Date

Speaker

September 15

Dynamical systems for metabolic networks
Nicola Vassena, Free University Berlin
Abstract: In this talk I will give an overview of one approach to the analysis of
metabolic networks, using dynamical systems. When considered in
applications, one of the main features of these networks is that the
interaction functions (reaction rates) are practically unknown. That is,
the most reliable data is the structure of network. For this reason, we
present here a qualitative approach based on the structure of the network,
only, where no quantitative information is needed. In particular,
following this approach, we introduce how to address some bifurcation
problems and sensitivity analysis.

September 22

Anderson localization for disordered trees
Selim Sukhtaiev, Auburn University
Abstract: In this talk, we will discuss a mathematical treatment of a disordered system modeling localization of quantum waves in random media. We will show that the transport properties of several natural Hamiltonians on metric and discrete trees with random branching numbers are suppressed by disorder. This phenomenon is called Anderson localization.

October 6

Epidemiological Forecasting with Simple Nonlinear Models
Joceline Lega, University of Arizona
Abstract: Every week, the CDC posts COVID19 death forecasts for the US and its states and territories. These estimates are created with an ensemble model that combines probabilistic predictions made by a variety of groups in the US and abroad. Our model, EpiCovDA, which is developed by mathematics graduate student Hannah Biegel and combines simple nonlinear modeling with data assimilation, is one of these contributions. In this talk, I will present a novel paradigm for epidemiological modeling that is based on a dynamical systems perspective, and which consists in describing an outbreak in terms of incidence versus cumulative case curves. I will then explain how this approach may be used for parameter estimation and how it is combined with data assimilation in EpiCovDA.

October 13

A coordinate transformation to highlight interesting flow features: local orthogonal rectification
Jonathan Rubin, University of Pittsburgh
Abstract: Following some pioneering earlier work, there has been an uptick in efforts to develop coordinate transformations that provide natural coordinate systems in which it becomes easier to study certain flow features. Many of these transformations are local or focus on periodic orbits and associated small perturbations. In this talk, I will introduce a new coordinate transformation, local orthogonal rectification (LOR), recently developed by my graduate student Ben Letson (SFL Scientific) and me. I will illustrate how LOR provides new insights about forms of transient dynamics including rivers, dynamics of trajectories as they approach periodic orbits, and canards, and represents a useful tool that others may wish to apply for the analysis of such phenomena.

October 20 Special time: 10:00 AM

Nonlinear stability of fast invading fronts in a GinzburgLandau equation with an additional conservation law
Bastian Hilder, University of Stuttgart
Abstract: In this talk, I consider the stability of traveling fronts connecting an invading state to an unstable ground state in a GinzburgLandau equation with an additional conservation law. This system appears generically as an amplitude equation for Turing pattern forming systems admitting a conservation law structure such as the BenardMarangoni convection problem. The main result is the nonlinear stability of sufficiently fast fronts with respect to perturbations which are exponentially localized ahead of the front. The proof is based on the use of exponential weights ahead of the front to stabilize the ground state. After presenting the general strategy, I discuss the specific challenges faced in the proof, namely the lack of a comparison principle and the fact that the invading state is only diffusively stable, i.e. perturbations of the invading state decay polynomially in time.

October 27

Dynamics on networks (and other things!)
Lee DeVille, University of Illinois at UrbanaChampaign
Abstract: We will introduce several models connected to applications and present several results, mostly analytic but also some numerical. These models will be defined on networks or higherorder objects (e.g. simplicial complexes). In many of the cases, the dynamical systems can be characterized as "nonlinear Laplacians"; as such, various classical and notsoclassical results about Laplacians will be the secret sauce that undergirds the results. We will also try to give some insight into the applications that give rise to the problems, as time permits.

November 3 Postponed to Spring 2021


November 17

Dynamics of curved travelling fronts on a twodimensional lattice
Mia Jukic, Leiden University
Abstract: In this talk I will introduce the AllenCahn lattice differential equation (LDE) posed on a two dimensional lattice. It is a wellknown result that this equation admits a traveling wave solution. In the first part, I will explain the most interesting differences between the traveling waves arising from PDEs and the traveling waves arising from LDEs, such as dependence of the wave profile and the wave speed on the direction of propagation. In the second part, I will present recent results on the stability of the traveling wave solutions propagating in rational directions, and show a connection between the solution of a discrete mean curvature flow with a drift term and the evolution of the interface region of a solution that starts as a bounded perturbation to the wave profile.
