Before joining Microsoft, Zhenghan was a professor of mathematics at Indiana University. His main interests are quantum topology, mathematical models of topological phases of matter, and their application to quantum computing.

Contact

Publications

• Classification of closed topological 4-manifolds with infinite cyclic fundamental group
  UC San Diego Ph.D thesis, 1993
  


• Mobius invariance of knot energy
  (with S. Bryson, M. Freedman and Z.-X. He)
  Bull. AMS 28, 99 (1993).
  


• Mobius energy of knots and unknots
  (with with M. Freedman and Z.-X. He)
  Ann. of Math .139, 1 (1994).
  


• CP2-stable theory
  (with M. Freedman)
  Math. Res. Letters 1, 45 (1994).
  


• On Bott polynomials
  J. of Knot Theory and Its Ramifications 3, 537 (1994).
  


• Classification of closed nonorientable 4-manifolds with infinite cyclic fundamental group
  Math. Res. Letters 2, 339 (1995).
  


• Controlled linear algebra
  (with M. Freedman)
  Prospects in Topology; Proceedings of a conference in honor of William Browder, ed. Frank Quinn
  Ann. of Math. Stud. 138, 138 (1995).
  


• Integral geometry of plane curves and knot invariants
  (with X.-S. Lin)
  J. Diff. Geom. 44, 74 (1996).
  


• Burau representation of string links and random walks
  (with X.-S. Lin and Feng Tian)
  Pac. J. of Math. 182, 289 (1998).
  


• On Ohtsuki invariants of integral homology 3-spheres
  (with X.-S. Lin)
  Acta. Math. Sinica 15, 293 (1999).
  


• Fermat limits, and a congruence of Ohtsuki's invariants
  (with X.-S. Lin)
  Proceedings of the Kirbyfest, edited by J. Hass and M. Scharlemann
  Geometry and Topology Monographs 2, 321 (1999).
  


• Self-homeomorphisms of 4-manifolds with fundamental group Z
  (with R. Stong)
  Topology Appl. 106, 49 (2000).
  


• The Gassner representation for string links
  (with P. Kirk and C. Livingston)
  Commun. Contemp. Math. 3, 87 (2001).
  


• Random walks on knot diagrams, colored Jones polynomials, and Ihara-Selberg type zeta functions
  (with X.-S. Lin) Knots, braids, and mapping class groups|papers dedicated to Joan S. Birman (New York, 1998), 107-121, AMS/IP, Stud. Adv. Math. 24 Providence, RI, (2001).
  


• Simulation of topological field theories by quantum computers
  (with M. Freedman and A. Kitaev)
  Commun. Math. Phys. 227, 587 (2002).
  


• A modular functor which is universal for quantum computation
  (with M. Freedman and M. Larsen)
  Commun. Math. Phys. 227, 605 (2002).
  


• The two-eigenvalue problem and density of Jones representation of braid groups
  (with M. Freedman and M. Larsen)
  Comm. Math. Phys. 228, 177 (2002).
  


• Quantum SU(2) faithfully detects mapping class groups modulo center
  (with M. Freedman and K. Walker)
  Geom. Topol. 6, 523 (2002).
  


• Topological quantum computation
  (with M. Freedman, A. Kitaev and M. Larsen) Mathematical challenges of the 21st century (Los Angeles, CA, 2000).
  Bull. Amer. Math. Soc. (N.S.) 40, 31 (2003).
  


• On Freedman's lattice model for topological phases
  (with J. Brink)
  Quantum Information Processing, 2, 81 (2003).
  


• A class of P,T invariant topological phases of interacting electrons
  (with M. Freedman, C. Nayak, K. Shentgel and K. Walker)
  Annals of Physics, 310, 428 (2004).
  


• Quantum logic as motivated by quantum computing
  (with J. Dunn, T. Hagge and L. Moss),
  Journal of Symbolic Logic, 70, iss. 2, 353 (2005).
  


