Past Events

Spring 2023


8:00 - 8:50am PST:

Olga Lukina (Leiden University)

Weyl groups in Cantor dynamics

Arboreal representations of absolute Galois groups of number fields are given by profinite groups of automorphisms of regular rooted trees, with the geometry of the tree determined by a polynomial which defines such a representation. Thus arboreal representations give rise to dynamical systems on a Cantor set, and allow to apply the methods of topological dynamics to study problems in number theory. In this talk we consider the conjecture of Boston and Jones, which states that the images of Frobenius elements under arboreal representations have a certain cycle structure. To study this conjecture, we borrow from the Lie group theory the concepts of maximal tori and Weyl groups, and introduce maximal tori and Weyl groups in the profinite setting. We then use this new technique to give a partial answer to the conjecture by Boston and Jones in the case when an arboreal representations is defined by a post-critically finite quadratic polynomial over a number field. Based on a joint work with Maria Isabel Cortez.


9:00 - 9:50am PST:

Alex Kapiamba (University of Michigan)

Elephants all the way down: the near-parabolic geometry of the Mandelbrot set

Understanding the geometry of the Mandelbrot set, which records dynamical information about every quadratic polynomial, has been a central task in holomorphic dynamics over the past forty years. Near parabolic parameters, the structure of the Mandelbrot set is asymptotically self-similar and resembles a parade of elephants.Near parabolic parameters on these "elephants'', the Mandelbrot set is again self-similar and resembles another parade of elephants. This phenomenon repeats infinitely, and we see different parades of elephants at each scale. In this talk, we will explore the implications of controlling the geometry of these elephants. In particular, we will partially answer Milnor's conjecture on the optimality of the Yoccoz inequality, and see potential connections to the local connectivity of the Mandelbrot set.


8:00 - 8:50am PST:

Tapio Rajala (University of Jyväskylä)

BV and Sobolev extensions in metric spaces

In general metric measure spaces we can move between studying functions of bounded variation and sets of finite perimeter via the coarea formula. To some extent we can also relate these two to the study of Sobolev W^{1,1}-functions. In this talk we will make these relations more precise for the question of extendability of functions and sets defined on open subsets of the space to the whole space. We will also look at what more can be said in the Euclidean setting and highlight some open problems. The talk is based on joint works with Miguel García-Bravo, and with Emanuele Caputo and Jesse Koivu.



9:00 - 9:50am PST:

Walter Parry (Eastern Michigan University)

Introduction to Pf4

My computer program Pf4 makes computations for finite branched covering maps from the 2-sphere to itself with exactly four postcritical points.  Output includes determination of rationality, all fixed (multi)curves, all mating equators, fundamental group wreath recursion, dynamic portrait, description of modular group liftables, picture of Thurston pullback map in special cases.  This talk will be an introduction to Pf4.



8:00 - 8:50am PST:

Caroline Davis (Indiana University)

A tour of Per_n(0)

In this talk, we survey recent work about the internal structure of the dynamical varieties Per_n(0) in the space of quadratic rational maps, varieties which are defined by marking a n-periodic critical point. We will discuss the structure of these spaces near their punctures as well as define the “mating” of two quadratic polynomials and see what application this has for elucidating global topological information about the varieties, such as connectivity


9:00 - 9:50am PST:

Alessandro Sisto (Heriot-Watt University)

Markov chains on groups and quasi-isometries

Random walks on groups provide a model for a "generic" element of a group, and they're very interesting and very well-studied. In geometric group theory it is natural to consider quasi-isometric groups, but unfortunately random walks are not compatible with quasi-isometries, in the sense that they cannot be "pushed forward" via quasi-isometries in any meaningful sense. To resolve this, in this talk I will propose the study of more general Markov processes on groups that are indeed "quasi-isometry compatible", and present the first results about them. In particular, I will discuss a central limit theorem for random walks  whose proof exploits this perspective of pushing forward Markov chains, and a potentially new quasi-isometry invariant.


8:00 - 8:50am PST:

Laura DeMarco (Harvard University)

Geometry of the PCF locus

A rational map on P^1 is postcritically finite (PCF) if each critical point has a finite forward orbit. They are special from a dynamical point of view, and there are many of them: they form a Zariski dense subset of the moduli space M_d of all maps on P^1 in every degree d>1. In fact, the closure of the PCF locus contains the support of the bifurcation measure (Buff-Epstein, 2009). A few months ago, Zhuchao Ji and Junyi Xie announced a proof of "DAO for curves" (https://arxiv.org/abs/2302.02583), a conjecture about the geometry of this PCF locus and, specifically, which algebraic curves in M_d contain infinitely many of these points. I will discuss some of the history of this question - especially how it connects complex dynamics and arithmetic geometry - and explain some ongoing work with Myrto Mavraki and Hexi Ye.


