Dynamical Systems Seminars

Spring 2025

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in CCDS 548 . There will be a brief tea in CCDS 548 at 3:45PM.

• January 27th Björn de Rijk (KIT)
  Title: Orbital stability of spatially periodic waves in the Klein-Gordon equation against localized perturbations Abstract: Nonlinear stability for spatially periodic waves in dispersive systems has so far been obtained against co-periodic or subharmonic perturbations by leveraging the Hamiltonian structure or other conserved quantities. However, nonlinear stability against localized perturbations remains largely unexplored. Since the perturbed wave is neither periodic nor localized in this case, the usual conservation laws are unavailable, thus obstructing a standard nonlinear stability argument. In this talk, I present a nonlinear orbital stability result for spatially periodic waves in the Klein-Gordon equation against $L^2$-localized perturbations. This is joint work with Emile Bukieda and Louis Garénaux.
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• February 3rd Mara Freilich (Brown)
  Title: Mixing of biological tracers: beyond turbulent diffusion
  Abstract: Ocean currents shape the distribution and magnitude of microbial populations with cascading influences on the global carbon cycle. In this talk, I will derive a parameterization for the turbulent flux of biological tracers and highlight the role of biological timescales. This parameterization reveals a growth-transport feedback that can generate diversity in phytoplankton community structure at fine scales (1-10 km) and higher net productivity in the presence of community diversity. I will then compare the theoretical results with ship-board observations to examine the interaction between ocean eddy processes and microbial communities revealing how frontal dynamics shape the upper ocean distribution of carbon and microbial communities.

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• February 10th Max Weinreich (Harvard)
  Title: Chaos and periodic orbits in complex billiards
  Abstract: Billiards is a mathematical model for anything that bounces: light, molecules, or the cue ball in the game of pool. In this talk, we will learn how to play billiards over the complex numbers. We prove two results on complex billiards in generic algebraic curves: first, the dynamical degree, or algebraic entropy, has an explicit lower bound; second, the set of periodic points has measure 0, confirming a classical conjecture of Ivrii and Weyl for generic algebraic plane domains.

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• February 17th No Seminar - President's Day (TBD)
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• February 24th Michelle Bartolo (Harvard)
  Title: Patient-Specific Mathematical Modeling of Pulmonary Hypertension
  Abstract: Patient-specific computational modeling provides an efficient, non-invasive method to study complex, multiscale pathologies such as pulmonary hypertension. By integrating medical imaging and hemodynamic data with computational fluid dynamics, we can uncover physiological mechanisms driving disease progression and quantify relationships not directly observable in vivo. In this study, we present a framework for generating patient-specific vascular models from segmentations of pulmonary arterial networks and examine the impact of uncertainty in medical image analysis. Using the one-dimensional cylindrical Navier-Stokes equations, we predict spatial and temporal variations in blood pressure, flow, and vessel deformation. Our model finds that network geometry significantly impacts pulmonary arterial pressure and flow dynamics, and that pulmonary hypertension is associated with decreased shear stress and increased cyclic stretch in the smallest blood vessels.

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• March 3rd Rebecca Winarski (College of the Holy Cross)
  Title: Thurston theory: unifying dynamical and topological
  Abstract: Thurston proved that a non-Lattés branched cover of the sphere to itself is either equivalent to a rational map (that is: conjugate via a mapping class), or has a topological obstruction. The Nielsen–Thurston classification of mapping classes is an analogous theorem in low-dimensional topology. We unify these two theorems with a single proof, further connecting techniques from surface topology and complex dynamics. Moreover, our proof gives a new framework for classifying self-covering spaces of the torus and Lattés maps. This is joint work with Jim Belk and Dan Margalit.

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• March 10th No Seminar - Spring Break (TBD)
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• March 17th Vanessa Matus de la Parra (SUNY-Stony Brook)
  Title: Dynamics of Covering Correspondences.
  Abstract: In this talk, we will describe the dynamics of compositions of deleted covering correspondences. These are a particular case of holomorphic correspondences on the Riemann sphere which yield interesting families studied by Bullett, Penrose and Lomonaco. These families lie in the gap between modularity and weak modularity, and turn out to be “matings” of rational maps and Kleinian groups. The main goal of this talk is to use the mating structure to deduce equidistribution results, as well as finding measures of maximal entropy.In this talk, we will describe the dynamics of compositions of deleted covering correspondences. These are a particular case of holomorphic correspondences on the Riemann sphere which yield interesting families studied by Bullett, Penrose and Lomonaco. These families lie in the gap between modularity and weak modularity, and turn out to be “matings” of rational maps and Kleinian groups. The main goal of this talk is to use the mating structure to deduce equidistribution results, as well as finding measures of maximal entropy.

