May 29, 2023

Fall 2005
Renormalization and Universality in Mathematics and Mathematical Physics

Spring 2006
Holomorphic Dynamics, Laminations, and Hyperbolic Geometry

Dynamical Systems Mini Courses
Wednesdays, 1:30 p.m.
& Fridays, 1:30 p.m. (if needed)

Fields Institute, Room 230
Audio of talks available here.

Wed. May 3 Johannes Rueckert (International University Bremen)
On Newton's method for transcendental functions
Newton's root finding method of a transcendental entire function is a transcendental meromorphic function (except in one special case). The Newton map of a transcendental function may have obstacles to root finding that do not exist in the polynomial case. We give a criterion under which these obstacles occur and show that in many cases, Newton's method still "tries" to find a root.

Fri. Apr. 28, 1:30PM
Fields Library

**(note special place)

Special Seminar
Bertrand Duplantier (Saclay)
SLE and Quantum Gravity
Abstract: TBA

Wed. Apr. 26 Rogelio Valdez (UAEM)
Mating a Siegel disk with a real quadratic polynomial
In this talk we show the construction of the mating between a quadratic polynomial with a Siegel disk and a real quadratic polynomial posessing a post-critical orbit that is semi-conjugate to a rotation with the same rotation number as the Siegel disk.
11:10 AM - 12:30 PM
Fields Library
** (note special time and place)
Jeremy Kahn
Mini-course on complex a-priori bounds
Starting this Thursday, and continuing for a few weeks, Jeremy Kahn will take over the lectures from Misha Lyubich's graduate course. He will present recent work on complex a-priori bounds, with applications to the renormalization theory of complex polynomials.
Wed. Apr. 19
Carlos Cabrera (Fields Institute)
Classification of regular spaces of hyperbolic quadratic polynomials
Regular spaces were introduced by Lyubich and Minsky in 1996, these are laminated spaces associated to rational functions. When $f$ is hyperbolic, the regular space of $f$ is a two dimensional lamination.
We prove that the topology of the regular space of a hyperbolic quadratic polynomial determines its combinatorial type.
Wed. Apr. 12

Mary Rees (University of Liverpool)
The Ending Laminations Theorem direct from Teichmuller Geodesics
Abstract: This talk will give an outline of a proof of the Ending Laminations Theorem direct from Teichmuller geodesics [2]. As with the Minsky-Brock-Canary proof, the broad strategy uses Minsky's approach, developed over the last decade and more, of constructing a geometric model for given ending lamination data, and proving that any hyperbolic manifold with this ending lamination data is Lipschitz equivalent to the model. But there are two main differences. The first is that the geometric model is constructed direct from Teichmuller geodesics, and the theory used is a theory of Teichmuller geodesics developed for a different purpose in [1]. The second is that the proof is more directly related to results that Minsky developed for the case of combinatorial bounded geometry, and mimics a proof in the bounded geometry case.

[1] Rees, M., Views of Parameter Space: Topographer and Resident.
Asterisque 288, 1983

[2] Rees, M., The Ending Laminations Theorem direct from Teichmuller

Wed. Apr. 5

Reza Chamanara (Fields Institute)
Affine automorphism groups of surfaces of infinite type.
Abstract: I will describe various translation surfaces of infinite
topological type with non-elementary affine automorphism groups. For a
family of these examples the affine automorphism group is a free group
generated by two parabolic elements. I will also present one example
with infinitely generated automorphism group and one example with purely
hyperbolic automorphism group.