9:00 - 9:50am PST:

Frank Trujillo (University of Zürich, Switzerland)

Hausdorff dimension for invariant measures of critical circle maps

Critical circle maps are a family of smooth circle homeomorphisms whose derivatives vanish at a non-empty finite set. Any sufficiently regular critical circle map with an irrational rotation number can be continuously conjugated to an irrational rotation. Furthermore, its unique invariant measure is singular with respect to the Lebesgue measure on the circle. In this talk, I will recall the notion of the Hausdorff dimension of a measure and give explicit bounds of its value for invariant measures of critical circle maps. As we shall see, these bounds will depend only on the arithmetic properties of the rotation number.


Fall 2022


Zachary B. Smith  (University of California, Los Angeles)

The Thurston pullback map 

Abstract:  A Thurston map f: (S^2, P_f) to (S^2, P_f) is a branched covering map of the 2-sphere which is not a homeomorphism and which has finite postcritical set P_f. Every such map induces a pullback operation on Jordan curves in S^2 minus P_f  by taking preimages. By pulling back complex structures through f, there is also an induced map on Teichmüller space. In this talk we will discuss the relationship between these two operations in the special case where |P_f|=4 and the orbifold of f is hyperbolic. After surveying some known results, we will compute the Thurston pullback map in some select examples.


9:00 - 9:50am PST:

Efstathios Konstantinos Chrontsios Garitsis  (University of Illinois Urbana-Champaign)

Assouad dimension and spectrum under quasiconformal mappings

Abstract:  Since Hausdorff dimension was introduced in 1918, many different notions of dimension have been defined, one of which is the Assouad dimension. It is natural to consider how these notions change under certain classes of maps, especially when questions of classification of sets up to said maps arise. Gehring and Väisälä showed in 1973 how the Hausdorff dimension of a set changes under quasiconformal maps, while Kaufman in 2000 proved the analogous result for  box-counting dimension. In this talk, an introduction to the different types of dimensions will be given, along with the results of Gehring, Väisälä and Kaufman. We will then discuss analogous theorems we proved for the Assouad dimension and spectrum, which describe how K-quasiconformal maps distort these notions for subsets of \mathbb{R}^n. We will conclude the talk by demonstrating how said theorems can be applied to quasiconformally classify certain spirals. 



8:00 - 8:50am PST:

Toni Ikonen  (University of Helsinki, Finland)

Weighted distances on the plane

Abstract:  The talk consists of two parts. In the first section we briefly describe the setting of interest: that of nonsmooth two-dimensional surfaces, called metric surfaces. We also recap some recent progress towards quasiconformal uniformization of such spaces. In the second part, we fix our setting to metric surfaces constructed from weighted distances on the plane. Given such a weight, one defines a length distance as in the classical Riemannian setting. We are interested in locally bounded lower semicontinuous weights that vanish on some totally disconnected compact set, or other nonseparating compact sets, in which case the construction yields a metric surface up to considering a monotone quotient map. We discuss several examples illustrating that the interaction of the geometry of the vanishing set and the decay rate of the weight typically create obstructions for the existence of a quasiconformal map from a Riemannian surface onto the constructed metric surface. It turns out that under a geometric assumption, the specific properties of the weight are immaterial: if the vanishing set is removable for conformal mappings, the aforementioned quotient map is always a quasiconformal homeomorphism. Recall that a compact set is removable for conformal mappings if conformal maps defined on its complement are restrictions of Möbius transformations. We also discuss a converse of this result.
The talk is based on the joint work "Quasiconformal geometry and removable sets for conformal mappings" (J. Anal. Math., 2022, to appear) with Matthew Romney.


9:00 - 9:50am PST:

Russell Lodge  (Indiana State University)

Global dynamics of the pullback on multicurves for Thurston maps

Abstract:  In his fundamental theorem of holomorphic dynamics, W. Thurston characterizes those postcritically finite branched covers from the two-sphere to itself that are "combinatorially" equivalent to rational maps. The theorem can be quite difficult to apply in practice, however, and the combinatorial equivalence was not even that well understood until Bartholdi and Nekrashevych applied self-similar group theory to solve the twisted rabbit problem about two decades later. I will show how this theory has further implications for some challenging problems, such as the enumeration of equivalence classes of Thurston maps and a conjecture of Pilgrim about the global dynamics of the pullback on multicurves.




8:00 - 8:50am PST:

Stefan Wenger  (University of Fribourg, Switzerland)

Existence of integral currents in metric manifolds, with applications to Poincaré inequalities

Abstract:  In the 1960s, Federer-Fleming developed their theory of normal and integral currents in Euclidean space, providing a suitable setting to study and solve Plateau's problem of finding area minimizing surfaces of any dimension with prescribed boundary. Around 20 years ago, Ambrosio-Kirchheim gave a vast generalization of Federer-Fleming's theory to the setting of metric spaces. In particular, they introduced integral currents in metric spaces, which can be thought of as measure theoretic generalizations of oriented surfaces for which there are natural notions of volume and boundary. In this talk we consider metric spaces homeomorphic to a closed, orientable smooth manifold. We study when such spaces (called metric manifolds) support a non-trivial integral current without boundary. The existence of such an object should be thought of as an analytic analog of the fundamental class of the metric manifold. As an application, we obtain a conceptually simple proof of a deep theorem of Semmes about the validity of a weak 1-Poincaré inequality in metric manifolds that are Ahlfors regular and linearly locally contractible. In the smooth case the idea for this simple proof goes back to Gromov. Based on joint work with G. Basso and D. Marti.


