Dynamical Systems Seminars

Spring 2018

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in MCS 148. Tea beforehand is at 3:45 PM in MCS 144.

- February 12: Jason Bramburger (Brown University)

Title: Snaking in the Swift-Hohenberg Equation in Dimension 1+Epsilon

Abstract: The Swift-Hohenberg equation is a widely studied partial differential equation which is known to support a variety of spatially localized structures. The one-dimensional equation exhibits spatially localized steady-state solutions which give way to a bifurcation structure known as snaking. That is, these solutions bounce between two different values of the bifurcation parameter while ascending in norm. The mechanism that drives snaking in one spatial dimension is now well-understood, but recent numerical investigations indicate that upon moving to two spatial dimensions, the related radially-symmetric spatially-localized solutions take on a significantly different snaking structure which consists of three major components. To understand this transition we apply a dimensional perturbation in an effort to use well-developed methods of perturbation theory and dynamical systems. In particular, we are able to identify key characteristics that lead to the segmentation of the snaking branch and therefore provide insight into how the bifurcation structure changes with the spatial dimension. - February 19: No Seminar - President's Day
- February 26:
Daniel Glasscock (Northeastern)

Title: Multiplicative richness of return times in topological dynamical systems

Abstract: An early result of ErdÅ‘s implies that any syndetic subset (one with bounded gaps) of the natural numbers contains a subset of the form {m,mn}, but we still do not know to this day if syndetic sets contain {m, mn^2}. Narrowing the class of syndetic sets, we can consider return times of points to open sets in minimal topological dynamical systems. In this talk, I will explain some recent partial progress on understanding the multiplicative richness of return times in minimal dynamical systems. In particular, I will show that the set of return times for almost all points to an open set in a totally minimal distal system contains arbitrarily long geometric progressions. This talk is based on ongoing joint work with Andreas Koutsogiannis and Florian Richter. - March 5: No Seminar - Spring Break
- March 12:
Kathryn Lindsey (Boston College)

Title: Shapes of polynomial Julia sets

Abstract: The filled Julia set of a complex polynomial P is the set of points whose orbit under iteration of the map P is bounded. W. Thurston asked "What are the possible shapes of polynomial Julia sets?" For example, is there a polynomial whose Julia set looks like a cat, or your silhouette, or spells out your name? It turns out the answer to all of these is "yes." I will characterize the shapes of polynomial Julia sets and present an algorithm for constructing polynomials whose Julia sets have desired shapes. - March 19: Chad Topaz
(Williams College)

Title: Topological data analysis of collective motion

Abstract: Biological aggregations such as bird flocks, fish schools, and insect swarms are striking examples of self-organized collective motion, and serve as the inspiration for algorithms in robotics, computer science, applied mathematics, and other fields. Aggregations give rise to massive amounts of data, for instance, the position and velocity of each group member at each moment in time during an field observation or numerical simulation. Interpreting this data to characterize the group's dynamics can be a challenge. To this end, we apply computational persistent homology - the workhorse of the field of topological data analysis - to the aggregation models of Vicsek et al. (1995) and D'Orsogna et al. (2006). We assign a topological signature to each set of simulation data. This signature identifies dynamical events that traditional methods do not. Time permitting, we pose open questions related to topological signatures averaged over many simulations of stochastic models, and we use topological signatures to choose between potential models of experimental data. - March 26: Eric Chang (BU)
- April 2: Mattia Serra
(Harvard)

- April 9:
- April 16: No Seminar - Patriot's Day
- April 23:
Qiliang Wu (Ohio University)
- April 30: Bob Devaney (Boston University)

### Directions to BU Math Dept.

### Speakers from previous years