Wed. Mar. 22 Arnaud Cheritat (Universite Paul Sabatier, France)
The conformal radius of Siegel disks and the size of parabolic points.
After defining the size of a parabolic point, I will explain why it is
kind of an analogue to the conformal radius of Siegel disks, in the
setting of quadratic map with an indifferent fixed point. If time allows,
I will explain how parabolic implosion gives information on the
fine behavior of conformal radii near a rational rotation number.
Wed. Mar. 15 Shaun Bullett (University of London)
Deformations of the modular group as a quasifuchsian correspondence.
The modular group $PSL_2(\mathbb Z)$ is rigid as a Kleinian group. However it has a space of deformations when it is regarded as a $(2:2)$ correspondence $z \to w$ (defined by the polynomial relation $(w-(z+1))(w(z+1)-z)=0$). We examine a slice of deformation space analogous to the Bers slice for Kleinian groups, the critical relations that occur at isolated interior points of the slice, and the degeneracies that occur at the boundary - one example is a continuous deformation of the action of the modular group on the upper half plane into the action of the polynomial $z \to z^2+1/4$ on its filled Julia set.
Fri., Mar. 3-
1:00 p.m.***
(note time change)
Adrien Douady (Ecole Normale Superior)
Carleson-Jones-Yoccoz without dynamics
Abstract: Consider an external ray of the Mandelbrot set passing through a point $c_1$ and landing at a point $c_0$ . By a theorem of Carleson, Jones and Yoccoz, the following conditions are equivalent:
(i) : $0$ is not recurrent under $f_{c_0}: z\mapsto z^2 + c_0$ ;
(ii) : The filled Julia set $K_{c_0}$ is a John dendrite.
Condition (i) can be reformulated as a condition (i') involving $K_{c_1}$ , and more specifically the rays in $\C - K_{c_1}$ descending from the critical points of the potential. Condition (i') makes sense for an arbitrary Dirichlet-regular Cantor set. Following an external ray of the Mandelbrot set is a particular case of the Branner Hubbard compression.
So it is reasonnable to think that, if $K$ is an arbitrary Cantor set satisfying (i') - and some regularity conditions -, the Branner-Hubbard compression leads to a John dendrite. We shall present this conjecture.
Wed., Mar. 1 Talk 1 at 1:10p.m.
Hiroyuki Inou (Kyoto)
Combinatorics of renormalizable cubic polynomials of capture type
We study combinatorics of renormalizable cubic polynomials of capture type and give a sufficient condition for the family of renormalizable cubic polynomials with a given combinatorics of capture type to be compact. In addition, we prove if such a family is compact, then the straightening map is bijective.
Talk 2-- 2:20 p.m.
Carsten Petersen (Universitetsvej 1)
Converting divergence to convergence
(joint work with Adam Epstein, Warwick)
Motivated by a divergence problem in holomorphic dynamics we prove compactness properties of the space \mathcal F of holomorphic maps f : C -> D which are branched double covers from an annulus C to a disk D with C \subset D, D\C connected, and which are close to the identity on the common boundary. These compactness properties serves to convert divergence properties of certain families of quadratic rational maps into convergence properties.
Wed., Feb. 15 Monica Moreno Rocha (Fields Institute)
Rational maps and Sierpinski gaskets
We study a family of rational maps acting on the Riemann sphere with a single preperiodic critical orbit. Using a generalization of the well-known Sierpinski gasket, a complete topological description of their Julia sets is provided. In addition, we deduce a combinatorial algorithm that allows us to show when two such Julia sets are not homeomorphic.
Wed., Feb. 8 Tomoki Kawahira (Fields Institute)
Tessellation and Lyubich-Minsky laminations associated with
quadratic maps (II)
Fri., Feb. 3 Alexey Glutsyuk (ENS-Lyon)
On density of horospheres in dynamical laminations
We consider horospheres in the factorized three-dimensional lamination associated to a rational function. We show that in many cases each horosphere is dense.
Wed., Feb. 1 Dierk Schleicher (I. U. Bremen)
Thurston's Theorem on Rational Maps and Spider Theory
This minicourse will be an introduction to "Thurston's characterization theorem of rational mappings as dynamical systems", which is sometimes called the "Fundamental Theorem of Complex Dynamics". I will try to motivate the theorem and explain its (rather complicated) statement in terms of "spiders" for polynomials. If time permits, I will indicate some ideas of its proof.

