December 2nd
Speaker: Dan Paraschiv (Institute of Mathematics of the Romanian Academy)
Title: Infinitely connected Fatou components in the Chebyshev-Halley family of numerical methods
Abstract: We study the Cebyshev-Halley methods applied to the family of polynomials $f_{n,c}(z)=z^n+c$, for $n\ge 2$ and $c\in \mathbb{C}^{*}$. We will discuss the existence of parameters such that the immediate basins of attraction corresponding to the roots of unity are infinitely connected. We also relate the main Julia set connected component to the Julia set of the map obtained by applying Newton's method to $f_{n,-1}$.
December 9th
Speaker: Caroline Davis (Stony Brook University)
Title: Baby mandelbrots for babies
Abstract: Celebrating mandelbrot set babies and a semester of our baby brot seminar :)
Part 1: An overview of mandelbrot set geometry and combinatorics. Takeaways here will be a high-level account of why we find baby brots and some language to speak about them and why they look how they do. Key words: rays, implosion, self-similarity, branch points, renormalization/tuning, decorations, wakes/veins/filaments, medallions, inhomogeneity. [email me before the seminar if theres anything in particular you want me to gloss!!]
Part 2: Practice fluency by playing the game MandelGuessr: vambrosi.github.io/MandelGuessrWeb/ (co-developed with V Ambrosi). Try it yourself before the seminar! From Part 1 we’ll know we can always win in theory. In Part 2, we’ll go over tips and tricks for winning in practice.
November 18th
Speaker: Rashmita (Tata Institute of Fundamental Research)
Title: Conformal matings by Schwarz reflections
Abstract: Inspired by the Fatou-Sullivan dictionary between Kleinian groups and rational dynamics, it is natural to think of iterations of anti-holomorphic rational maps on the Riemann sphere as the complex dynamics counterpart of actions of Kleinian reflection groups. Schwarz reflections associated with (disjoint unions of) quadrature domains generate a class of anti-holomorphic dynamical systems. In certain cases, such systems provide a framework for conformally combining (mating) the dynamics of anti-rational maps and that of reflection groups in the same dynamical plane. In this talk, we will demonstrate a general method of producing conformal matings between quadratic anti-holomorphic polynomials and reflection groups using Schwarz reflection maps, for the particular example of the deltoid reflection. If time permits, then we will see an overview of the converse, that is, construction of Schwarz reflections, and hence quadrature domains, from matings of two suitable maps.
November 4th
Speaker: Jossy Russell (Imperial College)
Title: An Introduction to Irrationally Indifferent Attractors
Abstract: When a holomorphic germ has a fixed point with derivative equal to an irrational rotation, it exhibits extremely delicate and complicated long term behaviour that depends on the arithmetic of the irrational rotation itself. Indeed many simple questions (such as a full classification of linearizable examples) remain open to this day. One of the best tools for understanding such functions turns out to be renormalisation, and in this talk I will introduce some of the basic intuitions and results that come as a consequence of this perspective.
October 28th
Speaker: Omar Neyra (Pontificia Universidad Católica de Chile)
Title: Transversality in spaces of rational maps.
Abstract: Results about transversality for families of holomorphic functions are of great interest. They can be used, for example, to give a (arguably) simpler proof of the renowned DHS theorem. I will showcase some elementary results about transversality for critical relations and for the multiplier map of periodic orbits in spaces of polinomial and rational maps.
October 21st
Speaker: Bernhard Reinke (Aix-Marseille Université)
Title: Transcendental Wandering Triangles
Abstract: A celebrated theorem of William Thurston states that every branch point of a locally connected Julia set of a quadratic polynomial is precritical or preperioidic. In fact, the theorem is usually phrased in the language of quadratic laminations as the "No Wandering Triangle Theorem".
Laminations are a tool in polynomial and nowadays also entire transcendental dynamics that capture the combinatorics of the landing behavior of dynamic rays (or filaments), a wandering triangle corresponds to three dynamic rays landing together at a point that is neither precritical nor preperiodic.
By the work of Blokh-Oversteegen and Buff-Canela-Roesch, it is known that there are cubic polynomials with wandering triangles. In this talk, I will give an overview of our joint project with Jordi Canela and Lasse Rempe of construction wandering triangles for entire transcendental functions, in particular in the family $a + b \cosh z$.
October 14th
Speaker: Eduardo Sodré (Brown University)
Title: Matings of Polynomials
Abstract: I will motivate and introduce the concept of polynomial matings in rational dynamics, with a focus on degree 2. There are several distinct definitions, and most of what is known is in the postcritically finite case; many questions still remain about matings of general polynomials. I hope to showcase some of these questions, and how mating finds applications in understanding certain moduli spaces of rational maps.
October 7th
Speaker: Richard Birkett (Brown University)
Title: Whirlwind Tour of Complex Dynamics in Two+ Dimensions.
Abstract: I will provide an introduction to essential dynamical features of rational maps in two (or more) dimensions. I will mention Fatou, Julia, measures, entropy, degrees, algebraic stability, geometry, blowups, and the recent trend of non-archimedean techniques.
September 30th
Speaker: Anna Jové (Universitat de Barcelona)
Title: What is a Baker domain? Introduction to transcendental dynamics
Abstract: Transcendental dynamics, in contrast with rational dynamics, present richer structures, even in the Fatou set, motivated by the presence of an essential singularity and the infinite degree of the iterated map. Essentially, such structures are wandering domains and Baker domains. In this talk, first I will present the definition of Baker domain and their classification, and then we will shift to their boundary dynamics, which is essentially chaotic, and a great source of open questions. Some conjectures and open problems will be presented, together with some new results.