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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 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.)

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 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.

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.