Abstracts and lecture notes for contributed talks:

• Margaret Beck (BU) Stability of patterns in reaction-diffusion equations
  
  
  • Reaction-diffusion equations model a wide variety of chemical and biological processes. Such systems are well known for exhibiting patterns, such as traveling waves and spatially- and/or temporally-periodic structures. One important property of such solutions is whether or not they are stable, which is im- portant because it is typically only the stable solutions that are observed in real world settings. In this talk, I will discus the difference between spectral, linear, and nonlinear stability, and highlight some key methods for analyzing stability.
  
  
• Jason Bramburger (Brown) Rotating Wave Solutions to Lattice Dynamical Systems
  
  
  • Examples of rotating waves abound in nature and have been an intense area of rigorous mathematical investigation for many decades now, particularly due to their association with electrophysiological pathologies. Much of the rigorous mathematical investigations of rotating waves rely on the model exhibiting a continuous Euclidean symmetry invariance, which is only present in a mathematically idealized situation. In this talk we investigate the existence of rotationally propagating solutions in a discrete spatial setting, for which typical symmetry methods cannot be applied. That is, we inspect an infinite system of coupled lambda-omega differential equations indexed by the two-dimensional integer lattice and demonstrate that there exists a rotating wave solution. It is demonstrated that the dimensionality of the problem poses unique problems in that traditional techniques from systems of finitely many differential equations cannot be directly applied. In particular, the existence proof requires extensive results from Banach space theory and an application of a non-standard implicit function theorem. Some brief remarks on the history and relevance of lambda-omega systems are also provided to properly frame the results.
  
  
• Eric Chang (BU) A Sierpinski Mandelbrot Spiral
  
  
  • The Sierpinski carpet fractal and Mandelbrot set are fascinating images with diverse applications such as transport through porous media, data compression, and measuring the coast of Brittany. We identify three new structures in the parameter plane for singularly perturbed rational maps in the family $z^n + \lambda / z^d$ for z and $\lambda$ complex, n at least 4 is even, and d at least 3 is odd. There exist two different types of arcs of infinitely many alternating Sierpinski holes and Mandelbrot sets. Furthermore, there exists a spiral of infinitely many arcs of one type passing through a single arc of the other type.
  
  
• Eric Cooper (BU) Selection of quasi-stationary states in 2d Stochastic Navier-Stokes on the symmetric and asymmetric torus
  
  
  • Recent numerical studies have revealed certain families of functions play a crucial role in the long time behavior of 2D Navier-Stokes with periodic boundary conditions. These functions, called bar and dipole states, exist as quasi-stationary solutions and attract trajectories of all other initial conditions exponentially fast. If the domain is the symmetric torus, then the dipole states dominate, while on the asymmetric torus the bar states dominate. Using the vorticity equation, we will formulate a mathematical framework to explain the evolution of solutions toward these metastable states in both a deterministic and stochastic setting.
  
  
• Tim Faver (Drexel) Nanopteron-stegoton traveling waves in mass and spring dimer Fermi-Pasta-Ulam-Tsingou lattices
  
  
  • The Fermi-Pasta-Ulam-Tsingou (FPUT) lattice is an infinite chain of particles connected by springs and constrained to move on a horizontal line. We present results on the existence and properties of traveling wave solutions to the lattice equations of motion for two classes of heterogeneous FPUT lattices: the mass dimer, consisting of alternating particles and identical springs, and the spring dimer, which has identical masses but alternating spring forces. For both dimers, our traveling waves are nanopterons; unlike the classical solitary wave that decays to zero at infinity, the nanopteron is asymptotic to periodic waves with extremely small amplitude. While we employ the same bifurcation and quantitative contraction mapping methods for both dimers, we encounter some subtle differences between the dimers in the precise techniques and results.
  
