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This seminar/project class is geared towards helping participants understand concepts and methods from differential geometry, in particular for 2 and 3-manifolds, in a discrete rather than discretized setup. Discrete differential geometry aims to preserve selected structure when going from a continuous abstraction to a finite representation for computational purposes. For example, for a piecewise linear approximation ("mesh") of a surface one may define Gaussian curvature in such a way that important theorems are preserved in the discrete setting. Observations like this and many others have been made independently in a variety of areas ranging from electromagnetism to discrete minimal surfaces theory.

Together we will study the basic ideas underlying these approaches and see how they can be applied to theoretical and algorithmic investigations. Subject matter for the course (may change , based on participant interests), includes but is not limited to: discrete exterior calculus; Whitney forms; deRham and Whitney complexes; Steiner polynomials, Hadwiger functionals, and Cauchy Quermass Integrale; Geodesics; Morse theory; computational and algebraic topology; discrete simulation: fluids, electro-magnetism, elasticity, minimal surfaces, and surface parameterization; Hodge star and Hodge decomposition; convergence and accuracy of discrete methods; higher order theories; geometric PDEs and flows.

This course contains a few homework assignments. Participants are also encouraged to pursue a small research project as part of the class and report on it to the class. Projects can be of a theoretical or algorithmic nature. For example, convergence analysis of a particular discrete method or the use of a particular method for computational purposes in the simulation of some relevant physical phenomenon.

Potential participants are invited to contact Mathieu Desbrun prior to the beginning of class to discuss interests and possible project directions. All class participants (undergraduate and graduate students, or even postdocs) are expected to actively contribute.