9:00 - 9:50am PST:

Rohini Ramadas  (University of Warwick, UK)

Complex dynamics, degenerations, and irreducibility problems

Abstract:  Per_n is a (nodal) Riemann surface parametrizing degree-2 rational functions with an n-periodic critical point. The n-th Gleason polynomial G_n is a polynomial in one variable with Z-coefficients, whose roots correspond to degree-2 polynomials with an n-periodic critical point (i.e. to the period-n components of the Mandelbrot set). 

Two long-standing open questions in complex dynamics are: (1) Is Per_n connected? (2) Is G_n is irreducible over Q? 

We show that if G_n is irreducible over Q, then Per_n is connected. In order to do this, we find a smooth point with Q-coordinates on a compactification of Per_n. This smooth Q-point represents a special degeneration of degree-2 rational maps, and as such admits an interpretation in terms of tropical geometry.





Fall 2021


Wei Qian (CNRS and Université Paris-Saclay)

Uniqueness of the welding problem with fractal interfaces

Abstract: We give a simple set of geometric conditions on curves  η, ῆ in H from 0 to infinity so that if  φ  : H → H is a homeomorphism which is conformal off eta with  φ(η) = ῆ then phi is a conformal automorphism of H. Our result applies to the setting where the interface eta is not the boundary of a Hölder domain or even a connected domain. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). This is based on a joint work with Jason Miller.


Ryosuke Shimizu (Kyoto University)

Construction of a canonical p-energy on the Sierpinski carpet

Abstract: It is known that the Brownian motion on the Sierpinski carpet (abb. SC) has sub-Gaussian behavior (it is also called anomalous diffusion), and so we can not develop the upper gradient based analysis on SC. In this talk, we give a construction of a ``canonical'' p-energy (formally written as  ∫|∇f |^p) on  SC as a scaling limit of discrete energies on approximating graphs. A motivation of this work is related to the study of Ahlfors regular conformal dimension of fractals.


Damaris Meier (University of Fribourg)

Quasiconformal almost parametrizations of metric surface

Abstract: The uniformization problem for metric spaces asks to find conditions on a given metric space X, homeomorphic to some model space, under which there exists a homeomorphism from the model space to X with good geometric and analytic properties

 We consider the case where X is a two-dimensional metric surface of locally finite Hausdorff-2-measure. By only assuming that X is locally geodesic, we show that any Jordan domain in X of finite boundary length admits a quasiconformal almost parametrization. This generalizes the uniformization results of Bonk and Kleiner as well as Rajala. The proof makes use of the theory of energy and area minimizing Sobolev discs developed by Lytchak and Wenger. A large part of this talk is devoted to the existence of Sobolev discs spanning a given Jordan curve in X. This is joint work with Stefan Wenger.


Slides available here.


Chenxi Wu (University of Wisconsin-Madison)

Galois conjugate for exponents of core entropy

Abstract: This is a collaboration with Kathryn Lindsey and Giulio Tiozzo. The Hubbard tree of a post-critically finite complex quadratic map is a finite simplicial tree that encodes the dynamics on its Julia set, and the core entropy is the topological entropy of the induced map on the Hubbard tree. Properties of core entropy have been extensively studied and they have important connections with the geometry of the Mandelbrot set. Using kneading theory and symbolic dynamics, we found new properties of the set of the Galois conjugates of the exponent of core entropy of superattracting points, which leads to a new necessary condition for an algebraic integer to be the exponent of core entropy of superattracting maps that lie in a specific vein. I will also discuss our ongoing work on generalizing these results to higher degree maps.



Ursula Hamenstädt (Universität Bonn, Germany)

Title: Random walks and boundaries of groups

Abstract: We explain how random walks on a hyperbolic group can be used to construct a hyperbolic flow from the group whose invariant measures correspond precisely to quasi-invariant measures on the boundary of the group. This construction gives rise to families of Sobolev spaces in the spirit of Bonk and Saksman whose properties can be related to topological properties of the boundary of the group.



Tatiana Smirnova-Nagnibeda (University of Geneva)

Title: Schreier graphs of self-similar groups as source of examples in spectral graph theory

Abstract: In this talk we will discuss some questions from spectral theory of infinite graphs and how to solve them by studying self-similar groups and their actions.



Summer 2021 Quasiworld Workshop

The 2021 Quasiworld workshop will take place on July 6th and July 7th through Zoom. The link will be sent out by email a few days prior. Please make sure you are signed up for the Quasiworld email list.



