Mondays at 2pm
Frank Adams 1, Alan Turing Building
Contacts Donald Robertson Yotam Smilansky
Absence of bounded-orbit wandering domains
Lasse Rempe
University of Manchester
Let f be an analytic self-map of either the complex plane or the Riemann sphere. We may consider f as describing the transition rule of a dynamical system, and study the behaviour of points under iteration (repeated application) of the function f. The Fatou set F(f) consists of those points near which the dynamics is stable under small perturbations of the starting value. A connected component of F(f) is called a Fatou component. A Fatou component is called wandering if its forward iterates are pairwise disjoint. A famous theorem of Sullivan from 1985 states that rational maps have no wandering domains. His proof used quasiconformal deformation theory, and relied crucially on the fact that rational maps of fixed degree depend only on finitely many parameters. Transcendental (i.e., non-polynomial) self-maps of the complex plane may have wandering domains. A long-standing open question asked whether it is possible for orbits in such a domain to remain bounded under iteration. We answer this question by proving that bounded-orbit wandering domains do not exist. (This is joint work with Drach, Pardo-Simón, Prochorov, Učakar and Waterman. Since our result was first announced, Ye also announced a proof of the main result.) This result was obtained with the help of generative AI, based on a new proof of Sullivan's theorem discovered by Ye, also using generative AI. We have also formalised the proof using the Lean proof assistant. I will discuss some thoughts on what the rapid advances in AI capabilities may mean for holomorphic dynamics (and mathematics) going forward.Constructing Non-Rectifiable Delone Sets
Ashwin Bhat
University of Birmingham
Delone sets, also known as separated nets, are point sets in Euclidean space that are both relatively dense and uniformly discrete. They model aperiodic structures, which have been of great interest since Dan Shechtman first observed quasicrystals, an achievement that ultimately won him the Nobel Prize in Chemistry in 2011. Since then, many lines of research have emerged regarding the structural properties of Delone sets. An interesting question posed independently by Furstenberg and Gromov asks whether all Delone sets in a given dimension are bi-Lipschitz equivalent. McMullen, as well as Burago and Kleiner, answered this question in the negative for dimensions two and above by transforming it into a continuous problem concerning the prescribed Jacobian equation. In this talk, we discuss their method of "encoding" a non-realisable density into a Delone set, and examine how modern versions of this technique are implemented in metric geometry.The independence polynomial for recursive graph sequences: the dynamical perspective
Misha Hlushchanka
University of Amsterdam
The distribution of zeros of partition functions on graphs is intimately related to the analyticity of physical quantities and their phase transitions. The independence polynomial of a finite graph is the generating function for the numbers of independent sets (i.e., subsets of pairwise non-adjacent vertices) of each size. It naturally originates in statistical physics as the partition function of the hard-core model for gases, where each particle occupies an exclusive region of space. It is known that the roots of the independence polynomial are dense outside a neighborhood of the origin for the family of bounded-degree graphs. However, the overall structure of this zero locus, as well as the corresponding zero locus for concrete sequences of graphs, may be quite intricate. Jointly with Han Peters (University of Amsterdam), we develop a unified dynamical framework for the analysis of these zero sets for recursive graph sequences. In the talk, I will report on our results concerning their structure, with particular emphasis on (the absence of) phase transitions.