New Perspectives on Difficult Mathematical Problems
Orrick Research investigates the mathematical structures behind motion, computation, and physical systems. Our work connects pure and applied mathematics, combining analytical reasoning with computational exploration to develop new ways of understanding difficult problems.
Led by Brian Orrick, the mathematics program examines how changes in representation, geometry, and mathematical formulation can reveal relationships that are difficult to see through conventional approaches.
Current research spans nonlinear dynamics, differential equations, gravitational systems, mathematical physics, and computational optimization.
Dynamical Systems and the Three-Body Problem
The motion of three interacting bodies presents a fundamental challenge in mathematical physics. Its equations are compact, yet their solutions can exhibit intricate dependencies on initial conditions, geometry, and the exchange of energy and momentum.
Our research explores alternative formulations of the Newtonian three-body problem, including a C-point force-center approach with candidate-reference state evaluation. This work examines whether a different representation of the system can expose useful relationships among forces, trajectories, and conserved quantities.
The investigation connects analytical development with computational comparisons, examining the conditions under which a formulation remains consistent with the original equations and provides useful insight into their solutions.
Nonlinear Equations and Fluid Dynamics
Nonlinear differential equations describe systems in which local interactions can produce complex behavior across many scales.
Our work includes investigations related to the Navier–Stokes equations, with attention to the mechanisms governing energy transfer, pressure, and the development of fine structure in fluid motion.
The research examines how estimates, geometric relationships, and interactions between scales may help characterize the behavior of solutions. A central objective is to identify precisely what follows from the equations, which assumptions are required, and where an argument needs further development.
Mathematical Foundations of Complex Mechanics
The mathematics program also supports Complex Mechanics, the theoretical framework developed within Orrick Research’s physics program.
This work investigates the mathematical relationships connecting geometry, dynamics, and observable physical quantities. It includes developing definitions, examining internal consistency, deriving consequences, and determining how proposed relationships connect with established mathematical and physical descriptions.
Each calculation raises a concrete question: what does the underlying structure require, and what predictions follow from it?
The connection between mathematics and physics provides opportunities in both directions. Physical questions motivate new mathematical investigations, while mathematical results clarify which physical interpretations are possible.
Mathematics of Computation and Optimization
Many computational problems become difficult because the number of possible configurations grows rapidly with problem size.
We investigate how structure within a problem can guide its solution. Areas of interest include alternative formulations, relationships among constraints, methods for reducing unnecessary computation, and the behavior of algorithms as problems become larger.
This work connects with our computer science program in optimization, scientific computing, and algorithm development. Mathematical analysis helps establish why a method works, where it applies, and how its performance should be evaluated.
From Exploration to Proof
Analytical work and computation play complementary roles in our research.
Symbolic calculations help develop and examine proposed relationships. Numerical experiments reveal patterns, test limiting cases, and identify counterexamples. Formal arguments establish the conditions under which a result holds.
Our objective is to develop research that others can examine in detail, with explicit assumptions, clear derivations, and supporting calculations. Conjectures, computational observations, and proved results are identified according to their status.
Research and Collaboration
Orrick Research welcomes collaboration with mathematicians, physicists, and computer scientists interested in nonlinear systems, mathematical mechanics, and the foundations of computation.
We welcome technical questions about specific arguments, independent examination of proposed results, and opportunities to develop research jointly.