Schedule  (in Pacific time)


Tuesday, July 6th

6:50 am Welcome

7 am   Misha Lyubich, Stony Brook University    

8 am   Dierk Schleicher, Aix-Marseille Université

9 am   Social hour**

10 am Frederik Viklund, KTH Royal Institute  of Technology

11 am James Lee, University of Washington 

12pm Social hour**

Wednesday, July 7th

7 am   Anna Erschler, École Normale Supérieure, France

8 am   Anke Pohl, University of Bremen

9 am   Social hour**

10 am Gareth Speight, University of Cincinnati

11 am Robert Young, Courant Institute, NYU

12pm Social hour**




** Social hour location:  https://www.wonder.me/r?id=d3898ea0-97a4-49c4-9854-fffa2509ecca

This webapp is best supported by Firefox and Chrome.  Safari is not supported.

  (Please access the social hour using your first and last name. The web app may ask you to take a picture. 

   You can "bypass" this by just covering the camera with your finger when taking your picture.)





Titles and Abstracts



















Spring 2021


Eden Prywes, Princeton University 

Quasiconformal Flows on Non-Conformally Flat Spheres

I will present integral curvature conditions for a Riemannian metric g on S^4 that quantify the best bilipschitz constant between (S^4,g) and the standard metric on S^4.  The condition relies on the integral of the QQ-curvature of the metric and the Weyl tensor.  This result was originally proven for the case when g=e^{2u}g_c, where g_c is the standard metric on S^4 by Bonk, Heinonen and Saksman.  I will show how to extend this result to a larger class of metrics that have a positive Yamabe constant.

This is joint work with Alice Chang and Paul Yang.


Vyron Vellis, University of Tennessee, Knoxville

Hölder curves and rectifiability

One of the most well studied problems in geometric measure theory is the problem of Lipschitz rectifiability: which metric spaces are Lipschitz images of the unit interval, and which spaces are Lipschitz images of a subset of the unit interval? The first question has a complete answer while the second has been solved in many settings such as the Euclidean spaces, the Hilbert spaces and Carnot groups. In this talk we discuss a notion of “fractional rectifiability” introduced my Martin and Mattila, where Lipschitz regularity is replaced by Holder regularity. One of the several obstacles in this study is the absence of differentiability. On the other hand, Holder curves are very interesting as they appear in many facets of analysis and include many exotic examples such as connected attractors of iterated function systems. The talk is based on joint papers with Matthew Badger and Lisa Naples.


Ilmari Kangasniemi, Syracuse University

Uniformly quasiregular maps and the manifolds (S^2 x S^2)#(S^2 x S^2)

Uniformly quasiregular (UQR) maps are quasiregular self-maps for which the distortion of the map remains bounded under iteration. UQR maps hence constitute a geometric higher-dimensional generalization of holomorphic dynamics. In this talk, we present a topological obstruction which shows that (S²xS²)#(S²xS²), the connected sum of two copies of S²xS², does not admit a non-constant non-injective UQR self-map. This specific manifold is notable as it's the main known non-trivial example of a quasiregularly elliptic manifold, to which the question of admitting such a UQR map is closely related. The proof reveals a connection to the theory of geometrically formal manifolds, which are manifolds admitting a Riemannian metric for which the wedge product of harmonic forms is harmonic. Another major component of the proof is the use of a conformally scaled Clifford product.


Alex Wright, University of Michigan

Towards optimal spectral gaps in large genus

I will discuss a recent preprint with Michael Lipnowski (arXiv:2103.07496), in which we show that typical high genus hyperbolic surfaces have first eigenvalue of the Laplacian at least 3/16 - epsilon. The key input is a geometric understanding of typical closed geodesics at length scales that grow slowly with the genus, building on work of Mirzakhani and Petri. This allows us to obtain bounds for the integral of the Selberg trace formula over the moduli space of Riemann surfaces. 


Daniel Meyer, University of Liverpool

The solenoid, the Chamanara space, and symbolic dynamics

The solenoid may be obtained as a hyperbolic attractor of a diffeomorphism of the solid torus. Alternatively it may be described as an inverse limit or as a mapping torus. The latter description is closely related to $p$-adic numbers. As a topological space it is indecomposable, homogeneous, and in fact a topological group. In this talk we present a 2-dimensional home for the solenoid: we show it fits, under a mild equivalence relation, into the Chamanara space, which is a surface with a single singular point, and that the solenoid homeomorphism descends to the baker's map on the Chamanara space. We also discuss how a further quotient then yields the tight horseshoe acting on the 2-sphere. The proofs are based on the usual symbolic descriptions of the solenoid and the horseshoe. This should be viewed as a lower dimensional analog of a construction, which is the subject of current work, that aims to extend expanding Thurston maps to generalized pseudo-Anosov maps acting on singular 3-manifolds with quotients to the 3-sphere. This is joint work with Andre de Carvalho.