  
• David Fried (BU) Riley groups and Caruso semigroups
  
  
  • The Julia set of a semigroup of Mobius transformations can sometimes be expressed as the limit set of a Kleinian group. We illustrate this with some two-generator semigroups introduced by Caruso, which describe a random dynamics variant of the Fibonacci sequence. This is joint work with Sebastian Marotta and Rich Stankewitz.
  
  
• Ryan Goh (BU) Pattern-forming fronts in a Swift-Hohenberg equation with directional quenching
  
  
  • Pattern formation in the wake of traveling inhomogeneities has become an interesting line of research in various fields, such as vegetation patterns, phase separation, and tissue patterning. Such "quenching" or "growth processes" can be encoded mathematically in a step-like parameter which allows patterns in a half plane, and suppresses them in a complement, while the boundary of the pattern forming region propogates with fixed normal velocity. A natural first question is: How does the speed of the boundary affect, or select, the pattern formed in its wake.

    In this talk, we will discuss such phenomena in the framework of the Swift-Hohenberg equation, a prototypical model of pattern formation. In one spatial dimension, we will show how techniques from infinite-dimension dynamics, such as center-manifold reduction and invariant foliations, as well as Melnikov integrals, can be used to prove existence of stripe-forming front solutions. Time permiting, we will also discuss how functional analytic techniques can be used to study pattern formation in two spatial dimensions.

  
  
• Mareike Haberichter (UMass Amherst) Squeezing Skyrmions
  
  
  • We investigate the role of pressure in a generalised Skyrme model. We introduce pressure as the trace of the spatial part of the energy-momentum tensor and show that it obeys the usual thermodynamical relation. Then, we compute analytically the mean-field equation of state in the high and medium pressure regimes by applying topological bounds on compact domains.

    The equation of state is further investigated numerically for the charge one skyrmions. We identify which term in a generalised Skyrme model is responsible for which part in the equation of state. This is further compared with the Walecka model.

  
  
• Kanji Inui (Kyoto U.) Values of Hausdorff measure and packing measure of the limit sets of infinite conformal IFSs related to complex continued fractions
  
  
  • Many famous fractal sets (for example, Cantor set, Sierpinski gasket and so on) are defined as the limit sets of contractive iterated function systems (for short IFSs) with finitely many mappings. But, recently D. Mauldin and M. Urbanski studied limit sets of conformal IFSs(for short CIFSs) with infinitely many mappings. And, they showed that there exists a CIFS such that the Hausdorff measure of the limit set corresponding to the Hausdorff dimension is zero and the packing measure of the limit set corresponding to the same dimension is positive. Note that the limit sets of CIFSs with finitely many mappings do not have the above properties. In this talk, we introduce an analytic family of CIFSs with infinitely many mappings related to complex continued fractions such that the limit set of each system in the family has the above strange properties and such that the Hausdorff dimension of the limit set is a real analytic and subharmonic function of the parameter. This study is a joint work with Hiroki SUMI (Kyoto University) and Hikaru OKADA (Osaka University).
  
  
• Isao Ishikawa (RIKEN Center for Advanced Intelligence Project) An invariant for the comparison of composition operators for reproducing kernel Hilbert spaces
  
  
  • Classifying time series datas (sequences in the Euclidean space) is important problem for the real world. Our task is giving a "good" mathematical invariant to classify them. Dynamical system is one of the effective mathematical model to analyze them, namely, we assume each time series data is generated by a dynamical system. In terms of the theory of reproducing kernel Hilbert spaces (RKHS), the time series datas and the dynamical systems are regarded as sequences of vectors and composition operators in RKHS, respectively. We define an invariant to measure "distance" between two dynamical systems (composition operators) in the framework of RKHS. This invariant has an explicit formula which can be easily implemented, on the other hand it includes mathematically difficult problems such as the bounded-ness of composition operators. This study is the joint work with Keisuke Fujii (RIKEN AIP), Masahiro Ikeda (RIKEN AIP/ Keio Univ), Yuka Hashimoto (RIKEN AIP/ Keio Univ), and Yoshinobu Kawahara (RIKEN AIP/ Osaka Univ.)
  
