Spatially localized states in pattern forming PDEs often result from bistability between a spatially homogeneous state and a spatially periodic state, and can be understood from the dynamical systems perspective. This talk focuses on the 1:1 forced complex Ginzburg-Landau equation, which is the normal form PDE for harmonically forced Hopf bifurcations in spatially extended systems.

It has long been observed that multi-scale systems, particularly those in climatology, exhibit behavior typical of stochastic models, most notably in the unpredictability and statistical variability of events. This is often in spite of the fact that the underlying physical model is completely deterministic.

One possible explanation for this stochastic behavior is deterministic chaotic effects. In fact, it has been well established that the statistical properties of chaotic systems can be well approximated by stochastic differential equations.

We consider scalar, bistable lattice differential equations on rectangular lattices in two space dimensions. We show (under mild conditions) that front-like solutions persist when holes are introduced into the domain by removing one or more lattice points. This is joint work with Hupkes and Van Vleck.

$p$-ground states are functions that minimize a quantity known as the $p$-Rayleigh quotient. These functions satisfy a nonlinear PDE that has an eigenvalue problem interpretation. We present two schemes that provide approximations of these functions. One is based on inverse iteration for square matrices and the other is a flow that is an appropriate generalization of the heat equation. In both cases, property scaled solutions converge to $p$-ground states.

Zebrafish (Danio rerio), a small fish with black and yellow stripes, has the ability to regenerate its stripes in response to growth or artificial disturbance. We simulate past laser ablation experiments using discrete (agent-based) and continuum (non-local conservation law) models. Both models consider two cell types diffusing and reacting based on rules of short range activation and long range inhibition. Our results suggest that the radius of interaction is a key determinant of stripe formation.

For dispersive PDE on spatially periodic domains, we formulate the time-periodic solutions problem as a fixed point problem. The operator in question is the composition of a linear operator and a nonlinear operator. The linear operator can be bounded with small divisor estimates, losing derivatives in the process. If the nonlinear operator can be shown to have smoothing properties, then the composition can be shown to be a local contraction. Thus, the nonexistence of nontrivial small-amplitude doubly periodic waves follows from dispersive smoothing estimates.

The problem of finite-time singularity versus global regularity for active scalar equations with nonlocal velocities (for instance, the vorticity equation of the Euler equations, the 2D quasi-geostrophic equations with or without dissipation, and the α-patch model) has attracted much attention in recent years. In this talk, I will discuss some recent results in this direction.

We consider the focusing 3D quantum many-body dynamic which models a dilute bose gas strongly confined in two spatial directions. We assume that the microscopic pair interaction is focusing and matches the Gross-Pitaevskii scaling condition. We carefully examine the effects of the fine interplay between the strength of the confining potential and the number of particles on the 3D N-body dynamic. We overcome the difficulties generated by the attractive interaction in 3D and establish new focusing energy estimates.

Abstract: Many biological rhythms evolve over multiple timescales, such as the beating of the heart or the electrical activity of neurons. Mathematical analysis of such multi-timescale problems has revealed special types of solutions, so-called canards, that play a crucial role in organising and shaping the dynamics. In this talk, we first give a brief overview of the current state of canard theory and its implications in mathematical neuroscience. We pay particular attention to two special classes of canard solutions: folded singularity canards and torus canards.

The motion of a vibrating material that is exposed to both deterministic and random forcing can be described by the damped semi-linear stochastic wave equation. As the mass density of the material converges to zero, the solutions of the damped semi-linear stochastic wave equations converge to the solution of a semi-linear stochastic heat equation uniformly on finite time intervals. This is called the Smoluchowski-Kramers approximation.

We present a new result, joint with Andrea Nahmod, Natasa Pavlovic, and Gigliola Staffilani, in which extremely low regularity flows are constructed globally in time almost surely for a family of modified SQG equations using Gibbs measures techniques; we also construct a Gibbs measure with respect to which these flows are invariant.