The pull of Earth's gravity on a satellite produces acceleration, which affects the satellite's velocity and position, making its orbit an ellipse. This is just one example of a mathematical phenomenon that arises from constraints on a physical system that can be modeled by differential or difference equations. This project will develop algorithms to automatically uncover the algebraic properties of functions defined by differential or difference equations by looking for symmetries. It will produce publicly-available implementations that will be useful to researchers in different areas, such as physics, number theory, and combinatorics. Key components of this project include the fostering of interactions with other scientific fields and the advancement of international collaborations.
The key to this project's approach for discovering the functional relations among the solutions of differential or difference equations is that these relations are encoded in the algebraic structure of a corresponding Galois group. This project will develop algorithms that determine, 1) for a linear differential equation with continuous or discrete parameters, which additional functional equations are satisfied by the solutions; 2) for a linear difference equation with respect to a shift, q-dilation, Mahler, or elliptic curve addition, which differential equations are satisfied by the solutions; and 3) for a linear difference equation with respect to a q-dilation or Mahler p-operator, which additional difference equations with respect to a q'-dilation or Mahler p'-operator, respectively, are satisfied by the solutions. In each of these three situations there is a Galois theory that encodes the sought properties of the solutions. Algorithms to compute the corresponding Galois groups lead directly to procedures to compute the functional relations.
This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.