  
• Hiroyasu Izeki (Keio) Equivariant harmonic maps and rigidity of discrete groups
  
  
  • Harmonic maps have been a usefule tool for the study of group actions on nonpositively curved spaces. In this talk I present a sort of gap theorem concerning energy growth of equivariant harmonic maps from discrete groups into nonpositively curved spaces. As an application, I will mention a rigidity result of higher rank lattices.
  
  
• Yuika Kajihara (Kyoto U.) Variational proof of the existence of brake orbits in the planer 2-center problem
  
  
  • The purpose of the talk is to show the existence of symmetric brake orbits in the 2-center problem. We first start with introduction to state the background and the main result. Next we provide the variational formulation and show the existence of a minimizer. Finally we show that the minimizer is neither trivial solutions nor collision solutions for some cases by investigating the properties of the value of the action functional.
  
  
• Tasso Kaper (BU) Torus canards and amplitude-modulated bursting
  
  
  • In this talk, the phenomena of canards in fast-slow (multi-scale) systems of ODEs, such as the van der Pol equation, will be reviewed briefly. Then, the relatively recent phenomena of torus canards, discovered by Mark Kramer, will be introduced. Torus canards arise naturally (and must occur topologically) in the transitions between periodic oscillations (spiking) and bursting in a broad array of fast-slow models in neuroscience, chemistry, and nonlinear systems. One of the highlights of torus canard theory is a novel class of bursting rhythms, called amplitude-modulated bursting (AMB), discovered by Theo Vo. AMB will be illustrated on a model of intracellular calcium dynamics, and it will be shown that there is a novel family of singularities, called toral folded singularities, which exist generically in slow/fast systems with two or more slow variables and which are the organizing centers for AMB. The results are based on joint articles with A. Barry, G.N. Benes, J. Burke, M. Desroches, M. Kramer, M. Krupa, and T. Vo.
  
  
• Tomoki Kawahira (Tokyo Inst. of Technology) Almost affine copies of the Julia sets in the Mandelbrot set
  
  
  • We show that there are quasiconformal copies of the Cantor Julia sets with parameters arbitrarily close to a parabolic or Misiurewicz parameter embedded in the boundary of the Mandelbrot set. In particular, these copies can be "superfine". More precisely, the dilatation of such a quasiconformal embedding can be arbitrarily close to one, and the inverse of the embedding can be arbitrarily close to an affine map. (This is partially a joint work with Masashi Kisaka at Kyoto University.)
  
  
• Kai Koike (Keio U.) Rigid body motion in a special Lorentz gas
  
  
  • Imagine a ball thrown in the air; the ball feels the resistance of the air. And the microscopic origin of the resistance is the collision between the air molecules and the ball. Our question is: can we relate this microscopic dynamics to the macroscopic dynamics of the ball? This is the theme of our study. We take the kinetic theory of gases viewpoint. In this talk, I present a recent result on the rigid body motion in a special Lorentz gas.
  
  
• Katsuhiro Miguchi (Osaka U.) Decorated enhanced Teichmuller space
  
  
  • Some extended Teichmuller spaces are studied to investigate the moduli spaces of hyperbolic structures for punctured surfaces. Decorated Teichmuller space is the moduli space respecting the decoration that is useful for investigating the original Teichmuller space. Enhanced Teichmuller space is the space of hyperbolic surfaces with boundaries which have signed length. In this talk I will define the decorated enhanced Teichmuller space, constract the lambda-length coordinates of this space, and describe its applications.
  
