Dynamical Systems Seminars

Fall 2022

The Dynamical Systems seminar is held on Monday afternoons at 4:00 PM in MCS B31 . There will be a brief tea in MCS B24 at 3:45PM. -->

• September 12 Fall Social Event
  All are welcome to come! Light refreshments will be provided.
  Abstract:

• September 19 Patrick Flynn (Brown University)
  Title: The scattering problem for Vlasov-Poisson
  Abstract: The Vlasov-Poisson system is a kinetic model for a continuous density of particles interacting through either Newtonian or Coulombic gravitation. I will describe the scattering problem for this equation, where one must find the asymptotic dynamic as time goes to infinity, and then connect the asymptotic behavior of the solution at time minus infinity to time plus infinity through the so-called scattering map. This model exhibits "modified" scattering, where the asymptotic dynamic is given by the linearized equation, plus an explicit nonlinear correction. To solve the scattering problem, we apply the pseudo-conformal transformation, more widely used in the study of the nonlinear Schrodinger equation. This transformation, which inverts time, allows us to reformulate the scattering problem as a Cauchy problem, which we then solve using Picard iteration. This talk is based on joint work with Benoit Pausader, Zhimeng Ouyang, and Klaus Widmayer.

• September 26 Bengier Ülgen Kılıç (SUNY Buffalo)
  Title: Thresholding and multi-body interactions orient cascades in spatially embedded networks
  Abstract: PDF here!

• October 3rd Natasa Dragovic (Tufts University)
  Title: The Perils Of Political Centrism
  Abstract: In this talk I will present a model of two competing political candidates who shift views opportunistically to maximize their share of the vote. We start with some observations about the model. First, the best strategy for a candidate is often to move towards the other candidate, eventually resulting in two centrists with coalescing views. Second, this strategy ceases to be optimal as soon as sufficiently many voters respond to their candidate’s opportunistic drift towards the center by staying away from the polls altogether. The surprise is that the change in the optimal candidate position can, in certain circumstances, be discontinuous. The underlying mathematical mechanism is a blue-sky bifurcation. This is a joint work with Bruce Boghosian, Christoph Börgers and Anna Haensch.

• October 10 No Seminar - Indiginous People's Day ()
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• October 17 Keith Promislow (Michigan State University)
  Title: Frustration in the Packing of Soft Materials
  Abstract: Many processes in material science involve entropic contributions from packing – the constraints imposed by volume occupied by other material. Diblock polymers offer a rich environment to study the packing of soft materials as gradient flows of a system energy. Ideas from $\Gamma$ convergence provide powerful tools to extract simplified models in certain singular limits. We present examples of packing dichotomies in both continuous and discrete formulations and identify cases in which limiting problems may be more complex. We present a derivation of a random phase reduction of self-consistent mean field models, identify regimes in which they converge to functionalized Cahn-Hilliard energy, and provide a discrete system for the packing of soft balls that exhibits large-system frustration: the inability of gradient flows to obtain the global energy minimum, that significantly complicates the extraction of limiting processes.

• October 24 Javier Gomez Serrano (Brown University/University of Barcelona)
  Title: GENERIC GLOBAL EXISTENCE FOR THE MODIFIED SQG EQUATION
  Abstract: In this talk we will present a construction of global existence of small solutions of the modified SQG equations, close to the disk. The proof uses KAM theory and a Nash-Moser argument, and does not involve any external parameters. We moreover prove that this phenomenon is generic: most solutions satisfy it. Joint work with Alex Ionescu and Jaemin Park.

• October 31 No Seminar ()
  Title:
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• November 7 No Seminar ()
  Please see the following lectures "A mathematical journey through literature" and "The mathematics of perspective in art" at the the Kilichand Honors College, given by Sarah Hart of Gresham College and Birbeck, University of London.
• November 14 Amin Rahman (University of Washington)
  Title: Spatio-temporal models of cancer treatment
  Abstract: Similar phenomena often appear across many different branches of Science. Cancer dynamics often involve interactions of various agents. Similar to Statistical Mechanics, these interactions appear random at the scale of a few agents, but average out at the scale of an entire population. Dynamical models for cancer often describe either transport (of cells, drugs, nutrients, etc.) or averaged population dynamics, but seldom both. By coupling the two processes we are able to model a complete treatment strategy. In this talk we first derive the simplest possible model for drug response. Then we carefully add more complexity as needed to capture realistic scenarios. In doing so we are able to simulate complete treatment strategies, which would be computationally prohibitive for agent-based models.

