In their proposed compactification of superstrings [4], Candelas, Horowitz, Strominger and Witten took the matric product of a maximal symmetric four dimensional spacetime M with a six dimensional CalabiYau vacua X as the ten dimensional spacetime; they identified the YangMills connection with the SU (3) connection of the CalabiYau metric and set the dilaton to be a constant. To make this theory compatible with the standard grand unified field theory, Witten [28] and HoravaWitten [20] proposed to use higher rank bundles for strong coupled heterotic string theory so that the gauge groups can be SU (4) or SU (5). Mathematically, this approach relies on UhlenbeckYaus theorem on constructing HermitianYangMills connections on stable bundles [27]. Many authors, including Friedman, Morgan and Witten [18]; Donagi, Ovrat, Pantev and Reinbacher [12]; Andreas [1], Kachru [21] and others, have worked

We give a brief summary of some of our work and our joint work with Stephan Tillmann on solving Thurstons equation and Haken equation on triangulated 3-manifolds in this paper. Several conjectures on the existence of solutions to Thurstons equation and Haken equation are made. Resolutions of these conjecture will lead to a new proof of the Poincar conjecture without using the Ricci flow. We approach these conjectures by a finite dimensional variational principle so that its critical points are related to solutions to Thurstons gluing equation and Hakens normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary.

Tautological systems are Picard-Fuchs type systems arising from varieties with large symmetry. In this survey, we discuss recent progress on the study of tautological systems. This includes tautological systems for vector bundles, a new construction of Jacobian rings for homogenous vector bundles, and relations between period integrals and zeta functions.

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