  
• Yoshitaka Saiki (Hitotsubashi U.) Machine-learning prediction of fluid variables
  
  
  • We predict both microscopic and macroscopic variables of a fluid flow using reservoir systems constructed from data. In our procedure of the prediction, we assume no prior knowledge of physical process of a fluid flow except that its behavior is complex but deterministic. We present two ways of prediction of the complex behavior; the first called partial-prediction requires continued knowledge of partial time-series data during the prediction as well as past time-series data, while the second called full-prediction requires only past time-series data as learning data. For the first case, we are able to predict a long-time motion of microscopic fluid variables. For the second case, we show that the reservoir dynamics constructed from only past data of energy functions can predict a future behavior of energy functions and reproduce the energy spectrum. This implies that the obtained reservoir system constructed without the knowledge of microscopic variables is almost equivalent to the dynamical system describing a macroscopic behavior of energy functions. This is the joint work with Kengo Nakai (The University of Tokyo).
  
  
• Farruh Shahidi (Penn State) Area preserving surface diffeomorphisms with polynomial decay of correlations are ubiquitous
  
  
  • We show that any surface admits an area preserving $C^{1+\alpha}$ diffeomorphism with non-zero Lyapunov exponents which is Bernoulli and has polynomial decay of correlations. We establish both upper and lower polynomial bounds on correlations. In addition, we show that this diffeomorphism satisfies the Central Limit Theorem and has the Large Deviation Property.
  
  
• Mitsuru Shibayama (Kyoto) Variational construction of periodic and connecting orbits in the planar Sitnikov problem
  
  
  • We consider the existence of solution in the planar Sitnikov problem, realizing given symbolic sequences by based on variational method. We also prove the existence of various periodic solutions and connecting orbits between them.
  
  
• Mao Shinoda (Keio) On non-convergence of equilibrium measures at zero temperature limit
  
  
  • We consider the sequence of equilibrium measures for a given potential parametrized by temperature. Temperature controls ordered and disordered powers, the potential and entropy. The lower temperature the goes, the more the potential effects strengthen. In this talk we pay attention to behavior of equilibrium measures as the temperature goes to zero. A fundamental problem in the zero temperature limit is the convergence of equilibrium measures. In the one-dimensional case, the sequence of equilibrium measures for a finite range function converges. However in the high-dimensional case, there exists a finite range function whose sequence of equilibrium measures does not converge. We construct such a finite range function in dimension two by imbedding a one-dimensional effective subshift into a two-dimensional subshift of finite type.
  
  
• Hayate Suda (U. Tokyo) Superdiffusion of energy in a chain of harmonic oscillators with noise in a magnetic field
  
  
  • Anomalous heat transport (violation of the FourierbOn non-convergence of equilibrium measures at zero temperature limits law) and the corresponding superdiffusion of energy have been observed numerically in a one-dimensional chains of anharmonic oscillators. There is a lot of numerical experiments and heuristic arguments about the anomalous behavior, but very few mathematical rigorous results have been obtained until now. Recently, it was rigorously shown that a chain of harmonic oscillators perturbed by a stochastic dynamics conserving total momentum and total energy has diverging thermal conductivity, which implies the anomalous heat transport. Currently, chains of harmonic oscillators with conservative stochastic noise are considered as good approximations of some anharmonic chains. In this talk, we consider a chain of harmonic oscillators with conservative stochastic noise in a magnetic field. We show that the anomalous heat transport and the superdiffusion of energy in this magnetic model. Especially, we prove that the time evolution of the macroscopic energy distribution is governed by some 5/6-fractional diffusion equation. This talk is based on joint works with Keiji Saito (Keio) and Makiko Sasada (U. Tokyo).
  
  
• Tomoharu Suda (U. Tokyo) Applications of the Helmholtz-Hodge decomposition to the study of vector fields
  
  
  • The Helmholtz-Hodge decomposition (HHD) is a decomposition of vector fields whereby they are expressed as the sum of a gradient vector field and a divergence-free vector field. In this talk, I will first summarize the definition and basic properties of HHD. Then I will introduce my work on its applications to the study of vector fields.
  