• November 21 Dallas Albritton (Princeton University)
  Title: Non-uniqueness of Leray solutions to the forced Navier-Stokes equations
  Abstract: In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative within the "forced" category, by exhibiting a one-parameter family of distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.

• November 28 Jonathan Touboul (Brandeis University)
  Title: Spatio-temporal dynamics in embryonic development and symmetry breaking: from mammalian brain functional organization to feather patterns in birds.
  Abstract: The development of embryos, historically the motivation behind Alan Turing's discovery of spatio-temporal pattern forming instability, remains the source of a variety of mathematical questions. In fact, novel biological observations keep challenging existing models and calling for a new mathematical understanding of the process of pattern formation in spatio-temporal systems with symmetry breaking. I will review some recent works in pattern formation motivated by a few new biological discoveries. In particular, I will show that homeoproteins, a class of newly discovered transcription factors with the ability to diffuse across membranes, may theoretically provide the nervous system with the ability to develop sharp and well-localized boundaries in the nervous system. Going beyond, we will show that this robustness may be broken up by cell adhesion and lead to the emergence of mixed patterns with dislocated boundaries. Going beyond accounting for a steady-state spatial patterns, I will present some recent works aiming at characterizing transients leading to the establishment of feather patterns in 5 species of birds, from the domestic chicken to the Gentoo penguin. In that context, I will introduce a unified model that allows reproducing all 5 phenotype and reveals the role of cell proliferation in the wave-like emergence of feather buds. This talk includes the contributions of various collaborators, particularly Cristobal Quiñinao, Richard Bailleul, Alain Prochiantz, Shen-Ju Chou and Marie Manceau.

• December 7 (Wednesday, MCS B39) Jezabel Curbelo ( Universitat Politècnica de Catalunya)
  Title: The role of invariant manifolds to analyze atmospheric dynamics
  Abstract: Understanding dynamic instabilities and turbulence is of capital importance to improve predictability on many environmental processes. Transport and mixing are key to the dynamics, chemistry, and predictability of the circulation of the stratosphere and upper troposphere including features such as the stratospheric polar vortex (SPV) and the subtropical jets. Our aim is to characterize the atmospheric transport in the stratospheric region following a dynamical system approach in the Lagrangian framework. The stratosphere exhibits large variations on multiple space and time scales, therefore, the study of transport processes there brings into the discussion additional complexities of Lagrangian structures. A full understanding of any transitional fluid flow requires analyzing and computing the underlying coherent structures that govern the dynamics and the description of atmospheric transport is challenging due to its complex nature.
  Following a dynamical system approach, in this talk we show several examples of transport and mixing processes in the stratosphere and upper troposphere where, aided by Lagrangian tools, we identify invariant manifolds which determine the deformation of the fluid, simplify the atmospheric dynamical description and make possible the characterization of the parcels evolution and transport pathways in the region.

• December 12 Panayotis Kevrekidis (University of Massachusetts Amherst)
  Title: On Some Select Klein-Gordon and Beam Problems: Internal Modes, Fat Tails, Wave Collisions and Beyond
  Abstract: In this talk, we revisit some very thoroughly studied (yet still quite entertaining!) problems involving kink-like structures in non-integrable systems starting with the Klein-Gordon models. We will start from the prototypical model of the \phi^4 class and discuss a bit its history of discoveries, successes and culprits as concerns the complex landscape of collisions of kinks and antikinks and its fractal features. We will briefly touch upon the recent developments in this vein, as well as the remarkable feature that after about 50 years of studies, there are still some fundamental questions remaining. We will then extend considerations to the case of higher order (\phi^6, 8, 10 and 12) models and present the particularities that each of these models bears, including the potential for numerous internal modes, fat tails and power law kink-antikink interactions among others. Time permitting, we 'll also open up the problem towards the possibility of higher order dispersion and motivate such considerations from the perspective of recent optics problems. We will see how such higher order dispersion also creates interesting possibilities such as oscillatory tails, numerous kink-antikink bound states and their own complex interaction landscapes.

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