Dynamical Systems Seminars

Fall 2015

The Dynamical Systems seminar is held on Monday afternoon at 4:00 PM in MCS 148. Tea beforehand at 3:45 PM in MCS 241.

• September 14: Blake Barker (Brown University)
  
  Title: Rigorous Verification of Stability of Traveling Waves Via Computer Assisted Proof
  
  Abstract: We discuss various aspects of numerical stability analysis of periodic roll wave solutions arising in equations of inclined thin film flow, with an eye toward the development of guaranteed error bounds. In particular, we rigorously verify stability of a family of periodic wave solutions arising in a generalized Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit, that is, in the limit viscosity goes to zero.

• September 21: Moon Duchin (Tufts)
  
  Title: Recent developments in rational billiards
  
  Abstract: A very classical question is to consider a polygon whose angles are rational multiples of pi and study the trajectories under the billiard flow (straight lines inside the polygon with optical reflection in the edges). An obvious question is which directions have periodic trajectories or, at the other extreme, are uniquely ergodic. Billiards tie in to Teichmüller theory (the study of varying geometric structures on a fixed topological surface) in a beautiful way, and some recent work in Teichmüller dynamics has been much celebrated (for instance, it's a primary interest for two of last summer's four Fields medalists!). I'll survey a small corner of this research area and discuss some of my own work in this direction.

• September 28: John Burke (Applied BioMath)
  
  Title: Quantitative Systems Pharmacology: Enabling drug invention from Research through Clinical Trials
  
  Abstract: Quantitative Systems Pharmacology (QSP) modeling has been used successfully in pharma. The cases include: feasibility assessments; prediction of best-in-class drug properties; and clinical trial support. In each case, systems approaches enabled decisions that traditional methods could not address as easily or as early in the pipeline. Unlike traditional PK/PD models, QSP models leverage known biophysical interactions and integrate data from a variety of sources (in vitro, in vivo, and clinical)and act as a central repository of data and knowledge of human pathomechanisms, allowing for the exploration of hypotheses that cannot be fully tested prior to dosing patients. These case studies show the potential of QSP approaches to enable earlier quantitative decision making with the end goal of reducing late stage attrition and accelerating premier therapeutics to address unmet medical need.

• October 5: Josep M. (Pitu) Cors (Universitat Politècnica de Catalunya)
  
  Title: Theory of sundials
  
  Abstract: 2015 is the international year of light. What better illustrates the use of light than a sundial? Sundial caches the apparent position of the Sun in the sky. Like a Pinhole Camera the light from the Sun passes through the gnomon, often a thin rod, and projects the shadow on a surface. All knowledge about Earth orbit (Kepler laws) and spherical astronomy or positional astronomy are used. After presenting some terminology and different types of sundials, we will talk about time (apparent solar time, mean solar time, ...), Celestial sphere (apparent motion of the Sun, ...) and Celestial coordinate (Horizontal, Equatorial, ...). All well mixed leading to the theory of sundials.

• October 12: Columbus Day, No Seminar

• October 19: Kostas Spiliopoulos (BU)
  
  Title: Hypoelliptic multiscale Langevin diffusions: Large deviations, invariant measures and small mass asymptotics
  
  Abstract: We consider a general class of hypoelliptic Langevin diffusions and study two related questions. The first one is large deviations for hypoelliptic multiscale diffusions. The second one is small mass asymptotics of the invariant measure corresponding to hypoelliptic Langevin operators and of related hypoelliptic PDEs. The invariant measure corresponding to the hypoelliptic problem and appropriate hypoelliptic PDEs enter the large deviations rate function due to the multiscale effects. Based on the small mass asymptotics we derive that the large deviations behavior of the multiscale hypoelliptic diffusion is consistent with the large deviations behavior of its overdamped counterpart. Additionally, we rigorously obtain an asymptotic expansion of the solution to the related hypoelliptic PDEs with respect to the mass parameter, characterizing the order of convergence. The proof of convergence of invariant measures is of independent interest, as it involves an improvement of the hypocoercivity result for the kinetic Fokker-Planck equation. We do not restrict attention to gradient drifts and our proof provides explicit information on the dependence of the bounds of interest in terms of the mass parameter.

• October 26: Scott Sutherland (Stony Brook)
  
  Title: On the measure of the Feigenbaum Julia set.
  