Ilya Gekhtman, Technion Israel Institute of Technology

Gibbs measures vs. random walks in negative curvature

Abstract: The ideal boundary of a negatively curved manifold naturally carries two types of measures. On the one hand, we have conditionals for equilibrium (Gibbs) states associated to Hoelder potentials; these include the Patterson-Sullivan measure and the Liouville measure. On the other hand, we have stationary measures coming from random walks on the fundamental group. We compare and contrast these two classes.First, we show that both of these of these measures can be associated to geodesic flow invariant measures on the unit tangent bundle, with respect to which closed geodesics satisfy different equidistribution properties. Second, we show that the absolute continuity between a harmonic measure and a Gibbs measure is equivalent to a relation between entropy, (generalized) drift and critical exponent, generalizing previous formulas of Guivarc’h, Ledrappier, and Blachere-Haissinsky-Mathieu. This shows that if the manifold (or more generally, a CAT(-1) quotient) is geometrically finite but not convex cocompact, stationary measures are always singular with respect to Gibbs measures. A major technical tool is a generalization of a deviation inequality due to Ancona saying the so called Green distance associated to the random walk is nearly additive along geodesics in the universal cover. Part of this is based on joint work with Gerasimov-Potyagailo-Yang and part on joint work with Tiozzo.


Joan Lind, University of Tennessee

Convergence of the Probabilistic Interpretation of Modulus

Abstract:  Given a Jordan domain and two arcs on boundary, the modulus of the curve family connecting the arcs is famously related, via a conformal map, to the corresponding modulus in a rectangle. Moreover, in the case of the rectangle the family of horizontal segments connecting the two sides has the same modulus as the entire connecting family. Pulling these segments back to the domain via the conformal map yields a family of extremal curves in the domain. We show that these extremal curves can be approximated by some discrete curves arising from an orthodiagonal approximation of the domain. Moreover, these curves carry a natural probability mass function (pmf) deriving from the theory of discrete modulus, and these pmf's converge to the uniform distribution on the set of extremal curves. One key ingredient is an algorithm that, for an embedded planar graph, takes the current flow between two sets of nodes, and produces a unique path decomposition with non-crossing paths.  This is joint work with Nathan Albin and Pietro Poggi-Corradini.


Meng Wu, University of Oulu

Furstenberg's intersection conjecture and some recent developments

Abstract: In the 1960s, H. Furstenberg proposed a conjecture about dimension of intersections of times 2 and times 3 invariant sets. This conjecture has been proved some years ago by Pablo Shmerkin and by me, independently. A nice short proof has been given recently by Tim Austin. I will talk about the intersection conjecture and some recent related work.


Livio Liechti, University of Fribourg

Strata of translation surfaces and trace field degrees

Abstract: The trace field of a translation surface is the field extension of the rationals generated by the traces of derivatives of the affine diffeomorphisms of the surface. It is a number field whose degree is known to be bounded from above by the genus of the surface. We show that in all strata of translation surfaces of genus g, the degree of the trace field realizes all values between 1 and g. Our result can be strengthened to work for each connected component of every stratum. This is joint work in progress with E. Lanneau.


Ilya Binder, University of Toronto

When can you compute Harmonic Measure

Abstract: What do you need to know about a domain to be able to compute its harmonic measure? Can you always do it by the same algorithm for all points of the domain? I will make these questions precise and partially answer them. I will also advertise some open computability questions in Complex Analysis. The talk is based on joint work with Adi Glucksam, Cristobal Rojas, and Michael Yampolsky.


No seminar on June 15th due to a concurrent workshop.


Anja Randecker, Heidelberg University

Saddle connection complex: coarse and fine

Translation surfaces arise naturally in many different contexts such as the theory of mathematical billiards, of Teichmüller spaces, or of stability conditions of categories. A new approach to study translation surfaces is to encode their geometry in a combinatorial object, called the saddle connection complex. For translation surfaces, this complex plays the same role as its more established cousin, the arc complex, does for topological surfaces.

In this talk, I will define translation surfaces and the saddle connection complex and introduce you to some properties of its fine geometry (in particular Ivanov-type rigidity) and its coarse geometry (in particular no quasi-isometric rigidity). Both is based on joint works with Valentina Disarlo, Huiping Pan, and Robert Tang.


Eveliina Peltola, University of Bonn

On Loewner evolutions with jumps 

Abstract: I discuss the behavior of Loewner evolutions driven by a Levy process. Schramm's celebrated version (Schramm-Loewner evolution), driven by standard Brownian motion, has been a great success for describing critical interfaces in statistical physics. Loewner evolutions with other random drivers have been proposed, for instance, as candidates for finding extremal multifractal spectra, and some tree-like growth processes in statistical physics. Questions on how the Loewner trace behaves, e.g., whether it is generated by a (discontinuous) curve, whether it is locally connected, tree-like, or forest-like, have been partially answered in the symmetric alpha-stable case. We consider the case of general Levy drivers. Joint work with Anne Schreuder (Cambridge).

