Thesis

Publications

- A note on special Kahler manifolds, Math. Ann., vol 313, 1999, pp 711-713
- On the geometry of classifying spaces and horizontal Slices, Amer. J. Math, vol 121, 1999, pp 177-198
- On the Futaki invariants of complete intersections, Duke J. Math, vol 100(2), 1999, pp 359-372
- On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math, vol 122(2), 2000, pp 235-273
- On the Hodge metric of the universal deformation space of Calabi-Yau threefolds, J. Geom. Anal., vol 11(1), 2001, pp 103-118
- On the curvature tensor of the Hodge metric of moduli space of polarized Calabi-Yau threefolds, J. Geom. Anal., vol 11(4), 2001, pp 635-647
- On the lower bound estimates of sections of the canonical bundle over a Riemanns surface, Inter. J. Math, vol 12(8), 2001, pp 235-273
- (with X. Sun) Weil-Petersson geometry on moduli space of polarized Calabi-Yau manifolds, J. Inst. Math. Jussieu, vol 3(2), 2004, pp 185-229
- (with G. Tian) The log term of Szego kernel, Duke J. Math, vol 125(2), 2004, pp 351-387
- K energy and K stability on hypersurfaces, Comm. Anal. Geom., vol 12(3), 2004, pp 599-628
- (with H. Fang) Generalized Hodge metrics and BCOV torsion on Calabi-Yau moduli, J. Reine Angew. Math., vol 588, 2005, 49-69
- (with X. Sun) On the Weil-Petersson volume of the moduli space of Calabi-Yau manifolds, Comm. Math. Phys., vol 261, 2006, 297-322
- (with C. Lin) On the discrete spectrum of generalized quantum tubes, Comm. PDE, vol 31, 2006, pp 1529-1546
- (with C. Lin) Existence of bound stated for layers built over hypersurfaces in Euclidean space, J. Funct. Anal., vol 244(1), 2007, pp 1-25
- (with C. Lin) Quantum layers over surfaces ruled outside a compact set, J. Math. Phys., vol 48, 2007, 053522
- (with T. Choi) On the DDVV Conjecture and the comass in Calibrated geometry (I), Math Z., vol 260, 2008, pp 409-429
- (with H. Fang and K. Yoshikawa) Asymptotic behavior of the BCOV torsion of Calabi-Yau moduli, J. Diff. Geom., vol 80, 2008, pp 175-259
- (with J. Ling) Bounds of eigenvalues on Riemannian manifolds, Trends in partial differential equations, 241–264, Adv. Lect. Math. (ALM), 10, Int. Press, Somerville, MA, 2010
- (with D. Zhou) On the essential spectrum of complete non-compact manifolds, J. Funct. Anal. vol 260, 2011, pp 3283–3298
- Normal scalar curvature conjecture and its applications, J. Funct. Anal, vol 261, 2011, pp 1284-1308
- (with P. Baldi) Complex-valued autoencoders, Neural Networks, vol 33, 2012, pp 136-147
- (with J. Rowlett) On the discrete spectrum of quantum layers, J. Math Phys., vol 53 (7), 2012, 073519
- (with J. Rowlett) Eigenvalues of collapsing domains and drift Laplacians, Math Res. Lett., vol 19(3), 2012, pp 627-648
- (with J. Rowlett) The fundamental gap of simplices, Comm. Math. Phys., vol 319, 2013, pp 111-145
- (with M. Douglas) Gauss-Bonnet-Chern theorem on moduli space, Math. Ann., vol 357, 2013, pp 469-511
- (with N. Charalambous)
**On the spectrum of the Laplacian**,*Math. Ann., vol 359, 2014, 211-238* - (with R. Seyyedali) Extremal metrics on ruled manifolds, Adv. Math., vol 258, 2014, 127-153
- (with D. Krejcirik) Location of the essential spectrum in curved quantum layers, J. Math Phys., vol 55, 2014, 083520
- (with N. Charalambous) Heat kernel estimates and the essential spectrum on weighted manifolds, J. Geom. Anal., vol 25, 2015, pp 536-563
- (with Bernard Shiffman) Asymptotic expansion of the off-diagonal Bergman kernel on compact Kahler manifolds, J. Geom. Anal., vol 25, 2015, pp 536-563
- (with C-J. Liu) Uniform asymptotic expansion on Riemann surfaces, Contemporary Mathematics, Analysis, Complex Geometry, and Mathematical Physics: a Conference in Honor of Duong H. Phong, vol 644, 2015, pp 159-173
- (with Steven Zelditch) Szego kernels and Poincare series, J. Anal. Math., vol 130, 2016, 167-184
- (with David Wenzel) The normal Ricci curvature inequality, Recent advances in the geometry of submanifolds---dedicated to the memory of Franki Dillen (1963--2013), Contemporary Mathematicsvol 674, 2016, 99-110
- (Joint with J. Rowlett), One can hear the corners of a drum, Bull. Lond. Math. Soc., vol 48, 2016, pp 85-93
- (with C-J. Liu) Abstract Bergman kernel expansion and its applications, Trans. Amer. Math. Soc., vol 368(2), 2016, pp 1467-1495
- (with D. Wenzel) Commutator Estimates Comprising the Frobenius Norm -- Looking Back and Forth, Operator Theory, vol 259, 2017, 533-559
- (with P. Baldi and P. Sadowski) Learning in the Machine: the Symmetries of the Deep Learning Channel, Neural Networks, vol 95, 2017, 110-133
- (with H. Hezari and J. Rowlett) The Neumann isospectral problem for trapezoids, Ann. Henri Poincare, vol 18(12) 2017, 3759–3792.
- (with Y. Wang) On locally conformally flat manifolds with finite total $Q$-curvature, Calc. Var. Partial Differential Equations , vol 56(4), 2017, Art. 98, 24 pp.
- (with P. Baldi and P. Sadowski) Learning in the machine: random backpropagation and the deep learning channel, Artificial Intelligence, vol 260, 2018, 1–35.