  Abstract: We discuss a computer-assisted proof that the Julia Set of the map $z \mapsto z^2 -1.4011551890\cdots$, which is the limit of period doubling bifurcations in the quadratic family, has zero Lebesgue measure. This is joint work with Artem Dudko.

• November 2:
  
  Title:
  Abstract:

• November 9: Monica Morena Rocha (CIMAT, Mexico and UNC Chapel Hill)
  
  Title: On the dynamics of perturbed Weierstrass P functions over real square lattices
  
  Abstract: Consider the Weierstrass P function defined over a fixed real square lattice. The dynamical system obtained by iteration of P can be completely characterized by the behavior of its single free critical orbit. In contrast, as soon as the function is perturbed by the addition of a complex parameter, the elliptic function P+b exhibits two free critical orbits, which in principle, complicates the study of the connectedness and bifurcation loci.
  
  This talk presents results that explain some of the rich structures found in parameter plane. In particular, we concentrate our study along horizontal lines of b-values where the family P+b exhibits a variety of dynamical behavior. Indeed, these lines may contain parameters for which the Julia set of P+b is either the whole complex plane, or it is connected with nonempty Fatou set, or it is totally disconnected. We also show that in the instance of a connected Julia set with nonempty Fatou set, varying the parameter along a particular horizontal line gives rise to both period-doubling and reversed period-doubling bifurcations.
  
  This is a joint work with Jane M. Hawkins, UNC-Chapel Hill.

• November 16: Russell Lodge (Jacobs University, Bremen)
  
  Title: A combinatorial classification of postcritically finite Newton maps

• November 23: Assaf Amitai (MIT)
  
  Title: The mean encounter time between two polymer sites: a Brownian search process in high dimensional manifolds
  
  Abstract: The encounter between two sites on chromosomes in the cell nucleus can trigger gene regulation, exchange of genetic material and repair of DNA breaks. By forming a DNA loop, a protein located on the DNA can trigger the expression of a gene located far along the chain. We computed asymptotically the mean time encounter time (MFET) between two polymer sites, using the classical Rouse polymer model, in which the polymer is described as a collection of bead monomers connected by harmonic springs. When two monomers come closer than a distance epsilon, the search process ends.
  
  The novel asymptotic relies on the expansion of the spectrum of the Fokker-Planck operator as a function of the small parameter epsilon. The key of the asymptotic is the explicit computation of the Riemannian volume for Chavel-Feldman formula, which gives the shift in the spectrum of the Laplacian operator when Dirichlet boundary conditions are imposed on the boundary of tubular neighborhood of a constraint manifold (removal of a manifold with a small volume). We have shown that the encounter time is Poissonian and obtained an asymptotic expression of the MFET as function of the polymer length N and epsilon. The MFET is inversely proportional to the first eigenvalue of the FP-operator but the expansion is not uniform in epsilon and N. Finally, we show that the asymptotic estimation of the eigenvalue can be obtained by a generalized boundary layer analysis in high dimension.

• November 30: John MacLean (UNC Chapel Hill)
  
  Title: Numerical Multiscale Methods and Lagrangian Data Assimilation
  
  Abstract: Many problems in science and engineering are modelled by time scale separated differential equations. These systems can rarely be solved analytically; at the same time, the fast processes restrict the time step of numerical integrators so that the (high dimensional) system can only be simulated on short time scales. We are interested in the dynamics of slow processes that evolve on long time scales and so cannot be approximated by standard analytical or numerical methods. This talk focusses on the heterogeneous multiscale and projective integration methods, which efficiently integrate such multiscale systems to simulate the behaviour of the slow variables. I will present convergence results for various formulations of both methods applied to stiff dissipative ordinary differential equations, highlighting recent contributions to the field. I conclude by discussing potential applications of these methods to Lagrangian Data Assimilation, and the pitfalls that must be overcome in order to do so.

• December 7: Tommy McCauley (BU)
  
  Title: The Maslov index and singularities of a matrix Riccati equation
  
  Abstract: In this talk we use the Maslov index as a tool to study stability of steady state solutions to reaction-diffusion equations in one spatial dimension. A steady state solution naturally defines a path of Lagrangian subspaces in $\mathbb{R}^{2n}$, the path of unstable subspaces, and the Maslov index of this path detects spectral instability of the steady state solution. Moreover, we show that this path of subspaces is governed by a matrix Riccati equation. We discuss recent results suggesting the Maslov index can be computed by the singularities of solutions to this matrix Riccati equation.

  

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