Past Seminars by Series

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2020
Tue Nov 17

Dynamical Systems

2:30pm - Via Zoom See abstract
Dynamics of curved travelling fronts on a two-dimensional lattice
Mia Jukic, Leiden University

In this talk I will introduce the Allen-Cahn lattice differential equation (LDE) posed on a two dimensional lattice. It is a well-known 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.Zoom Link: https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Tue Oct 27

Dynamical Systems

2:30pm - Via Zoom Link - see abstract
Dynamics on networks (and other things!)
Lee DeVille, University of Illinois

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 higher-order objects (e.g. simplicial complexes).  In many of the cases, the dynamical systems can be characterized as “nonlinear Laplacians”; as such, various classical and not-so-classical 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.

Zoom Link: https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Tue Oct 20

Dynamical Systems

10:00am - Zoom Link - See Abstract
Nonlinear stability of fast invading fronts in a Ginzburg-Landau equation with an additional conservation law
 Bastian Hilder , University of Stuttgart  

In this talk, I consider the stability of traveling fronts connecting an invading state to an unstable ground state in a Ginzburg-Landau 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 Bénard-Marangoni 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.

Zoom Link: https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Tue Oct 13

Dynamical Systems

2:30pm - Zoom link: See abstract
A coordinate transformation to highlight interesting flow features: local orthogonal rectification
Jonathan Rubin, University of Pittsburgh

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.

Zoom Link: https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Tue Oct 06

Dynamical Systems

2:30pm - Zoom link: See abstract
Epidemiological Forecasting with Simple Nonlinear Models
 Joceline Lega , University of Arizona

Every week, the CDC posts COVID-19 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. Zoom Link: https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Tue Sep 22

Dynamical Systems

2:30pm - Zoom - see link below
Anderson localization for disordered trees
Selim Sukhtaiev, Auburn University

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.

https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Tue Sep 15

Dynamical Systems

2:30pm - Zoom - see link below
Dynamical systems for metabolic networks
Nicola Vassena, Free University Berlin

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.

https://umn.zoom.us/meeting/register/tJ0lc-CsrjIqHN1xLg-ljWWlDIYBIUwKwJK-

Tue Feb 25

Dynamical Systems

2:30pm - Vincent Hall 213
Hyperbolic scattering in the N-body problem
Rick Moeckel, University of Minnesota

It is a classical result that in the N-body problem with positive energy, all solutions are unbounded in both forward and backward time. If all of the mutual distances between the particles tend to infinity with nonzero speed, the solution in called purely hyperbolic. In this case there is a well-defined asymptotic shape of the configuration of N points. We consider the scattering problem for solutions which are purely hyperbolic in both forward and backward time: given an initial shape at time minus infinity, which final shapes at time plus infinity can be reached via purely hyperbolic motions ? I will describe some recent work on this problem using a variation on McGehee's blow-up technique. After a change of coordinates and timescale we obtain a well-defined limiting flow at infinity and use it to get Chazy-type asymptotic estimates on the positions of the bodies and to study scattering solutions near infinity. This is joint work with G. Yu, R. Montgomery and N. Duignan.

Tue Feb 04

Dynamical Systems

2:30pm - Vincent Hall 213
Spectral Stability, the Maslov Index, and Spatial Dynamics
Margaret Beck, Boston University

Understanding the spectral stability of solutions to partial differential equations is an important step in predicting long-time dynamics. Recently, it has been shown that a topological invariant known as the Maslov Index can play an important role in determining spectral stability for systems that have a symplectic structure. In addition, related ideas have lead to a suggested generalization of the notion of spatial dynamics to general, multidimensional spatial domains. In this talk, the notions of spectral stability, the Maslov Index, and spatial dynamics will be introduced and an overview of recent results will be given.

2019
Fri Nov 15

Dynamical Systems

2:30pm - Vincent Hall 20
The mathematics of taffy pulling
Jean-Luc Thiffeault, University of Wisconsin

Taffy is a type of candy made by repeated 'pulling' (stretching andfolding) a mass of heated sugar. The purpose of pulling is to get air
bubbles into the taffy, which gives it a nicer texture. Until the
late 19th century, taffy was pulled by hand, an arduous task. The
early 20th century saw an avalanche of new devices to mechanize the
process. These devices have fascinating connections to the
topological dynamics of surfaces, in particular with pseudo-Anosov
maps. Special algebraic integers such as the Golden ratio and the
lesser-known Silver ratio make an appearance, as well as more exotic
numbers. We examine different designs from a mathematical
perspective, and discuss their efficiency. This will be a "colloquium
style" talk that should be accessible to graduate students.

