Bridges between subriemannian geometry and algebraic geometry: Now and then

Alex L  Castro; Wyatt  Howard; Corey Shanbrom

doi:10.3934/proc.2015.0239

Article Contents

2015, Volume 2015: 239-247. Doi: 10.3934/proc.2015.0239 open access

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Bridges between subriemannian geometry and algebraic geometry: Now and then

1.
Department of Mathematics, Imperial College, London, 180 Queen's Gate, London SW7 2AZ
2.
Mathematics and Computer Science, Santa Clara University, Santa Clara, CA
3.
Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA

Received: August 31, 2014

Revised: January 31, 2015

Published: October 2015

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Abstract

We consider how the problem of determining normal forms for a specific class of nonholonomic systems leads to various interesting and concrete bridges between two apparently unrelated themes. Various ideas that traditionally pertain to the field of algebraic geometry emerge here organically in an attempt to elucidate the geometric structures underlying a large class of nonholonomic distributions known as Goursat constraints. Among our new results is a regularization theorem for curves stated and proved using tools exclusively from nonholonomic geometry, and a computation of topological invariants that answer a question on the global topology of our classifying space. Last but not least we present for the first time some experimental results connecting the discrete invariants of nonholonomic plane fields such as the RVT code and the Milnor number of complex plane algebraic curves.

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Mathematics Subject Classification: 53C17, 32S05, 58K40, 14N10.

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