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
  


• On exotic modular tensor categories
  (with S. Hong, E. Rowell)
  Commun. Contemp. Math. (accepted), 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 (accepted), arXiv:0803.1016
  


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


• From string nets to nonabelions
  (with L. Fidkowski, M. Freedman, C. Nayak, K. Walker)
  cond-mat/0610583
  


• On classification of modular tensor categories
  (with E. Rowell, R. Stong)
  math.QA/0712.1377
  


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


• Topology driven quantum phase transitions in anyonic quantum liquids
  (with C. Gils, S. Trebst, A. Kitaev, A. Ludwig, M. Troyer), submitted
  


• Topological stability of q-deformed quantum spin chains
  (with C. Gils, E. Ardonne, S. Trebst, A. Ludwig, M. Troyer), submitted