Winter 2021

Chris Bishop, Stony Brook University

      Weil-Petersson curves, traveling salesman theorems, and minimal surfaces

Abstract: I will describe several new characterizations of Weil-Petersson curves. These curves are the closure of the smooth planar closed curves for the Weil-Petersson metric on universal Teichmuller space defined by Takhtajan and Teo. Their work was motivated by problems in string theory, but the same class arises naturally in geometric function theory, Mumford's work on computer vision, and the theory of Schramm-Loewner evolutions (SLE). The new characterizations include quantities such as Sobolev smoothness, Mobius energy, fixed curves of biLipschitz involutions, Peter Jones's beta-numbers, the thickness of hyperbolic convex hulls, the total curvature of minimal surfaces in hyperbolic space, and the renormalized area of these surfaces. Moreover, these characterizations extend to higher dimensions and remain equivalent there. 

Talk slides: here. 


Sarah Maloni, University of Virginia

Convex hulls of quasicircles in hyperbolic and anti-de Sitter space

Abstract: Thurston conjectured that quasi-Fuchsian manifolds are determined by the induced hyperbolic metrics on the boundary of their convex core and Mess generalized those conjectures to the context of globally hyperbolic AdS spacetimes. In this talk I will discuss a universal version of these conjectures and prove the existence part by considering convex sets spanning quasicircles in the boundary at infinity of hyperbolic and anti-de Sitter space. This work generalizes Alexandrov and Pogorelov's results about the characterization of the metrics induced on the boundary of a compact convex subset of hyperbolic space. This is joint work with Bonsante, Danciger and Schlenker.


Matthew Romney, Stony Brook University

The branch set of minimal disks in metric spaces

Abstract: A theory of minimal surfaces in metric spaces has recently been developed by Lytchak–Wenger using the quadratic isoperimetric inequality as a basic axiom. This assumption is quite broad, being satisfied, for example, by compact Finsler manifolds, complete CAT(k) spaces, and all Banach spaces. In this talk, we discuss several new contributions to this theory related to the structure of the branch set of a minimal surface, i.e., the set of points where an energy-minimizing mapping is not a local homeomorphism. This is joint work with Paul Creutz.


Pekka Pankka, University of Helsinki

Quasiregular curves

Abstract: The analytic definition of quasiconformal, and more generally quasiregular, mappings between Riemannian manifolds requires that the domain and range of the map have the same dimension. This equidimensionality presents itself also in the local topological properties on quasiregular mappings such as discreteness and openness. In this talk, I will discuss an extension of quasiregular mappings which covers also holomorphic curves, called quasiregular curves, for which the range may have higher dimension than the domain. 


Wenbo Li, University of Toronto

Conformal dimension and minimality of stochastic objects 

Abstract: In this talk, we discuss the conformal dimension of some stochastic objects. The conformal dimension of a metric space is the infimum of the Hausdorff dimension of all its quasisymmetric images. We call a metric space minimal if its conformal dimension equals its Hausdorff dimension. We begin with a construction of a graph of a random function which is minimal. Inspired by this, we apply the same techniques to the study of 1-dimensional Brownian graphs. The main tool is the Fuglede modulus. This is a joint work with Ilia Binder and Hrant Hakobyan.


Jayadev S. Athreya,  University of Washington, Seattle

Square-integrability of the Mirzakhani function and statistics of simple closed geodesics on hyperbolic surfaces

Abstract:  We study the global behavior of the Mirzakhani function B which assigns to a hyperbolic surface X the Thurston measure of the set of measured geodesic laminations on X of hyperbolic length at most 1. We improve bounds of Mirzakhani describing the behavior of this function near the cusp of moduli space and deduce that B is square-integrable with respect to the Weil-Petersson volume form. We relate this knowledge of B to statistics of counting problems for simple closed hyperbolic geodesics. This is joint work with Francisco Arana-Herrera.


Paul Creutz, University of Cologne

   Maximal metric surfaces

By the strengthened Bonk-Kleiner-Theorem of Lytchak-Wenger, every Ahlfors 2-regular, linearly locally connected metric space homeomorphic to the sphere admits a canonical quasisymmetric parametrization. As a consequence to every such quasisphere one can associate a unique Finsler metric on the standard sphere. Two quasispheres are called analytically equivalent if they determine the same Finsler metric. In the talk we will see that every equivalence class of quasispheres contains a unique maximal representative. This maximal quasisphere can also be characterized in terms of thick geodecity, volume rigidity or the Sobolev-to-Lipschitz property.


Kevin Pilgrim, Indiana University Bloomington

On the action of a Thurston map on the curve complex

Abstract: Via taking preimages, a Thurston map f : (S^2, P) → (S^2, P) with postcritical set P induces a map on the set of simplices of an associated curve complex. This action is closely related to the corresponding pullback map on Teichmueller space. I will survey some known results.  When #P=4, the Teichmueller metric is the hyperbolic metric, and study of the geometry of continued fractions provides insight into relationships between the complexity of f and the complexity of its fixed-curves, including potential obstructions.


