In Press
Jaffe A, Liu Z. A Mathematical Picture Language Project. 2019 ICCM Conference Proceedings. In Press.
Jaffe A, Liu Z. Levin-Wen models and reflection positivity. Expositiones Mathematicae, a special issue dedicated to Dick Kadison [Internet]. In Press. arXiv Version
Liu Z, Ryba C. The Grothendieck Ring of a Family of Spherical Categories. [Internet]. Submitted. arXiv VersionAbstract
The first author constructed a q-parameterized spherical category $\mathscr{C}$ over ℂ(q) in [Liu15], whose simple objects are labelled by all Young diagrams. In this paper, we compute closed-form expressions for the fusion rule of $\mathscr{C}$, using Littlewood-Richardson coefficients, as well as the characters (including a generating function), using symmetric functions with infinite variables.
Liu Z. Yang-Baxter relation planar algebras. [Internet]. Submitted. arXiv Version
Liu Z, Wu J. Antisymmetric characters and Fourier duality. [Internet]. Submitted. arXiv Version
Liu Z. Quantized Graphs and Quantum Error Correction. [Internet]. Submitted. arXiv Version
Liu Z, Palcoux S, Wu J. Fusion Bialgebras and Fourier Analysis. [Internet]. Submitted. arXiv Version
Gao X, Liu Z. Universal Quantum Computation by a Single Photon. [Internet]. Submitted. arXiv Version
Liu Z, Norledge W, Ocneanu A. The adjoint braid arrangement as a combinatorial Lie algebra via the Steinmann relations. [Internet]. Submitted. arXiv's Version
Jaffe A, Jiang C, Liu Z, Ren Y, Wu J. Quantum Fourier Analysis. PNAS [Internet]. 2020. Published Version Online
Liu Z, Wu J. Non-Commutative Renyi entropic uncertainty principles. Science China Mathematics [Internet]. 2020. Published Version Online
Bu K, Jaffe A, Liu Z, Wu J. De Finetti theorems for braided parafermions. Communications in Mathematical Physics [Internet]. 2020;373 :435-456. Published Version Online
Liu Z, Morrison S, Penneys D. Lifting shadings on symmetrically self-dual subfactor planar algebras. Contemporary Mathematics [Internet]. 2020;747. Published Version OnlineAbstract
In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This finishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.
Bakshi K, Das S, Liu Z, Ren Y. An angle between intermediate subfactors and its rigidity. Transactions of the AMS [Internet]. 2019;371 : 5973-5991. Publisher's Version Online
Liu Z. Quon language: surface algebras and Fourier duality. Communications in Mathematical Physics [Internet]. 2019;366 :865-894. Publisher's Version OnlineAbstract
Quon language is a 3D picture language that we can apply to simulate mathematical concepts. We introduce the surface algebras as an extension of the notion of planar algebras to higher genus surface. We prove that there is a unique one-parameter extension. The 2D defects on the surfaces are quons, and surface tangles are transformations. We use quon language to simulate graphic states that appear in quantum information, and to simulate interesting quantities in modular tensor categories. This simulation relates the pictorial Fourier duality of surface tangles and the algebraic Fourier duality induced by the S matrix of the modular tensor category. The pictorial Fourier duality also coincides with the graphic duality on the sphere. For each pair of dual graphs, we obtain an algebraic identity related to the S matrix. These identities include well-known ones, such as the Verlinde formula; partially known ones, such as the 6j-symbol self-duality; and completely new ones.
Jiang C, Liu Z, Wu J. Block maps and Fourier analysis. Science China Mathematics [Internet]. 2019;62 (8) :1585-1614. publisher's Version Onlinen
Liu Z, Xu F. Jones-Wassermann subfactors for modular tensor categories. Advances in Mathematics [Internet]. 2019;355 (15) :106775. Published Version Online
Jones C, Liu Z, Ren Y. Classification of Thurston-Relation subfactor planar algebras . Quantum Topology [Internet]. 2019;10 (3) :441-480. Published Version OnlineAbstract

Bisch and Jones suggested the skein theoretic classification of planar algebras and investigated the ones generated by 2-boxes with the second author. In this paper, we consider 3-box generators and classify subfactor planar algebras generated by a non-trivial 3-box satisfying a relation proposed by Thurston. The subfactor planar algebras in the classification are either E6 or the ones from representations of quantum SU(N). We introduce a new method to determine positivity of planar algebras and new techniques to reduce the complexity of computations.

Liu Z, Wu J. Extremal pairs of Young’s inequality for Kac algebras. Pacific Journal of Mathemati [Internet]. 2018;295 (1) :103-121. Publisher's Version
Jaffe A, Liu Z. Mathematical picture language program. PNAS [Internet]. 2018;115 (1) :81-86. Publisher's VersionAbstract
We give an overview of our philosophy of pictures in mathematics. We emphasize a bidirectional process between picture language and mathematical concepts: abstraction and simulation. This motivates a program to understand different subjects, using virtual and real mathematical concepts simulated by pictures.