Montie Avery (BU)
Universality in spreading into unstable states
Front propagation into unstable states plays an important role in organizing structure formation in many spatially extended systems. When a trivial background state is pointwise unstable, localized perturbations typically grow and spread with a selected speed, leaving behind a selected state in their wake. A fundamental question of interest is to predict the propagation speed and the state selected in the wake. The marginal stability conjecture postulates that speeds can be universally predicted via a marginal spectral stability criterion. In this talk, we will present background on the marginal stability conjecture and present some ideas of our recent conceptual proof of the conjecture in a model-independent framework focusing on systems of parabolic equations.
Sathyanarayanan Chandramouli (UMass Amherst)
Dispersive hydrodynamics and Riemann problems in a non-Hermitian, nonlinear Schrödinger equation
We recently introduced a class of non-centered Riemann problems to an inhomogeneous and non-Hermitian nonlinear Schrödinger equation. The dynamics of these Riemann problems revealed a connection to the wave patterns witnessed in the classical transcritical flow problem. These classical transcritical flow problems were first studied in the context of surface water waves by Grimshaw & Smyth (1986) and in the context of superfluidic condensate flows by Hakim (1997) and more recently Leszczyn et. al. (2009). We point to research directions that are being currently pursued to investigate this connection to classical transcritical flow further.
Keynote: Carina Curto (Penn State)
Threshold-linear networks, attractors, and oriented matroids
Threshold-linear networks (TLNs) are common firing rate models in theoretical neuroscience that are useful for modeling neural activity and computation in the brain. They are simple, recurrently-connected networks with a rich repertoire of nonlinear dynamics including multistability, limit cycles, quasiperiodic attractors, and chaos. Over the past few years, we have developed a mathematical theory relating stable and unstable fixed points of TLNs to combinatorial properties of an underlying graph. The resulting “graph rules” and “gluing rules” provide a direct link between network architecture and key features of the dynamics. Many open questions remain, however, such as: How do changes in network parameters, such as connectivity, transition a TLN from one dynamic regime to another? In this talk, I present some new results in the bifurcation theory of TLNs, with an eye towards understanding how these networks may evolve via training (or learning). It turns out that a certain family of determinants related to the underlying hyperplane arrangement of a TLN is key. In particular, the theory of oriented matroids and Grassmann-Plucker relations provide valuable insights into the allowed fixed point configurations of TLNs, as well as the allowed bifurcations.
Xuwen Zhu (Northeastern)
Degeneration of hyperbolic surfaces and spectral gaps for large genus
The study of “small” eigenvalues of the Laplacian on hyperbolic surfaces has a long history and has recently seen many developments. In this talk I will focus on the recent work (joint with Yunhui Wu and Haohao Zhang) on the higher spectral gaps, where we study the differences of consecutive eigenvalues up to 
Katie Slyman (Brown)
Noisy tipping in nonautonomous systems
Rate-induced tipping occurs when a time-dependent ramp parameter changes rapidly enough to cause the system to tip between co-existing, attracting states. We demonstrate a prototypical example of rate-induced tipping influenced by stochastic forcing. We show that the addition of stochastic forcing to the system can cause it to tip well below the critical rate at which rate-induced tipping would occur. Moreover, it does so with significantly increased probability over the noise acting alone. We achieve this by finding a global minimizer in a canonical problem of the Freidlin-Wentzell action functional of large deviation theory that represents the most probable path for tipping. This is realized as a heteroclinic connection for the Euler-Lagrange system associated with the Freidlin-Wentzell action and we find it exists for all rates less than or equal to the critical rate. Its role as the most probable path is corroborated by direct Monte Carlo simulations.
Felipe Hernández (MIT)
The semiclassical limit of noisy quantum systems
In chaotic systems, the predictions of quantum mechanics and classical mechanics may disagree substantially after the Ehrenfest time, which is logarithmic in the semiclassical parameter h. Physicists have long suggested that the presence of a small amount of noise from an external environment may effectively eliminate such disagreement. In this talk I will describe a mathematical setup for this problem in which the quantum system is described by a Lindblad equation and the classical system by a Fokker-Planck equation. In this setup we see agreement between the quantum and classical evolutions for times which are polynomial in 1/h even when the noise vanishes in the semiclassical limit. This is based on joint work with Daniel Ranard and Jess Riedel.