Séverine Rigot, Université Côte d'Azur, Nice, France

Monotone subsets of step-2 Carnot algebras

Monotone subsets of the Heisenberg group have been first introduced by J. Cheeger and B. Kleiner. Namely, they proved that the fact that the Heisenberg group does not admit a bi-Lipschitz embedding in L^1 can be reduced to a classification of its monotone subsets. I will discuss in this talk the generalization of such a class of sets in the wider framework of step-2 Carnot algebras. As I shall explain, some motivations for studying monotone sets in this setting stem from their relationship with minimal hypersurfaces, subharmonic maps, and some notions of uniform rectifiability. I will present recent results that give a classification of precisely monotone subsets in rank-3 step-2 Carnot algebras and give some hints towards a classification in arbitrary step-2 Carnot algebras. Based on joint works with E. Le Donne and D. Morbidelli.




Fall 2020

Annina Iseli, University of California Los Angeles

Eliminating obstructions for Thurston maps with four postcritical points

 A Thurston map is a branched covering map of the 2-sphere which is not a homeomorphism and for which every critical point has a finite orbit under iteration of the map. Frequently, a Thurston map admits a description in purely combinatorial-topological terms. In this context it is an interesting question whether a given map can (in a suitable sense) be realized by a rational map with the same combinatorics. This question was answered by Thurston in the 1980's in his celebrated characterization of rational maps. Thurston's Theorem roughly says that a Thurston map is realized if and only if it does not admit a Thurston obstruction, which is an invariant multicurve that satisfies a certain growth condition. However, in practice it can be very hard to verify whether a given map has no Thurston obstruction, because, in principle, one would need to check the growth condition for infinitely many curves.

In this talk, we will consider a specific family of Thurston maps with four postcritical points that arises from Schwarz reflections on flapped pillows. Using a counting argument, we will prove a very simple necessary and sufficient condition for a map in this family to be realized by a rational map. In the last part of the talk, we will discuss a generalization of this result which states that, given an obstructed Thurston map with four postcritical points, one can eliminate obstructions by applying a so-called blowing up operation. These results are joint with M. Bonk and M. Hlushchanka.


Malik Younsi, University of Hawaii

Holomorphic motions, capacity and conformal welding

The notion of a holomorphic motion was introduced by Mané, Sad and Sullivan in the 1980's, motivated by the observation that Julia sets of rational maps often move holomorphically with holomorphic variations of the parameters. In the years that followed, the study of the behavior of various set-functions under holomorphic motions became an area of significant interest. For instance, holomorphic motions played a central role in the work of Astala on distortion of Hausdorff dimension and area under quasiconformal mappings.

In this talk, I will first review the basic notions and results related to holomorphic motions, including the extended lambda lemma. I will then present some recent results on the behavior of logarithmic capacity and analytic capacity under holomorphic motions. The proofs involve different notions such as conformal welding, quadratic Julia sets and harmonic measure. This is joint work with Tom Ransford and Wen-Hui Ai.

Link to Malik's slides 

Kostya Drach, Aix-Marseille University

Dynamical Rigidity of Rational Maps 

How complicated the dynamics of a rational map can be? In the talk, we will explain why for large families of rational maps of arbitrary degree their dynamics is no more complicated than the dynamics of complex polynomials, in a very precise sense. For these families, the orbit of every point in the Julia set is either rigid, i.e. can be distinguished in some combinatorial terms from all other orbits, or it lands in the Julia set of a polynomial-like restriction of the original rational map. This is a statement of dynamical rigidity. It can also be phrased in a less dynamical way as a result on local connectivity of Julia sets along rigid orbits. In the talk, a guiding example for us will be rigidity of Newton maps which was recently established jointly with Dierk Schleicher. If time permits, we will also talk about related questions on rigidity in parameter spaces.  


Nageswari Shanmugalingam, University of Cincinnati,

Prime end boundaries and their applications in potential theory and BQS maps

In considering continuous extensions of conformal mappings between planar domains to their boundaries, no such extension is guaranteed; by constructing an alternate notion of prime ends for such domains, Caratheodory demonstrated that conformal maps between simply connected planar domains do extend as homeomorphisms to their respective prime end boundaries. For non-simply connected domains Cartheordory's construction need not work. In this talk we will discuss an alternative notion of prime ends and discuss its difference from the Caratheodory prime ends, and its application to potential theory and branched quasisymmetric maps.


Dimitrios Ntalampekos, Stony Brook University

Rigidity theorems for circle domains

A circle domain Ω in the Riemann sphere is a domain each of whose boundary components is either a circle or a point. A circle domain Ω is called conformally rigid if every conformal map from Ω onto another circle domain is the restriction of a Möbius transformation. In this talk I will present some new rigidity theorems for circle domains satisfying a certain quasihyperbolic condition. As a corollary, John and Hölder circle domains are rigid. This provides new evidence for a conjecture of He and Schramm, relating rigidity and conformal removability. This talk is based on joint work with Malik Younsi.



Sergei Tabachnikov, Pennsylvania State University

Cross-ratio dynamics on ideal polygons

Define a relation between labeled ideal polygons in the hyperbolic space by requiring that the complex distances (a combination of the distance and the angle) between their respective sides equal c; the complex number c is a parameter of the relation. This defines a 1-parameter family of maps on the moduli space of ideal polygons in the hyperbolic space (or, in its real version, in the hyperbolic plane). I shall discuss complete integrability of this family of maps and related topics, including its connection with the Korteweg-de Vries equation.



