A set is strongly invariant for a dynamical system if all solutions to the dynamics, from that set, remain in that set. When the dynamics is somewhat irregular, for example given by a non-Lipschitz differential equation or by a differential inclusion, properties of the dynamics on the set are not enough to ensure strong invariance. Known conditions for strong invariance involve "subtangential" velocities, which can be expressed in terms of tangent cones to the set, and Lipschitz continuity of the possibly-multivalued dynamics near the set.
This talk presents novel sufficient conditions for strong forward invariance of compact sets, given in terms of Lyapunov-like functions. These functions attain their minimum on the invariant set, are reasonably smooth, and --- usually --- don't increase too fast along the solutions. The setting is of differential inclusions, where uniqueness of solutions need not be ensured. Connections of these functions to the usual Lyapunov functions, for differential inclusions and hybrid dynamics, and to barrier functions that exclude finite-time blow up of solutions, are explored in the proofs. The results come from joint work with A. Teel and R. Sanfelice.

If you take a rectangular strip of paper, perform a half twist of one of the short ends, and then join the short ends together you will obtain a Möbius band. How can you figure out what this shape will be? In my talk I’ll approach this problem from a mechanics perspective. The equilibrium shape, ignoring gravity, will be the one that minimizes its bending energy. As a material like paper can bend, but not stretch, I’ll assume that the material surface used to form the band can only undergo isometric deformations. This allows for the bending energy, which is proportional to the integral of the square of the mean curvature over the deformed surface, to be dimensionally reduced to a line integral. This is accomplished by using some facts about developable surface, which are surfaces with zero Gaussian curvature. (Basic definitions and results related to the curvature of surfaces will be covered in the talk.) The fact that the bending energy can be written as a line integral means that the corresponding Euler—Lagrange equations are ODEs, rather than PDEs. Thus, making the problem of finding the equilibrium configuration much more tractable. No effort to solve the ODEs will be given in this talk. This is joint work with Yi-chao Chen and Eliot Fried.

The Fourier transform is a fundamental and ubiquitous mathematical tool for writing an arbitrary function in terms of an infinite series of sines and cosines. In optics, Fourier transforms describe how light is focused by a lens. Rigorously, light traveling through an object t(x) thick is focused into bright spots whose intensities depend on the Fourier transform of e^{it(x)}=cos(t(x))+isin(t(x)). These second "highly oscillating" functions are difficult to analyze, so the physics literature contains very few closed-form results regarding this class of functions. An exciting exception is known procedures for producing functions t(x) which optimize their output into exactly two, or three, or n equal beams.
Symmetric 2D outputs, like four points which are the vertices of a square, are optimized by asymmetric functions t(x,y). It is often forgotten that we do not know if the symmetric 1D functions called "optimal" are truly the best possible. I will discuss our work searching for "loophole" beamsplitters, asymmetric functions which outperform the best-known symmetric functions for optimizing intensity into n points. I will additionally show five novel highly oscillating functions we have found closed-form results for.

Phase transitions are a fundamental phenomenon in nature and play a critical role in various physical and industrial applications. The modeling of these transitions and the understanding of phase boundary formation have been central problems in mathematical, physical, and engineering research for several decades.
In this talk, we will revisit the classical theory of liquid-liquid phase transitions and introduce a phase field approximation in the form of a Modica–Mortola-type energy functional incorporating a space-dependent double-well potential. This model captures scenarios in which, due to non-isothermality, the preferred phases vary across the domain. Using variational techniques, we will study the sharp interface limit, revealing that this is a surface energy with density given by a degenerate metric induced by the potential.

In the context of nonlinear dynamics on a finite-dimensional state space, we will examine information requirements for control-enabling tasks such as state estimation, model detection, localization and mapping in an unknown environment. We will discuss the role of topological entropy in determining minimal data rates necessary for solving these tasks. Through examples, we will study simultaneous localization and mapping based on a binary signal generated by an unknown landmark. Anyone having basic familiarity with ordinary differential equations should be able to follow the talk.

Glaucoma is the leading cause of blindness worldwide and is treated by lowering intraocular pressure. However, some people who have high intraocular pressure do not develop glaucoma and others with average intraocular pressure develop glaucoma. We would like to understand the dynamics related to the development of glaucoma better through modeling. We consider an elliptic-parabolic coupled partial differential equation connected to a nonlinear ODE through an interface. The model describes fluid flowing through biological tissues with the ODE accounting for global features of blood flow that affect the tissue. We discuss a modeling choice on the interface coming from the fact that the ODE is 0D and the PDE is 3D and show existence of a solution to this problem using a fixed point method.

We mainly discuss a model of a complex fluid describing thixotropic yield stress behavior in the limit of large relaxation time.  We use the multiscales to describe separations of regimes of large amplitude oscillatory shear flows. Regimes of similar fast, slow and yielded dynamics are identified and similar methodologies can be used to estimating width shear banding on steady (non-oscillatory) Poiseuille flow. There are some simulations which can agree with corresponding multiscale analysis.

The Kimura equation is a degenerate diffusion equation that plays an important role in population genetics, which can be viewed as a large population approximation for N goes to infinity of discrete Moran-Wright-Fisher models. To address the boundary issues, we develop a Lagrangian particle based operator splitting scheme for a truncated Kimura equation.

This talk presents a study of the Poisson-Nernst-Planck (PNP) equation equipped with a dynamical boundary condition to model the adsorption phenomenon at electrodes. The dynamical boundary condition allows for a more realistic representation of ion interaction at the electrode surface, which is critical for understanding electrochemical processes. Additionally, I will review previous models addressing similar phenomena and delve into the mathematical analysis of these equations, emphasizing their well-posedness. This comprehensive exploration aims to enhance our understanding of electrochemical systems and the rigorous mathematical frameworks that describe them.

In the past couple of decades, mathematical fluid dynamics has made significant strides with numerous constructions of solutions to fluid equations that exhibit pathological or wild behaviors. These include the loss of the energy balance, non-uniqueness, singularity formation, and dissipation anomaly. Interesting from the mathematical point of view, providing counterexamples to various well-posedness results in supercritical spaces, such constructions are becoming more and more relevant from the physical point of view as well. Indeed, a fundamental physical property of turbulent flows is the existence of the energy cascade. Conjectured by Kolmogorov, it has been observed both experimentally and numerically but had been difficult to produce analytically. In this talk, I will overview new developments in discovering not only pathological mathematically, but also physically realistic solutions of fluid equations. I will focus on dissipation anomaly for viscous fluid flows as well as anomalous dissipation for the limiting inviscid flows. These two intrinsically linked laws of turbulence are postulated by Kolmogorov and Onsager’s empirical theories built on the persistence of the energy flux through the inertial range. I will first analyze these phenomena on a finite time interval and prove the existence of various scenarios in the limit of vanishing viscosity, ranging from the total dissipation anomaly to a pathological one where inviscid anomalous dissipation occurs without viscous dissipation anomaly, as well as the existence of infinitely many limiting solutions of the Euler equations in the limit of vanishing viscosity. Finally, I will show the existence of dissipation anomaly for long time averages, relevant for turbulent flows, proving that the Doering-Foias upper bound is sharp.