Thematic Program on Complex and Algebraic Dynamics in One Dimension
January 1 - June 30, 2028
Description
The study of dynamical systems over the complex numbers began late in the 19th century with work of Fatou and Julia, who applied the newly-formulated notion of normal families to distinguish between the tame and chaotic dynamical behavior of analytic functions under iteration. The study was galvanized in the latter half of the 20th century along with advances in hyperbolic and algebraic geometry, and also crucially with advances in computer technology late in the 20th century.
The Mandelbrot set, which parameterizes quadratic polynomials whose critical point stays bounded under iteration, is widely familiar as a computer generated picture of a curious fractal object in the complex plane, exhibiting mesmerizing patterns along its boundary as one zooms in. A prominent open question, known as MLC, asks if the Mandelbrot set, proved to be connected by a seminal result of J. Hubbard and A. Douady in the 1970s, is locally connected. The question has been one of the impetuses for the influential and ongoing study of renormalization. Among many other benefits, an affirmative answer would imply a pleasingly simple combinatorial description of the
Mandelbrot set, by work of W. Thurston.
Questions about the Mandelbrot set naturally extend to the study of the geometry and topology of various other moduli spaces of holomorphic dynamical systems. A key object of interest is the moduli space $M_d$ of degree $d$ complex rational maps, defined as the quotient of the space $Rat_d$ of degree $d$ rational maps by $PSL_2$($C$) acting by conjugation. The space $M_d$ contains subspaces of fundamental interest: hyperbolic components, which, conjecturally, are maximal domains of structural stability, and critical orbit subvarieties, which are algebraic varieties determined by coincidences of critical orbits. More recently, moduli spaces of algebraic correspondences, such as matings between polynomials and Kleinian groups, have been introduced and studied extensively.
The variety $M_d$ is not compact and determining how hyperbolic components and critical orbit subvarieties can meet the degeneracy locus at infinity has been the subject of much recent work. Various dynamical compactifications of these moduli spaces have been introduced, using tools such as Berkovich spaces, trees of spheres, and rescaling limits, to analyze how the dynamics degenerate. These degenerations reveal rich new structures of moduli spaces, and draw techniques from algebraic geometry, arithmetic dynamics, and Teichmuller theory. Degenerate rational dynamics is a theme that invites researchers working across these perspectives to unify frameworks, share insights, and explore new tools for analyzing limiting dynamics in these moduli spaces.
Relations between the complex theory and classical algebraic geometry emerge from questions about moduli, degenerations and compactifications, and it becomes natural to apply the tools and insights to more general algebraic dynamical systems, and in the process gain new insights back in the complex setting. Arithmetic dynamics blends ideas from dynamical systems and number theory, guided by analogies between classical arithmetic questions. Central themes are the study of postcritically finite maps (PCF) as the dynamical counterpart of complex multiplication points in arithmetic geometry, and preperiodic points as analogues of torsion points in Abelian varieties.
In the past decade, substantial progress has been achieved in “unlikely intersection” problems within this framework. An important example is the Dynamical Andre-Oort Conjecture which aims to classify algebraic subvarieties of dynamical moduli spaces containing a Zariski dense subset of PCF parameters. Advances in unlikely intersection questions rely on arithmetic equidistribution theorems and complex analytic techniques, including the geometric study of invariant and bifurcation measures. This portion of the proposal will further stimulate the exchange of ideas between researchers with various backgrounds to address fundamental questions of arithmetic nature in complex dynamics.
The program will be organized around three week-long thematic workshops. For longer term participants, there will be activities to promote both subject preparation and facility with tools to engage in active research, collaborations, and idea sharing for participants at all levels.
