## Dynamical Systems Seminars Spring 2019

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 4 No seminar

• February 11 Shugan Ji
Title: Time periodic solutions of nonlinear wave equation with x-dependent coefficients
Abstract: In this talk, we consider the time periodic solutions of nonlinear wave equation with x-dependent coefficients. Such a model arises from the forced vibrations of an inhomogeneous string and the propagation of seismic waves in nonisotropic media. We shall talk about some existence results on the time periodic solutions of such a model with different types of nonlinearity.

• February 18 No Seminar, President's Day

• February 25 Seminar Cancelled
Title: TBA
Abstract: TBA

• March 4 Michael Bialy (Tel Aviv University); Note: The tea and seminar will be held in MCS B33.
Title: Around Birkhoff's conjecture for convex and other billiards
Abstract: Birkhoff's conjecture states that the only integrable billiards in the plane are ellipses. I am going to give a survey of recent progress in this conjecture and to discuss geometric results and questions around it. Based on joint works with Andrey E. Mironov, Maxim Arnold. No prior knowledge of the subject will be assumed.
• March 11 No Seminar, Spring Break
• March 18 Jonathan Jacquette (Brandeis)
Title: Computer assisted proofs of Wright's and Jones' Conjectures: Counting and discounting slowly oscillating periodic solutions to a delay differential equation
Abstract: A classical example of a nonlinear delay differential equations is Wright's equation: $y'(t) = \alpha y(t-1) [1+y(t)]$, considering $\alpha >0$ and $y(t)>-1$. This talk discusses two conjectures associated with this equation: Wright's conjecture, which states that the origin is the global attractor for all $\alpha \in ( 0 , \pi /2 ]$; and Jones' conjecture, which states that there is a unique slowly oscillating periodic solution for $\alpha > \pi / 2$. To prove Wright's conjecture our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at $\alpha = \pi /2$. Using a rigorous numerical integrator we characterize slowly oscillating periodic solutions and calculate their stability, proving Jones' conjecture for $\alpha \in [1.9,6.0]$ and thereby all $\alpha\geq 1.9$. We complete the proof of Jones conjecture using global optimization methods, extended to treat infinite dimensional problems.

• March 25 Matt Holzer (George Mason University)
Title: Invasion fronts in spatially extended systems
Abstract: Invasion fronts refer to fronts propagating into unstable states. An important characteristic of such fronts is the speed at which they propagate. In this talk, I will review wavespeed selection principles and discuss two recent research projects related to wavespeed selection. In the first, we study the emergence of locked fronts in a class of two component reaction diffusion equations. In the second, a model of global epidemics is considered and predictions for arrival times are obtained based upon the linearization about the unstable steady state.

• April 1 Margaret Beck (Boston University)
Title: Spectral Stability, Spatial Dynamics, and the Maslov Index
Abstract: Understanding the stability of solutions to PDEs is important, because it is typically only stable solutions which are observable. For many PDEs in one spatial dimension, stability is well-understood, largely due to a formulation of the problem in terms of so-called spatial dynamics, where one views the single spatial variable as a time-like evolution variable. This allows for many powerful techniques from the theory of dynamical systems to be applied. In higher spatial dimensions, this perspective is not clearly applicable. In this talk, I will discuss recent work that suggests both that the Maslov index could be a important tool for understanding stability when the system has a symplectic structure, particularly in the multi-dimensional setting, and also suggests a possible analogue of spatial dynamics in the multi-dimensional setting.

• April 8 Cynthia A. Bradham (Boston University)
Title: The Molecular Basis of Skeletal Patterning
Abstract: During embryonic development, a single cell, the fertilized egg, gives rise to a complete animal. Understanding how tissues are sculpted and patterned during embryonic development is important for the prevention of birth defects and to inform tissue engineering and regenerative medicine, but the mechanisms that drive developmental patterning processes remain largely unknown due to the complexity of the question. Because the mechanisms underlying development are well-conserved in evolution, much can be learned from studying the patterning mechanisms of simple organisms such as sea urchins. The sea urchin larval skeleton serves as a simple model of patterning. The skeleton is secreted by primary mesenchymal cells (PMCs), while the patterning information is contained within the ectoderm, and sensed by the migrating PMCs. Using high throughput sequencing, we performed a screen to identify novel skeletal patterning cues that are expressed by the ectoderm. Collectively, we’ve discovered conserved cues required for patterning most of the major skeletal regions. We are now addressing how the PMCs interpret the ectodermal patterning cues using high throughput sequencing at the single cell resolution.
• April 15 No Seminar - Patriot's Day
• April 22 Luigi Chierchia (University of Roma Tre)
Title: Generic topological and geometrical structure of nearly-integrable Hamiltonian systems at simple resonances
Abstract: As well known, the dynamics of nearly-integrable Hamiltonian systems is strongly affected by resonances. Particularly relevant (e.g., in the study of Arnold diffusion, or in the metric theory of regular motions) are simple resonances. We describe the analytic, geometric and topological structure of simple resonances for nearly-integrable analytic Hamiltonian systems, establishing, in particular, "sharp" analytic normal forms for generic holomorphic perturbations.
• April 29 Bastian Hilder (Universität Stuttgart)
Title: Modulating traveling fronts in pattern forming systems with conserved quantities
Abstract: In this talk, I will consider a Swift-Hohenberg equation coupled to a conservation law. As a parameter increases this system undergoes a Turing bifurcation and small periodic solutions emerge. The first part of the talk is concerned with the existence of modulating traveling fronts near the bifurcation. These fronts describe an invasion of the unstable ground state by the periodic pattern and provide a mechanism of pattern formation. In the second part, I will discuss the stability of fronts in the corresponding amplitude equation, a modified Ginzburg-Landau system with a conservation law. This is a natural first step towards a stability result for the full problem.
Please also consider attending the Brown/BU/UMassAmherst seminar in PDE and Dynamics on May 3rd.