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• March 24th Fei Cao (U Mass-Amherst)
  Title: Derivation of wealth distributions from biased exchange of money --- Models and results
  Abstract: In this talk, we will illustrate the use of kinetic theory to better understand the time evolution of wealth distribution and their large-scale behavior such as the limiting money distribution and the evolution of wealth inequality (e.g. Gini index). We investigate three types of dynamics denoted unbiased, poor-biased and rich-biased exchange models. At the individual level, one agent is picked randomly based on its wealth and one of its dollars is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individual approaches infinity using both coupling techniques and martingale-based approaches. Equipped with the limit equation, we identify and prove the convergence to specific equilibrium for both the unbiased and poor-biased dynamics. In the rich-biased dynamics however, we observe a more complex behavior where a dispersive wave emerges. Although the dispersive wave is vanishing in time, it's also accumulates all the wealth leading to a Gini approaching 1 (its maximum value). We characterize numerically the behavior of dispersive waves but further analytic investigation is needed to derive such dispersive waves directly from the dynamics. If time allows, we also talk about the possibility of allowing agents to go into debt where a central bank is introduced.

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• March 31st Nicole Buczkowski (WPI)
  Title: Aspects of Two Nonlocal Biharmonic Operators
  Abstract: Nonlocal operators are advantageous in modeling due to their flexibility in handling discontinuities, incorporating nonlocal effects, and modeling a range of interactions through different choices for kernels. Using these operators in models has several applications, notably peridynamics (fracture mechanics). The biharmonic operator appears in many models including deformations of beams and plates. The nonlocal biharmonic operator can be formulated in at least two ways: using a fourth difference operator or iterating the nonlocal Laplacian. In this talk, we discuss various aspects of the two operators, including convergence of the operators to their classical counterparts and the extraction of the ``hinged" boundary value conditions.

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• April 7th No Seminar ()
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• April 14th Jia Shi (MIT)
  Title: Constructing finite time singularities: Non-radial implosion for com- pressible Euler, Navier-Stokes and defocusing NLS in $T^d$ and $R^d$.
  Abstract: In this talk I will introduce the implosion blow-up results for the compressible Euler, the compressible Navier-Stokes equations and the defocusing nonlinear Schrodinger equation. We will discuss the existence of self-similar solutions and the stability near those solutions. During the talk I will mention our work with Gonzalo Cao-Labora, Javier G´omez-Serrano and Gigliola Staffilani on the first non-radial implosion result for those three equations.

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• April 23rd (BU Monday Schedule) Roland Welter (Hamburg University)
  Title: Rotating Rayleigh-Benard Convection: Energetic Consistency, Attractors and Heat Transport Via a Galerkin Hierarchy
  Abstract: Motivated by the need for energetically consistent parameterizations of convective heat transport, in this presentation I will present recent results which develop a mathematically rigorous framework for studying vertical heat transport in turbulent rotating convection. Starting from the paradigmatic Boussinesq-Oberbeck equations with a Coriolis force, heat transport is investigated via the HKC hierarchy of Galerkin truncated ODE models of increasing dimension. The dynamics of these models are studied for varying Rayleigh and rotation numbers, and particular attention is given to stable values of heat transport, as well as the convergence across models where the models accurately represent the PDE. Implications for energetically consistent parameterization of convection will then be discussed.

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• April 28th C. Eugene Wayne (BU) - The seminar will start early, at 3:30pm.
  Title: Breathers in nonlinear wave equations and "hard" implicit function theorems.
  Abstract: Breathers are temporally periodic, spatially localized, solutions of nonlinear wave equations or other, spatially extended, infinite dimensional dynamical systems. They are rare, because localized oscillations in such systems tend to be destroyed by resonances with the dispersive modes in the system. While breathers have been shown to exist some nonlinear wave equations with spatially periodic coefficients, these examples require rather special conditions on the coefficients. I'll explain why Nash-Moser, or "hard", implicit function theorems are a natural tool to try to expand the class of equations for which breathers exist.

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• May 1st (special time/day, still in CDS 548) Yankl Mazor (Stony Brook U)
  Title: Mating Kleinian groups and (anti-)rational maps via Schwarz reflections and correspondences
  Abstract: In the field of holomorphic dynamics a common problem is that of realizing topological data by holomorphic actions. Mating, or combination, theorems give a class of answers to this general problem, in which one glues together the dynamics of two different actions. Some well known results of this flavor are the Klein and Maskit combination theorems, Bers' simultaneous uniformization theorem, Douady and Hubbard's straightening theorem, and W. Thurston's double limit theorem.

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  I will discuss a joint work with Mikhail Lyubich and Sabyasachi Mukherjee in which we mate degree $d$ (anti-)polynomials with the anti-Hecke group (the group generated by reflections across the sides of an ideal, regular, hyperbolic $d+1$-gon together with rotations of angle $2\pi/d+1$). We will see how to even understand such matings, both as iterated branches of an anti-holomorphic correspondence and as iterations of a Schwarz reflection map. This work forms an anti-holomorphic analogue to holomorphic results which S. Bullett, C. Penrose, L. Lomonaco, and others produced for $d=2$.

  Directions to BU Math Dept.

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  Speakers from previous years

  

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