We derive a sharp, localized version of elliptic type gradient estimates for positive solutions (bounded or not) to the heat equation. These estimates are akin to the Cheng-Yau estimate for the Laplace equation and Hamilton’s estimate for bounded solutions to the heat equation on compact manifolds. As applications, we generalize Yau’s celebrated Liouville theorem for positive harmonic functions...

somatosensory cortex 9 Yu-Cheng Pei, Peter V. Denchev, Steven S. Hsiao, James C. Craig and Sliman J. Bensmaia 10 11 12 1 Krieger Mind/Brain Institute, Johns Hopkins University, Baltimore, MD 21218 13 2 Department of Neuroscience, Johns Hopkins University, Baltimore, MD 21218 14 3 Department of Psychological and Brain Sciences, Indiana University, IN 47405 15 4 Department of Physical Medicine an...

In joint work with S.-T. Yau, we construct new cohomologies of differential forms and elliptic operators on symplectic manifolds. Their construction can be described simply following a symplectic decomposition of the exterior derivative operator into two first-order differential operators, which are analogous to the Dolbeault operators in complex geometry. These first-order operators lead to ne...

I will discuss results of three different types in geometry and topology. (1) General vanishing and rigidity theorems of elliptic genera proved by using modular forms, Kac-Moody algebras and vertex operator algebras. (2) The computations of intersection numbers of the moduli spaces of flat connections on a Riemann surface by using heat kernels. (3) The mirror principle about counting curves in ...

5 CA 4 1. Institute of Neuroscience, National Yang-Ming University, Taipei, Taiwan 5 2. Department of Surgery, National Yang-Ming University Hospital, Yilan, Taiwan 6 3. Department of Education and Research, Taipei City Hospital, Taipei, Taiwan 7 4. Institute of Microbiology and Immunology, National Yang-Ming University, Taipei, 8 Taiwan 9 5. Department of Rehabilitation, National Yang-Ming Uni...

We construct a (0, 2), D = 2 gauged linear sigma model on a supermanifold in both the Abelian gauge group and the non-Abelian gauge group. For the purpose of checking the exact supersymmetric (SUSY) invariance of the Lagrangian density, it is convenient to introduce a new operatorˆU for the Abelian gauge group. Thê U operator provides consistency conditions for satisfying the SUSY invariance. C...

The Strominger system is a system of partial differential equations describing the geometry of compactifications of heterotic superstrings with flux. Mathematically it can be viewed as a generalization of Ricci-flat metrics on non-Kähler Calabi-Yau 3folds. In this thesis, I will present some explicit solutions to the Strominger system on a class of noncompact Calabi-Yau 3-folds. These spaces in...

Abstract: We prove that for a Dirac operator, with no resonance at thresholds nor eigenvalue at thresholds, the propagator satisfies propagation and dispersive estimates. When this linear operator has only two simple eigenvalues sufficiently close to each other, we study an associated class of nonlinear Dirac equations which have stationary solutions. As an application of our decay estimates, w...