  
• Hiroki Takahasi (Keio U.) Large Deviation Principle for arithmetic functions in continued fraction expansion
  
  
  • Khinchin proved that the arithmetic mean of continued fraction digits of Lebesgue almost every irrational number in $(0,1)$ diverges to infinity. Hence, none of the classical limit theorems such as the weak and strong laws of large numbers or central limit theorems hold. Nevertheless, we prove the existence of a large deviations rate function which estimates exponential probabilities with which the arithmetic mean of digits stays away from infinity. This leads us to a contradiction to the widely-shared view that the LDP is a refinement of the weak or strong laws of large numbers: the former can be more universal than the latter.
  
  
• Hisayoshi Toyokawa (Hokkaido) Asymptotic behavior of the iterate of weakly almost periodic Markov operators and invariant densities
  
  
  • For a nonsingular transformation on a probability space, we establish equivalent conditions for the existence of an absolutely continuous $\sigma$-finite invariant measure whose support eventually contains almost all trajectories. Further, we extend this result to certain random dynamics.
  
  
• Nick Wadleigh (Technion) Shrinking targets and Diophantine approximation
  
  
  • Dirichlet's Theorem states that for a real mxn matrix A, ||Aq+p||^m \leq t, ||q||^n < t has nontrivial integer solutions for all t > 1. Davenport and Schmidt have observed that if 1/t is replaced with c/t, c<1, almost no A has the property that there exist solutions for all sufficiently large t. Replacing c/t with an arbitrary function, it's natural to ask when precisely does the decay of the function prohibit almost all A from having this property. In the case m=1=n, we give an answer using dynamics of continued fractions. We then discuss an approach to higher dimensions based on dynamics on the space of lattices. Where this approach meets an obstruction, a similar approach to the analogous *inhomogeneous* approximation problem will succeed. Joint work with Dmitry Kleinbock.
  
  
• Takayuki Watanabe (Kyoto U.) Random holomorphic dynamics of Markov systems
  
  
  • We consider random holomorphic dynamical systems on the Riemann sphere whose choices of maps are related to a Markov chain. Our motivation is to generalize the facts which hold in i.i.d. random holomorphic dynamical systems. Specifically, we focus on the function T which represents the probability of tending to infinity. We show some sufficient conditions which make T continuous on the whole space and we characterize the Julia sets in terms of the function T under some assumptions. This is a joint work with Hiroki Sumi (Kyoto University).
  
  
• Gene Wayne (BU) Dynamical systems and the asymptotics of the Navier-Stokes equations
  Also, click here for a survey article.
  
  
  • In this lecture I will describe how one can use ideas of dynamical systems theory to give a quite complete picture of the long time asymptotics of solutions of the two-dimensional Navier-Stokes equation. I will discuss the existence and properties of invariant manifolds for dynamical systems defined on Banach spaces and review the theory of Lyapunov functions, again concentrating on the aspects of the theory most relevant to infinite dimensional dynamics. I will then explain how one can apply both of these techniques to the two-dimensional Navier- Stokes equation to prove that any solution with integrable initial vorticity will will be asymptotic to a single, explicitly computable solution known as an Oseen vortex equations.
  
  
• Roland Welter (BU) Decay profiles of a linear system associated with the compressible Navier Stokes equations
  
  
  • In their 1995 paper, Hoff and Zumbrun showed that the leading order asymptotic behavior of suitably small solutions to the compressible Navier Stokes equations is governed by a much simpler linear artificial viscosity system. We give an asymptotic expansion for solutions of the artificial viscosity system to any inverse power of time by combining techniques of Gallay and Wayne with those of Constantin.
  
  
• Xueying Yu (UMass Amherst) Global well-posedness and scattering for the quintic NLS in two dimensions
  
  
  • We consider the Cauchy initial value problem for the defocusing quintic nonlinear Schroedinger equation in R^2 with general data in the critical space \dot H^{1/2} (R^2). We show that if a solution remains bounded in \dot H^{1/2} (R^2) in its maximal interval of existence, then the interval is infinite and the solution scatters.