Wed., Jan. 25 Victor Sirvent (Universidad Simon Bolivar)
Symbolic dynamics and geodesic laminations
For a class of symbolic dynamical systems we give some geometrical models as dynamical systems defined on geodesic laminations on the hyperbolic disc. The symbolic systems studied here come from a family of minimal sequences on a $3$-symbol alphabet with complexity $2n+1$, which satisfy a special combinatorial property. These sequences were originally defined by P. Arnoux and G. Rauzy as a generalization of the binary sturmian sequences. We show some applications of these results to Rauzy fractals.
Wed., Jan. 18 Victor Sirvent (Universidad Simon Bolivar)
Space filling curves and geodesic laminations
In this talk we shall associate space filling curves to connected fractals, obtained as the fixed point of an iterated function systems (IFS) satisfying the common point property and other conditions. These curves are H\"older continuous and measure preserving. To these space filling curves we associate geodesic laminations satisfying among other properties that points joined by geodesics have the same image in the fractal under the space filling curve. The laminations help us to understand the geometry of the curves. We define an expanding dynamical system on the laminations.
Fall 2005
Tuesdays & Thursdays, 1:10 p.m.
Wednesdays, 2:10 p.m.
Thurs., Nov. 17 Young-Heon Kim (University of Toronto)
Determinants of Laplacians and their holomorphic extensions
We consider the determinant of the Laplacian as a function on the Teichmueller space of a closed surface S of genus > 1 and show that it has a unique holomorphic extension to the quasifuchsian space. To realize this holomorphic extension as determinants of actual differential operators on S, we construct holomorphic extensions of Laplace operators of hyperbolic metrics on S using Ahlfors-Bers theory of quasiconformal mappings, and we study their determinants.
Wed., Nov. 16 Marco Martens (University of Groeningen).
Henon renormalization (III)
Tues., Nov. 15 Marco Martens (University of Groeningen).
Henon renormalization (II)
Thurs., Nov. 10 Hans Koch (University of Texas, Austin).
Renormalization of vector fields (IV)
Tues., Nov. 8 Hans Koch (University of Texas, Austin).
Renormalization of vector fields (III)
Fri, Nov. 4
**talk in Bahen Building, BA3004
D. Beliaev (Princeton University / IAS)
Spectrum of harmonic measure on SLE
Thurs., Nov. 3
** talk in room 230 **
Hans Koch (University of Texas, Austin)
Renormalization of vector fields (II)
Tues., Nov. 1
** talk in room 230 **
Young-Heon Kim (Fields Institute)
A Lower Bound Estimate on Quasiconformal Maps
Based upon the classical work of Ahfors and Bers on quasiconformal maps as solutions of Beltrami differential equations, we will derive a lower bound for the first derivatives of quasiconformal maps by some bounds on their corresponding Beltrami differentials. A brief review of Ahlfors' and Bers' work will be given as well.
Oct. 27, 12:30 p.m.
* Note: Special time
** talk in room 230 **
Leo Kadanoff (University of Chicago)
Universality, Scaling, and Renormaization: the View from the Real Space
II. Correlation Functions and Universality

