Thursday, December 3, 2020
Modeling, Computation, Nonlinearity, Randomness and Waves Seminar Analysis of the “Unruly” Breakdown of Phase Reduction Theory: an Application of Markov Renewal Processes to Noise-driven Oscillators
Series: Modeling, Computation, Nonlinearity, Randomness and Waves Seminar
Presenter: Avinash Karamchandani, Department of Mathematics, University of Arizona
Abstract: We investigate the noise response properties of nonlinear, dissipative oscillators, i.e. systems of ODEs that have a stable limit cycle in the absence of noise. We find that the response of such oscillators to noise is often quite “unruly”: measures of the output diffusion are non-monotonic in the input noise strength and greatly enhanced as compared to a linear prediction from the standard phase reduction analysis.
The dynamics of the principal neurons of the mammalian olfactory bulb serves as a motivating example. The cell's membrane potential exhibits mixed-mode oscillations (MMOs) with both the large amplitude, action potential “spikes” typical of neurons as well as small amplitude, sub-threshold oscillations (STOs). Noisy input from upstream neurons can strongly perturb the patterns of spikes and STOs displayed, as is reflected in an unruly diffusion coefficient for spike times. The noise-driven perturbations can greatly affect the spike-driven interactions between cells, and we note the effect that the enhanced diffusion has on coherent, population-level behavior.
Finally, we propose a technique for the analysis of noise-driven oscillators that accounts for unruly corrections to the standard phase reduction. We center our analysis on linearized, random Poincare map dynamics. Taking the segregation of MMOs into spikes and STOs as motivation, we label each crossing of the Poincare section as either an event of interest (e.g. a spike) or a non-event. Using Markov renewal theory, we compute the diffusion coefficient of the resulting point process for the occurrence and timing of events. We argue that for many oscillator models, the corresponding point process exhibits unruly diffusion. In particular, we offer a thorough analysis in the case of planar oscillators, which exhibit unruliness in a finite region of a natural parameter space.
Password: “arizona” (all lower case)
Mathematics Colloquium Localization and unique continuation on the integer lattice
Series: Mathematics Colloquium
Presenter: Charles Smart, University of Chicago
Abstract: I will discuss results on localization for the Anderson--Bernoulli model. This will include my work with Ding as well as work by Li--Zhang. Both develop new unique continuation results for the Laplacian on the integer lattice.
Friday, December 4, 2020
Program in Applied Mathematics Brown Bag Seminar Uncertainty and Optimization in Finance
Series: Program in Applied Mathematics Brown Bag Seminar
Presenter: Ammon Washburn, Corporate Treasury Strategist, Goldman Sachs and Program in Applied Mathematics, Alumni
Will showcase two or three sample projects that highlight the need to understand uncertainty and optimization in the finance world. First will highlight how to conservatively hedge any misprojection while incorporating any possible mitigating factors. The second will highlight how optimization and deep understanding of optimization plays a key role in determining resources. If time permits, will talk about a third project
Tuesday, December 8, 2020
Hamiltonian Systems Weak and strong Riemannian metrics: theory and examples
Series: Hamiltonian Systems
Presenter: Martin Bauer, Florida State University
In finite-dimensional geometry the Riemannian metric is an isomorphism between the tangent and the cotangent bundle. In infinite dimensions the metric is always injective but may fail to be surjective. Accordingly, one distinguishes between two classes of Riemannian metrics: weak and strong ones. In this talk I will discuss several differences between weak and strong Riemannian metrics in terms of existence of the geodesic equation, properties of the geodesic distance, and the theorem of Hopf-Rinow. I will present both the the general theory and some specific examples, including l^2 spaces and diffeomorphism groups, in order to demonstrate the large variety (and pitfalls) of this infinite dimensional setting.
The talk will be via Zoom at: https://utoronto.zoom.us/j/99576627828 Passcode: 448487
Algebra and Number Theory Seminar Non-Abelian simple groups act with almost all signatures
Series: Algebra and Number Theory Seminar
Presenter: Aaron Wootton, University of Arizona
The topological data of a finite group $G$ acting conformally on a compact Riemann surface is often encoded using a tuple of non-negative integers $(h;m_1,\ldots ,m_s)$ called its signature, where the $m_i$ are orders of non-trivial elements in the group. There are two easily verifiable arithmetic conditions on a tuple which are necessary for it to be a signature of some group action. We derive necessary and sufficient conditions on a group for the situation where all but finitely many tuples that satisfy these arithmetic conditions actually occur as the signature for an action of $G$ on some Riemann surface. As a consequence, we show that all non-abelian finite simple groups exhibit this property.