# Jeffrey L. Jauregui

Harvey Mudd College Mathematics 2005

Thesis Proposal: | Volume Minimizing Cycles in G-Manifolds_{2} |
---|---|

Thesis Advisor: | Prof. Weiqing Gu |

Second Reader: | Prof. Jon Jacobsen |

E-Mail: | jjauregui@hmc.edu |

Final Talk: | HMC Presentation Days 2005 Slides |

Senior Thesis: | Solving for Volume-Minimizing Cycles in G Manifolds_{2} |

## Abstract

M-theory, a generalization of string theory, motivates the search
for examples of volume-minimizing cycles in Riemannian manifolds
of *G _{2}*-holonomy. Methods of calibrated geometry lead to a
system of four coupled nonlinear partial differential equations
whose solutions correspond to associative submanifolds of

*R*, which are 3-dimensional and minimize volume in their real homology classes. Several approaches to finding new solutions are investigated, the most interesting of which exploits the quaternionic structure of the PDE system.

^{7}