions in subfactor theory. Motivated by them, we introduce planar anionic algebras and answer a couple of questions. We give new methods to construct subfactors using planar anionic algebras. Furthermore, we talk about parafermion algebras which are planar anionic algebras satisfying additional axioms, namely planar para algebras. These examples lead to holographic software for quantum information and surprising applications from subfactor theory to quantum information and backwards.

New diagrammatic insights in mathematical physics , at ETH , Thursday, March 3, 2016

Mathematical physics seminar

Abstract: We interpret some classical concepts in mathematical physics in a diagrammatic way, such as the Fourier transform and reflection positivity. We give one new family of algebras to realize this idea. As an application, we give some diagrammatic models for quantum information, which unite the topological aspect of qudits and their entanglement.

Parafermion algebras, subfactors and reflection positivity, at UCLA, Wednesday, January 20, 2016
Functional Analysis Seminar
Abstract: We will talk about parafermion algebras and the parasymmetry for subfactors. We introduce the Fourier transform on parafermion algebras by subfactor theory. Based on the Fourier transform, we prove the reflection positivity for Hamiltonians on the parafermion algebras. We give some generalizations of this result.
Yang-Baxter relation planar algebras, at Chongqing University, Monday, December 21, 2015
The International Conference on Non-commutative Geometry and K-Theory
Abstract: We introduce the Yang-Baxter relation as a generalization of the Yang-Baxter equation. 
We classify singly generated Yang-Baxter relation planar algebras. This can be interpreted as an initial step toward Bisch and Jones suggested skein theoretic classification. The classification yields Bisch-Jones planar algebras, Birman-Wenzl-Murakami planar algebras, as well as a new one-parameter family of planar algebras C(q). Our new planar algebras...
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Fourier Analysis on subfactors, at Chongqing Normal University, Monday, December 21, 2015

Abstract: Subfactors are wildly used to study quantum symmetry. We talk about the Fourier transform on subfactors and its applications. We formalize and prove the Hardy uncertainty principle for finite index subfactors.

Parasymmetry for subfactors, at Vanderbilt University, Friday, October 23, 2015

Subfactor seminar

Abstract: The universal skein theory provides a general framework to study subfactors and planar algebra. I will talk about different generalizations based on some examples. Motivated by the example from parafermions, I introduce the parasymmetry for subfactors and generalize some concepcts in subfactor theory, such as standard invariants, paragroups, $\lambda$ lattices, and planar algebras. As an application, infinitely many finite/infinite depth subfactors are constructed.

Skein theory for subfactors, at Memphis, Sunday, October 18, 2015



We talk about the classication and construction of subfactors. We provide a universal skein theory to construct small index subfactors and a different type skein relation, called Yang-Baxter relation, to construct new subfactors with large index.

Invariants, subfactors and quantum groups, at UC Riverside, Wednesday, October 14, 2015


Abstract: I will talk about the appearance of the Jones polynomial from subfactors and quantum SU(2). For quantum SU(N), O(N) and Sp(2N), the corresponding knot/three manifold invariants and subfactors were well studied around 1990. I give a classification of the next complicated subfactors and discover a new one parameter family. Infinitely many new subfactors and unitary fusion categories, therefore three manifold invariants, are derived from this family. In particular, two families of those fusion categories...
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Skein theory for subfactors, at UCSD, Saturday, October 10, 2015


Abstract: We provide two different skein theories to construct subfactors with small and large indices respectively, and we construct a new family of subfactors whose indices approach infinity. The idea of the "universal skein theory" is to simplify the construction of subfactors by the knowledge of the Temperley-Lieb-Jones algebra. When the index is small, we can construct subfactors by solving some simple equations. When the index is large, this method in no longer efficient. Instead, we suggest a different type of skein theory motivated by the Yang-...

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