Department of Mathematics

Title: A Universal Description of Stochastic Oscillators

Abstract: Many systems in physics, chemistry and biology exhibit oscillations with a pronounced random component. Such stochastic oscillations can emerge via different mechanisms, for example linear dynamics of a stable focus with fluctuations, limit-cycle systems perturbed by noise, or excitable systems in which random inputs lead to a train of pulses. Despite their diverse origins, the phenomenology of random oscillations can be strikingly similar. In joint work with Alberto Perez, Benjamin Lindner, and Boris Gutkin, we have introduced a nonlinear transformation of stochastic oscillators via a complex-valued function, Q, that greatly simplifies and unifies the mathematical description of the oscillator’s spontaneous activity, its response to external time-dependent perturbation, and the correlation statistics of different oscillators that are weakly coupled. The Q function is the eigenfunction of Kolmogorov’s backward operator (also called the stochastic Koopman operator) with the least negative (but non-vanishing) eigenvalue $\lambda_1=\mu_1+i\omega_1$. The resulting power spectrum of the complex-valued function is exactly given by a Lorentz spectrum with peak frequency $\omega_1$ and half-width $\mu_1$; its susceptibility with respect to a weak external forcing is given by a simple one-pole filter, centered around $\omega_1$; and the cross-spectrum between two coupled oscillators can be easily expressed by a combination of the spontaneous power spectra of the uncoupled systems and their susceptibilities. Our approach makes qualitatively different stochastic oscillators comparable, provides simple characteristics for the coherence of the random oscillation, and gives a framework for the description of both weakly and strongly coupled stochastic oscillators.

Joint work with:
Alberto Perez-Cervera (Universitat Politècnica de Catalunya, Barcelona)
Boris Gutkin (Ecole Normale Supérieure, Paris)
Benjamin Lindner (Humboldt University, Berlin)
Max Kreider (Penn State University)

Title: More unlikely intersections on elliptic surfaces
(w/ G. Urzua and F. Voloch)

Abstract: In arithmetic geometry, one encounters situations where an intersection is “unlikely”, for example on dimension grounds, and where one wants to give the unlikely thing infinitely many chances to happen (say by varying some parameter). A statement that the intersection happens only finitely many times is then interesting (and a bound is even better). Families of elliptic curves host many examples of this type of problem. We discuss one such situation where the main result is a beautiful bound of topological nature, but whose proof, somewhat surprisingly, requires us to briefly leave the world of algebraic geometry for a real-analytic context. Time permitting, I’ll also mention a connection to a famous homomorphism of Manin and point out some inaccuracies in the literature about it.