Tue Oct 15

Dynamical Systems

2:30pm - Vincent Hall 209
Forecasting U.S. elections with compartmental models of infection
Alexandria Volkening, Northwestern University

U.S. election forecasting involves polling likely voters, making assumptions about voter turnout, and accounting for various features such as state demographics and voting history. While political elections in the United States are decided at the state level, errors in forecasting are correlated between states. With the goal of shedding light on the forecasting process and exploring how states influence each other, we develop a framework for forecasting elections in the U.S. from the perspective of dynamical systems. Through a simple approach that borrows ideas from epidemiology, we show how to combine a compartmental model with public polling data from HuffPost and RealClearPolitics to forecast gubernatorial, senatorial, and presidential elections at the state level. Our results for the 2012 and 2016 U.S. races are largely in agreement with those of popular pollsters, and we use our new model to explore how subjective choices about uncertainty impact results. We conclude by comparing our forecasts for the senatorial and gubernatorial races in the U.S. midterm elections of 6 November 2018 with those of popular pollsters. This is joint work with Daniel Linder (Augusta Univ.), Mason Porter (UCLA), and Grzegorz Rempala (Ohio State Univ.)

Tue Oct 01

Dynamical Systems

2:30pm - Vincent Hall 209
Relative equilibrium configurations of gravitationally interacting rigid bodies
Rick Moeckel, University of Minnesota

Consider a collection of n rigid, massive bodies interacting according to their mutual gravitational attraction. A relative equilibrium motion is one where the entire configuration rotates rigidly and uniformly about a fixed axis — all of the bodies are phase locked. Such a motion is possible only for special positions and orientations of the bodies. A minimal energy motion is one which has the minimum possible energy in its fixed angular momentum level. While every minimal energy motion is a relative equilibrium motion, the main result here is that a relative equilibrium motion of n >= 3 disjoint rigid bodies is never an energy minimizer. Since energy minimizers are the expected final states produced by tidal interactions, phase locking of 3 or more bodies will not occur.

Tue Apr 16

Dynamical Systems

2:30pm - Vincent Hall 313
The Origins of the Reaction Zone in Microtornado Experiments
Patrick Shipman, Colorado State University

In experimental systems involving diffusing and convecting vapors that react to form solid particulates, a complex sequence of nucleation and growth reactions produces pulsing charged crystals, oscillating fronts, and patterns such as beautiful 3-dimensional structures that we call “microtornadoes”, “microstalagtites”, and “microhurricanes”. We will review the rich history of these experiments, starting with a counterdiffusional experiment that figures in the pioneering work on diffusion of Dalton, Graham, Fick, and Stefan. Mathematical analysis will progress from a reaction-diffusion model for the origins of the initial reaction zone, to an analysis of oscillations and particle size distributions, to a fluid dynamical model. The insights carry over to similar structures in protein crystallization experiments and the formation of periodic structures in plants.

Tue Mar 26

Dynamical Systems

2:30pm - Vincent Hall 313
Ice issues in conceptual climate models
Alice Nadeau, University of Minnesota

Conceptual climate models are a necessary tool for scientists trying to understand Earth and other rocky planets because they can provide insight on predominant forces affecting a planet's climate. This talk will focus on Budyko-Sellers type energy balance models, a particular class of conceptual models used to study ice-albedo feedback in the climate system. In this talk I will discuss the different ways one can represent ice in these models, including some of my recent results in extending these models to other planets.

Tue Mar 05

Dynamical Systems

2:30pm - Vincent Hall 313
Measuring attractor strength using bounded, nonautonomous control
Kate Meyer, UMN

A topological definition of an attractor leaves out metric information relevant to modeling real-world systems, particularly how far the attractor persists against perturbations and error. This talk will review some existing approaches to measuring the strength of an attractor in metric terms and will introduce the quantity “intensity” to generalize basin steepness to systems of autonomous ODEs in arbitrary dimension. One can compute an attractor’s intensity by probing a domain of attraction with bounded, non-autonomous control and tracking the sets reachable from the attractor. A connection between reachable sets and isolating blocks implies that an attractor’s intensity not only reflects its capacity to retain solutions under time-varying perturbations, but also gives a lower bound on the distance the attractor continues in the space of vector fields.
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Tue Feb 26

Dynamical Systems

2:30pm - Vincent Hall 313
Spectral Consequences of Hidden Symmetry in Network Dynamical Systems
Eddie Nijholt, UIUC

Network dynamical systems play an important role in many fields of science; whenever there are agents whose time evolution is linked through some interaction structure, we may view the system as a network and model it accordingly. However, despite their prevalence, network dynamical systems are in general not well understood. One can identify two reasons for this. First of all, many coordinate changes and other transformations from well-known dynamical systems techniques do not respect the underlying network structure. Second of all, despite this somewhat `ethereal' character, systems with a network structure often display behavior that is highly anomalous for general dynamical systems. Examples of this include very unusual bifurcation scenarios and high spectral degeneracies. As a possible explanation of this, it can be shown that a large class of network ODEs admit hidden symmetry, which may be discovered through the so-called fundamental network construction. In most cases, this underlying symmetry does not come from a group though, but rather from a more general algebraic structure such as a monoid or category. I will show how the fundamental network allows one to adapt techniques from dynamical systems theory to a network setting, and how some of the more unusual properties of networks may be explained. In doing so, I will mostly focus on spectral properties of linear network maps.
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