Dynamical Systems Seminars

Fall 2024

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.

• September 9th Social Event - BU PUB, 3:45pm
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• September 16th Malavika Mukundan (BU)
  Title: On PSF branched self-covers of the plane
  Abstract: In this talk we will start with this simple question: which branched self-covers of the plane are holomorphic? We will focus on a class of branched covers that satisfy the post-singular finiteness (PSF) property, and consider again the question of when such maps are holomorphic. Such maps are known to have vital importance in complex dynamics, and we will explore this connection.

• September 23rd Karol Bacik (MIT)
  Title: Hidden order in chaotic crowds
  Abstract: At first glance, a crowd of pedestrians crossing a busy plaza may seem disordered, but if you look closely, you can discover hidden patterns. For example, two groups moving in opposite directions on a crosswalk spontaneously segregate into contraflowing lanes. In my talk, I will introduce a new kinetic theory that gives insight into the physical origin of lane formation and predicts some new dynamical phenomena. Mathematically, the discussion will amount to a sequence of stability problems of increasing complexity. To complement the theoretical analysis, I will also discuss a suite of experiments with human crowds - pretty videos of people wearing little paper hats will be provided.

• September 30th Boris Hasselblatt (Tufts University)
  Title: Escaping billiard horns with angular momentum.
  Abstract: While cusps in billiards have been known to eject a particle after finitely many collisions, this was not known for "horns", where both cheeks are on the same side of the common tangent. It is now. And it turns out that an analog of angular momentum serves as an adiabatic invariant. (Joint with Mark Levi.)

• October 7th Willie Rush Lim (Brown University)
  Title: Critical quasicircle maps
  Abstract: There is an essentially complete renormalization theory for analytic critical circle maps which serves to justify the golden mean universality and other conjectures by physicists. Its development ran parallel with the universality of Feigenbaum constants in the framework of unimodal and quadratic-like maps. In this talk, I will introduce a generalization called "critical quasicircle maps", i.e. analytic self homeomorphisms of a quasicircle with a single critical point. We will sketch the realization of such maps with any prescribed combinatorics not necessarily equivalent to critical circle maps. We will then discuss the state of the art renormalization picture together with its implications on rigidity, universality, and regularity of conjugacy classes.

• October 14th No Seminar - Indigenous People's Day (Brown University)
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• October 21st Rob Benedetto (Amherst College)
  (NOTE this is a joint seminar, held with, and organized by, the BU Number Theory group, it will still be held in CDS 548 at the same time) Title: Aboreal Galois groups with colliding critical points
  Abstract: see here.

• October 28th Katie Slyman (Boston College)
  Title: Tipping in a carbon cycle model
  Abstract: Rate-induced tipping (R-tipping) occurs when a ramp parameter changes rapidly enough to cause the system to tip between co-existing, attracting states, while noise-induced tipping (N-tipping) occurs when there are random transitions between two attractors of the underlying deterministic system. We investigate R-tipping and N-tipping events in a carbonate system in the upper ocean, in which the key objective is understanding how the system undergoes tipping away from a stable fixed point in a bistable regime. While R-tipping away from the fixed point is straightforward, N-tipping poses challenges due to a periodic orbit forming the basin boundary for the attracting fixed point of the underlying deterministic system. Furthermore, in the case of N-tipping, we are interested in the case where noise is away from the small noise limit, as it is more appropriate for the application. We compute the most probable escape path (MPEP) for our system, resulting in a firm grasp on the least action path in an asymmetric system of higher scale. Our analysis shows that the carbon cycle model is susceptible to both tipping mechanisms.

• November 4th Mark van den Bosch (Leiden University)
  Title: Multidimensional Stability of Planar Travelling Waves for Stochastically Perturbed Reaction-Diffusion Systems
  Abstract: Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns? We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations; based on applications. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. In the deterministic setting, multidimensional stability of planar waves on the whole space has been obtained, and we show to what extend we can do this in the stochastic case. The metastability result above is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long-time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

• November 11th Ryan Creedon (Brown University)
  Title: Transverse Instability of Stokes Waves
  Abstract: In 1847, George Stokes derived asymptotic expansions for the small-amplitude, periodic traveling-wave solutions of the full water wave equations. In the years since his work, much has been studied about these waves, including their stability (or lack thereof). Among the foundational results concerning the stability of Stokes waves was discovered by McLean in 1981, who showed via numerical methods that these waves are unstable with respect to transverse perturbations of the initial data. A rigorous proof of this fact has been missing ever since. In this talk, I present the first ever proof of the existence of transverse instabilities of Stokes waves in both finite and infinite depth. In particular, it will be shown that the linearization of the full water wave equations about the Stokes waves admit solutions that grow exponentially in time. The proof is joint work with Huy Nguyen at the University of Maryland and Walter Strauss at Brown University.

• November 12th (Special Department Colloquium) Laura DeMarco (Harvard University)
  Title: The Mandelbrot set: geometry and arithmetic
  Abstract: The Mandelbrot set M has been studied for many years but continues to baffle mathematicians. By definition, M is the set of all complex numbers c for which the orbit of 0 is bounded under iteration of f(z) = z^2 + c. Within M, there is a distinguished subset of "special" parameters, where the orbit of 0 is finite. These parameters are special from both a dynamical and - somewhat surprisingly - a number-theoretic point of view. In this talk, we'll explore how the geometry of these special parameters is controlled by the number theory, and how this relates to the big remaining open questions about M. This is joint work with Myrto Mavraki.

• November 18th Mitch Curran (Auburn University)
  Title: Counting eigenvalues in the cubic fourth-order Nonlinear Schrödinger equation with a non-regular Maslov box.
  Abstract: As Arnol'd pointed out, Sturm's 19th century theorem regarding the oscillation of solutions to a second-order selfadjoint ODE has a topological nature: it describes the rotation of a straight line in the phase space of the equation. The topological ingredient here is the Maslov index, a homotopy invariant which counts signed intersections of a path of Lagrangian planes with a codimension-one subset of the Lagrangian Grassmannian. This talk will be divided into two parts. In the first, I'll explain Arnol'd's interpretation of Sturm's theorem, the main idea being that one can glean spectral information from the geometric structure of an eigenfunction! I'll also provide some background material on the Maslov index, along with a little-known formulation in terms of higher order crossing forms due to Piccione and Tausk. In the second part of the talk, I'll show how these ideas can be used to study the spectral (in)stability of standing wave solutions to a fourth order nonlinear Schrödinger equation with cubic nonlinearity. The main result is a lower bound for the number of real unstable eigenvalues of the linearised operator. Along the way we encounter multiple instances of non-regular crossings with crossing forms of varying ranks of degeneracy; we circumvent this issue directly (without relying on homotopy invariance!) via the aforementioned higher order crossing forms. Based on joint work with Robby Marangell.

• November 25th Ludwig Hoffman (Harvard University)
  Title: Local and global control in biological development
  Abstract: Biological systems are inherently noisy; therefore, mechanisms are necessary to regulate development and ensure robust and reproducible growth. Identifying and understanding these mechanisms is a major task in developmental biology, which has more recently attracted the attention of the (bio)physics community. I will provide a general introduction to these questions and then present two specific theoretical models to understand different aspects of morphogenesis: active topological defects and mechanochemical feedback.

• December 2nd Alex Kapiamba (Brown University)
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• December 9th Insung Park (Stony Brook University)
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