Physical & Biological Sciences Division
Professor
Faculty
McHenry Library
McHenry Building Room 4122
Mathematics Department
B.A., Applied Mathematics, University of California, Berkeley, 1981
Ph.D., Mathematics, University of California, Berkeley, 1987
Debra Lewis is an expert in relative equilibria for Hamiltonian systems and their stability, as well as bifurcations of relative equilibria. A relative equilibrium for a dynamical system is a solution that is an equilibrium modulo a symmetry group action. Her work is an interplay between group theory and symplectic geometry, and uses a good deal of symbolic manipulation. Her research focuses on geometric mechanics and optimal control theory, including models influenced by psychological and social pressures. She is presently involved with math placement and preparation, where she uses learning analytics to assess and increase equity in math access and outcomes
Debra Lewis's research focuses on geometric mechanics, particularly Hamiltonian and Lagrangian systems with symmetry. Inviscid fluids, hyperelastic materials, and systems of coupled rigid bodies are a few important examples of Hamiltonian and Lagrangian systems. Fundamental properties of these systems, e.g. conservation of total energy and momentum, or a variational formulation, facilitate the analysis of crucial features of the dynamics.
Lewis is interested in the design of algorithms for the numerical integration of conservative systems. Symmetries of mechanical systems and the associated conservation laws, such as conservation of linear and angular momentum, are typically not respected by conventional numerical schemes. Key features of the dynamics, such as equilibria, separatrices, and periodic orbits, may be lost or artificially introduced unless methods designed to preserve the underlying structures are used.
Lewis is currently working on the extension of key constructs and results in geometric mechanics to biological systems, particularly biomechanical control systems and population dynamics. The guiding dogma of geometric mechanics - that nature "optimizes" and "balances" - is as relevant to biological processes as it is to physical ones, but the development and analysis of biologically meaningful cost functions requires new insights and techniques.
- D. Lewis, T. Mizoguchi, and S. McCurdy. Multiple model transition and anomaly detection in direct and indirect population data. Preprint.
- D. Lewis. A soothing invisible hand: moderation potentials in optimal control, preprint.
- D. Lewis. Relative critical points. SIGMA 9, 038 (2013).
- B. Hamzi, J. Lamb, and D. Lewis. A characterization of normal forms for control systems. Preprint.
- D. Lewis. Optimal control with moderation incentives. Preprint.
- D. Lewis. Trivializations, factorizations, and geometric integration for pseudo-rigid bodies. Group theory and numerical analysis, CRM Proceedings Lecture Notes 39, AMS (2005), 191-205.
- D. Lewis, N. Nigam, and P. Olver. Connections for general group actions, Communications in Contemporary Mathematics, 7 No. 7 (2005), 341-374.
- P. Chossat, D. Lewis, J.-P. Ortega, and T. Ratiu. Bifurcation of relative equilibria in Hamiltonian systems with symmetry. Advances in Applied Mathematics 31 (2003), 10-45.
- D. Lewis and P. Olver. Geometric Integration Algorithms on Homogeneous Manifolds, Foundations of Computational Mechanics, 2 No. 4 (2002), 363-392.
- F. Fasso and D. Lewis. Stability Properties of the Riemann Ellipsoids, Archive for Rational Mechanics and Analysis 158 (2001), 259-292.
- D. Lewis and M. Shub. The distribution of the maximum condition number on great circles through a fixed 2 x 2 real matrix, Linear Algebra and its Applications 297 (1999), 193-202.
