Xiang Tang Professor of Mathematics Office: Cupples I, 112B Phone: (314)9359488 Fax: (314)9356839 Email address:Post:  Department of Mathematics            Washington University            1 brookings Dr.              St. Louis, MO, 63130 Teaching: (Spring 2016) 1. Math 5042 Geometry II Research Interest: noncommutative geometry, symplectic geometry, and mathematical physics. Publications: Thermodynamics of the general diffusion process: time-reversibility and entropy production (with H. Qian & M. Qian),Journal of Statistical Physics, Vol. 107, Nos. 5/6, June (2002). Quantization and Morita equivalence for constant Dirac structures on tori (with A. Weinstein), Ann. Inst. Fourier, 54, 5 (2004). A counter example of invariant deformation quantization, Travaux mathematiques. Fasc. XVI, 273-283, av. Math., XVI Univ. Luxemb., Luxembourg, (2005). Homology of formal deformations of proper etale Lie groupoids (with N. Neumaier, M. Pflaum, and H. Posthum), J. Reine Angew. Math. 593 (2006), 117-168. Deformation Quantization of Pseudo Symplectic(Poisson) Groupoids, Geom. Funct. Anal.16 (2006), no. 3, 731-766. A note on Q-algebra and quantization, Lett. Math. Phys. 75 (2006), no. 1, 49--61. Hopfish algebras (with A. Weinstein, and C. Zhu), Pacific J. Math. 231 (2007), no. 1, 193--216. An algebraic index theorem for orbifolds (with M. Pflaum, and H. Posthuma), Adv. Math. 210 (2007), no. 1, 83--121. Poisson geometry and deformation quantization near a pseudoconvex boundary (with E. Leichtnam, and A. Weinstein), J. Eur. Math. Soc. (JEMS) 9 (2007), no. 4, 681--704. Rankin-Cohen brackets and formal quantization (with P. Bieliavsky, and Y. Yao), Adv. Math. 212 (2007), no. 1, 293--314. Hopfish structure and modules over irrational rotation algebras (with C. Blohmann and A.Weinstein), Noncommutative Geometry and Mathematical Physics, Contemporary Mathematics 2008, Volume: 462. Quantization and holomorphic anomaly (with A. Schwarz), J. High Energy Phys., 03 (2007). Gelfand-Fuchs cohomology of invariant formal vector fields (with I. Shapiro), Math. Res. Lett. 15 (2008), no. 1, 129--148. Hopf algebroids and secondary characteristic classes (with J. Kaminker), J. Noncommut. Geom. (2009), no. 1, 1-25. A universal deformation formula for $\mathcal{H}_1$ without projectivity assumption (with Y. Yao), J. Noncommut. Geom. 3 (2009), no. 2, 151--179. On the algebraic index for riemannian \'etale groupoids (with M. Pflaum and H. Posthuma), Lett. Math. Phys. 90 (2009), no. 1-3, 287--310. Noncommutative Poisson structures on orbifolds (with G. Halbout),  Trans. Amer. Math. Soc. 362 (2010), no. 5, 2249--2277. Quantization of Poisson-Hopf stacks associated with group Lie bialgebras (with G. Halbout), Pacific J. Math. 245 (2010), no. 1, 99--118. Equivariant Lefschetz number of differential operators (with G. Felder), Math. Z. 266 (2010), no.2, 451-470. Cyclic cocycles on deformation quantizations and higher index theorems (with M. Pflaum and H. Posthuma), Adv. Math. 223 (2010), no. 6, 1958--2021. Asymptotic equivariant index of Toeplitz operators and relative index of CR structures (with L. Boutet de Monvel, E. Leichtnam, and A. Weinstein), Geometric aspects of analysis and mechanics, 57--79, Progr. Math., 292, Birkhause/Springer, New York, (2011). Orbifold cup products and ring structures on Hochschild cohomologies (with M. Pflaum, H. Posthuma, and H. Tseng), Comm. Comtemp. Math. 13 (2011), no. 13, 123--182. Deformations of orbifolds with noncommutative  linear Poisson structures (with G. Halbout and J. Oudom), Int. Math. Res. Not. (2011), no. 1, 1--39. A survey on Rankin-Cohen deformations (with R. Rochberg and Y. Yao), Perspectives on noncommutative geometry, 133--151, Fields Inst. Commun., 61, Amer. Math. Soc., Providence, RI, (2011). From double Lie groupoids to local Lie 2-groupoids (with R. Mehta), Bull. Braz. Math. Soc. (N.S.), 42, (2011), no. 4, 651--681.  Hochschild  (co)homology of the Dunkl operator quantization of  $\mathbb{Z}_2$-singularity (with A. Ramadoss), Int. Math. Res. Not.  (2012), no. 9, 2123--2162. Quantization of Whitney functions (with M. Pflaum and H. Posthuma),  Travaus Mathematique, 20 (2012), 153--165. Dunkl operator and quantization of $\mathbb{Z}_2$-singularity (with G. Halbout), J. Reine Angew. Math.  673 (2012), 209--235.  The Picard group of  a noncommutative algebraic quantum torus (with Y. Berest, A. Ramadoss), J. Noncommut. Geom. 7 (2013), no. 2, 335--356. Hopf cyclic cohomology and Hodge theory for proper actions (with Y. Yao, W. Zhang), J. Noncommut. Geom. 7 (2013), no.3, 885--905.    Hochschild Lefschetz class for D-modules (with A. Ramadoss and H. Tseng), Math. Z. 275 (2013), no. 1-2, 367--388. Duality theorems of \'etale gerbes on orbifolds (with H. Tseng), Adv. Math. 250 (2014), 496--569. K-theory of equivariant quantization (with Y. Yao), J. Funct. Ana. 266 (2014), no. 2, 478-486. The impossibility of exactly flat non-trivial Chern bands in strictly local periodic tight binding models (with L. Chen, T. Mazaheri, and A. Seidel), J. of Physics A: Math. and Theor. 47, (2014), 152001. Geometry of orbit spaces of proper Lie groupoids (with M. Pflaum and H. Posthuma), J. Reine Angew. Math. 694 (2014), 49-84.  Conjugacy classes and characters for extensions of finite groups (with H. Tseng), Chinese Annals of Mathematics, Series B, 35 (2014), no. 5, 743-750. Quantization of Whitney functions and reduction (with M. Pflaum and H. Posthuma), J. of Singul 13 (2015), 217-228.  The index of geometric operators on Lie groupoids (with M. Pflaum and H. Posthuma), Indag. Math (N.S.) 25 (2014), no.5, 1135-1153. The transverse index theorem for proper cocompact actions of Lie groupoids (with M. Pflaum and H. Posthuma), J. of Differential Geometry, 99 (2015), no.3, 443-472. On the relative dual of an $S^1$-gerbe over an orbifold (with I. Shapiro and H. Tseng), Adv. Math. 270 (2015), 1-20. The localized longitudinal index theorem for Lie groupoids and the van Est map (with M. Pflaum and H. Posthuma), Adv. Math. 270 (2015), no.3, 223-262. Preprints: Symplectic structures on the integration of exact Courant algebroids (with R. Mehta), preprint: arXiv:1310.6587. An analytic Grothendieck Riemann Roch Theorem (with R. Douglas, and G. Yu), preprint: arXiv:1404.4396. Mechanices on fibered manifolds (with S. Li, and A. Stern), preprint: arXiv:1511.00061. A quantum Leray-Hirsch theorem for banded gerbes (with H. Tseng), preprint: arXiv:1602.03564.