• Density of the SO(3) TQFT representation of mapping class groups
  (with M. Larsen)
  Comm. Math. Phys., 260, 641 (2005), math.GT/0408161
  


• Universal manifold pairings and positivity
  (with M. Freedman, A. Kitaev, C. Nayak, J. Slingerland, K. Walker)
  Geometry and Topology, 9 53, 2303 (2005), math.GT/0503054
  


• The N-eigenvalue problem and two applications
  (with M. Larsen and E. Rowell)
  Math. Research Notices, 2005, iss. 64, 3987 (2005), math.RT/0506025
  


• Extraspecial 2-groups and the images of braid group representations
  (with J. Franko, and E. Rowell)
  J. of Knot Theory and Its Ramifications, 15, 1 (2006), math.RT/0503435
  


• Topologization of electron liquids with Chern-Simons theory and quantum computation
  Nankai Tracts in Mathematics, 10, (2006), Editors Mo-lin Ge and Weiping Zhang, World Scientific, pg. 106-120, cond-mat/0601285.
  


• Large quantum Fourier transforms are never exactly realized by braiding conformal blocks
  (with M. Freedman)
  Phys. Rev. A, 75, 032322 (2007).
  


• Interacting anyons in topological quantum liquids: The golden chain
  (with A. Feiguin, S. Trebst, A. W. W. Ludwig, M. Troyer, A. Kitaev, M. Freedman)
  Phys. Rev. Lett., 98, 160409 (2007); cond-mat/0612341
  


• A classification of symmetric polynomials of infinite variables- a construction of Abelian and non- Abelian quantum Hall states
  (with X.-G. Wen)
  Phys. Rev. B, 77, 235108 (2008); cond-mat/0801.329
  


• On (2+1)-picture TQFTs
  (with M. Freedman, C. Nayak, and K. Walker)
  Topology and Physics - Proceedings of the Nankai Conference in Memory of Xiao-Song Lin, Nankai Tracts in Mathematics, 12, 19 (2007) World Scientific Company (September 2008)
  


• On exotic modular tensor categories
  (with S. Hong, E. Rowell)
  Commun. Contemp. Math. Supplementary Issue 1 (2008) pp. 1049-1074, math.GT/0710.5761
  


• Topological properties of Abelian and non-Abelian quantum Hall states from the pattern of zeros
  (with X.-G. Wen)
  Phys. Rev. B 78, 155109 (2008), arXiv:0803.1016
  


• A short introduction to Fibonacci anyon models
  (with S. Trebst, M. Troyer, A. Ludwig)
  Prog. Theor. Phys. Suppl. 176, 384 (2008).
  


• Emergence of non-abelian statistics from an abelian model
  (with J. R. Wootton, V. Lahtinen, and J. K. Pachos)
  Phys. Rev. B 78, 161102(R) (2008), arXiv:0804.0931
  


• From string nets to nonabelions
  (with L. Fidkowski, M. Freedman, C. Nayak, K. Walker)
  Commun. Math. Phys. 287, 805-827 (2009), cond-mat/0610583
  


• On classification of modular tensor categories
  (with E. Rowell, R. Stong)
  Commun. Math. Phys. 292, 343-389 (2009), math.QA/0712.1377
  


• Topology driven quantum phase transitions in anyonic quantum liquids
  (with C. Gils, S. Trebst, A. Kitaev, A. Ludwig, M. Troyer)
  Nature Physics 5, 834 (2009), arXiv:0906.1579
  


• Collective States of Interacting Anyons, Edge States, and the Nucleation of Topological Liquids
  (with C. Gils, E. Ardonne, S. Trebst, A. Ludwig, M. Troyer)
  Phys. Rev. Lett. 103, 070401 (2009).
  


• Anyonic quantum walks
  (with G. Brennen, D. Ellinas, V. Kendon, J. Pachos, I. Tsohantjis)
  Annals of Physics 325, 664 (2010), arxiv:0910.2974
  


• Non-Abelian quantum Hall states and their quasiparticles: From the pattern of zeros to vertex algebra
  (with Y. Lu, X.-G. Wen, Z. Wang)
  Phys. Rev. B 81, 115124 (2010), arxiv:0910.3988
  


• Projective Ribbon Permutation Statistics: a Remnant of non-Abelian Braiding in Higher Dimensions
  (with M. Freedman, M. B. Hastings, C. Nayak, X.-L. Qi, K. Walker)
  arXiv:1005.0583

Books

Topology and Physics

Proceedings of the Nankai International Conference in Memory of Xiao-Song Lin
Tianjin, China 27-31 July 2007 (Nankai Tracts in Mathematics)
Kevin Lin (Editor), Zhenghan Wang (Editor), Weiping Zhang (Editor)
World Scientific Publishing Company (September 2008)


Topological Quantum Computation

CBMS Regional Conference Series in Mathematics, vol. 112
American Mathematical Society (May 2010)

Writings in Chinese