Correlation Function Ideas are explored using the two-dimensional Ising Model as an example Operator Algebras/Short Distance Expansions are explored as a technique for discussing invariance properties of critical models, including scale and conformal invariance. The effect of marginal operators is considered in detail. An explicit form for the spin correlation functions of the two dimensional Ising model at criticality is exhibited. This is then used to show the non-existence of marginal operators coupled to the spin. These operators would make the spin correlations non-universal. Universality is then proven for this model in a manner which is "good enough for physicists work..."
Oct. 26, 1:10 p.m.
* Note: Special time
** talk in room 230 **
Leo Kadanoff (University of Chicago)
Universality, Scaling, and Renormaization: the View from the Real Space
I. Universality and the RG
The basic ideas of critical phenomena theory including Universality and Scaling are Reviewed. These are tied up in a neat package via the Renormalization Group idea a la K. W. Wilson. This method is described in one concrete example, its application to Ising models. A formalism for renormalization is set up. The concept of a fixed point is explored and shown to give rise to behaviors described by the words "scaling" and "universality" as well as "relevant", "marginal", and "irrelevant". An explicit example of an approximate renormalization scheme is developed using "lower bound" approximation techniques.
Oct. 25, 1:10 p.m.
** talk in room 230 **
Robert S. MacKay (Warwick University)
Renormalisation, breakup of invariant circles for area-preserving maps, and incommensurate structures (II)
Oct. 17, 1:10 p.m. Dirk Kreimer (Institut des Hautes Études Scientifiques, France)
Renormalization in Quantum Field Theory (II)
Oct. 13, 1:10 p.m Pavel Bleher (Indiana University-Purdue University Indianapolis)
Scaling and Universality in Random Matrix Models
Lecture 4: Large N asymptotics of the free energy of random matrix models
.Oct. 12, 2:10 p.m. Robert S. MacKay (Warwick University)
Renormalisation, breakup of invariant circles for area-preserving maps, and incommensurate structures
I'll survey the history of how renormalisation ideas entered the study of breakup of invariant tori for Hamiltonian dynamics, including contributions by Greene, Chirikov, Escande & Doveil, Kadanoff & Shenker, myself, Aubry & Peyrard, Khanin & Sinai, Haydn, Wilbrink, Gallavotti & Benfatto, Stirnemann, Morrison, Jauslin and Koch, and describe the associated interpretation in terms of decimation of incommensurate structures for Frenkel-Kontorova (FK) chains. The result is a beautiful extension of KAM theory (which gives conditions for persistence of invariant tori for perturbations of integrable systems, equivalently sliding states for FK chains) to understand the boundary of the KAM domain and beyond (cantori, pinned states). Nevertheless, many of the consequences of the renormalisation view remain to be explored; some of the main questions will be highlighted.
Oct. 11, 2:10 p.m Pavel Bleher (Indiana University-Purdue University Indianapolis)
Scaling and Universality in Random Matrix Models
Lecture 3: Double scaling limits and universality at critical points
Oct. 11, 1:10 p.m. Dirk Kreimer (Institut des Hautes Études Scientifiques, France)
Renormalization in Quantum Field Theory (I)
We discuss renormalization from an algebraic viewpoint emphasizing the Hopf algebra structure underlying perturbative quantum field theory. The first lecture will introduce the necessary algebraic background and procede by example, the second lecture investigates how the structure of Dyson-Schwinger equations fits into this framework.
Oct. 4 & 6

** talks in room 230 **

Pavel Bleher (Indiana University-Purdue University Indianapolis)
Scaling and Universality in Random Matrix Models
Lecture 1: General introduction to random matrix models.
Lecture 2: The Riemann-Hilbert approach to the large N asymptotics of orthogonal polynomials and random matrix models. Scaling limits and universality in the bulk of the spectrum and at the end-points.
Abstract. Partition functions of random matrix models provide generating functions for a number of combinatorial and physical problems: Enumeration of graphs on Riemannian surfaces and quantum gravity, models of statistical mechanics on random surfaces, enumeration of knots and links, meanders, and others. Critical points and double scaling limits of random matrix models determine in this context large N asymptotics of the quantities under consideration. In this introductory series of 4 lectures we will discuss ensembles of random matrix models, integrable structures for correlation functions of eigenvalues of random matrices and their relation to orthogonal polynomials. We will consider the Riemann-Hilbert approach to semiclassical asymptotics of orthogonal polynomials, the Deift-Zhou nonlinear steepest descent method, and scaling limits and universality of the eigenvalue correlation functions. Finally, we will discuss large N asymptotics of the free energy of the ensemble of random matrices, critical asymptotics and double scaling limits at critical points.

Sept. 28 Michael Yampolsky (University of Toronto)
Sept. 27 & 29 Roland Roeder (Fields Institute)
Newton's method in C^2: first excursions into the topology for the basins of attraction.
Abstract: The Newton map N for finding the roots of two polynomial equations P(x,y)=0, Q(x,y)=0 has incredibly complicated dynamics because the system simultaneously has points of indeterminacy and topological degree > 1. In this mini-course we will examine the simplest non-trivial case, the Newton map to find the roots of x(x-1)=0 and y^2+Bx-y=0.
Theorem: If |B-1|>1, then the basins of attraction for the roots (0,0) and (0,1) have infinitely generated first homology and the basins of attraction for the roots (1,0) and (1,1-B) have either trivial first homology, or infinitely generated first homology, depending on the parameter B.
The proof of this theorem uses many different methods/techniques including blow-ups, Morse theory, linking numbers, and invariant currents. We will introduce the necessary details about these techniques in the mini-course.

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