Dzmitry Dudko, Stony Brook University

Expanding and relatively expanding maps

A non-invertible Thurston map f is isotopic to an expanding map if and only if f admits no Levy cycle (unless an ``exceptional'' torus endomorphism doubly covers f). Even if there is a Levy cycle, f may still have an expanding ``cactoid'' quotient. We will discuss different properties of expanding and relatively expanding maps and their relations to matings and Thurston decidability problems. 


InSung Park, Indiana University Bloomington  

Julia sets with Ahlfors regular conformal dimension one

Ahlfors regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasisymmetric class of Ahlfors regular metric spaces. As a fractal embedded in the Riemann sphere, the Julia set of a post-critically finite rational map has ARconfdim between 1 and 2. The Julia set has ARconfdim 2 if and only if it is the whole sphere. However, the other extreme case, when ARconfdim=1, contains diverse Julia sets, including all Julia sets of post-critically finite polynomials or Newton maps, and critically fixed rational or anti-rational maps. In this talk, we show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there is a f-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero. The latter condition is also a defining property of a crochet map, in the sense of Dudko-Hlushchanka-Schleicher.

This work involves several past talks in the Quasiworld seminar. The proof relies on the results talked by D.Thurston and M.Hlushchanka. Critically fixed anti-rational maps, discussed by L.Geyer and S.Mukherjee, are examples of ARconfdim=1. In the spirit of Sullivan's dictionary, this result can also be compared with the talk by J.Mackay.


Jack Burkart, Stony Brook University

Transcendental Julia sets with Fractional Packing Dimension

In this talk, we will define and compare different definitions of dimension (Hausdorff, Minkowski, and packing) used to analyze fractal sets. Then we will give a survey of several results about these dimensions in the context of Julia sets of entire functions. The type of entire function we are studying makes a tremendous difference; the results are quite different if the Julia set is associated with a polynomial versus a Julia set associated with a transcendental (non-polynomial) entire function.  We will conclude by discussing my recent construction of Julia sets of transcendental entire functions with packing dimension strictly between one and two. 



Katrin Fässler, University of Jyväskylä

The Loomis-Whitney inequality in the Heisenberg group

Abstract: The classical Loomis-Whitney inequality in R^3 bounds the volume of a set K in R^3 by the areas of its orthogonal projections to the three coordinate planes. I will discuss a counterpart for this inequality in a noncommutative group, the first Heisenberg group H^1. Now the relevant "coordinate projections" are two nonlinear maps, and I will show how the associated Loomis-Whitney inequality can be proven by solving an incidence geometry problem in R^2. As a corollary, we obtain a refined version of the classical geometric Sobolev inequality in H^1. This is joint work with Tuomas Orponen and Andrea Pinamonti.



Yusheng Luo, University of Michigan

Cusps, pinching and self-bump

Abstract: In this talk, we will discuss how degenerations of Blaschke products allows us to study the boundaries of hyperbolic components. We give a classification on the geometrically finite polynomials on the boundary of the main hyperbolic component H_d containing z^d. In particular, this gives a classification of `satellite components' of H_d. We prove a pinching theorem for quasi-Blaschke products QB_d which allows us to show two geometrically finite polynomials on the boundary of H_d are mateable if and only if there is no topological obstruction. Using related methods, we also construct a self-bump on the boundary of H_d and QB_d, showing the closures H_d and QB_d are not topological manifolds with boundary for d ≥ 4.

Our theory runs parallel with the developments in Teichmuller theory and Kleinian groups. We will also talk about some comparisons of these results.

 


Anton Lukyanenko, George Mason University

Computational Tools and the Quasiworld Community

A computational approach provides a rich direction for mathematics research, providing intuition, experimental evidence for conjectures, and connections to applications. We will start by discussing some interactions between the quasiworld community and computation, ranging from early hyperbolic space visualizations at the Geometry Center to recent work in modeling human vision via sub-Riemannian geometry. We will then talk about a few different tools available for mathematical computation, and finish with a Mathematica tutorial showing how to draw von Koch snowflakes and snowballs.


Summer 2020 Quasiworld Workshop

We hosted a July workshop to conclude our spring and summer seminar series. See the email list for details. We will add the abstracts and titles here shortly. The schedule is below. All times in the schedule are Pacific Standard Time (GMT-7).


Abstract:  Each negatively curved homogeneous space has the structure of a Lie group that is the semidirect product of a nilpotent group and the real line. The visual boundary of such a space can be identified with the one-point compactification of the nilpotent group. In this talk we focus on the situation where such a nilpotent group is a reducible Carnot group and we will prove that every self-quasi-isometry on such negatively curved homogeneous spaces   is a rough isometry. In addition, we will also show that every two Riemannian distances on each of these groups modelling negatively curved homogeneous spaces are roughly similar.  This is a joint collaboration with Gabriel Pallier and Xiangdong Xie.


Abstract: Thurston's Master Teapot is the closure of the set of all points (z,λ)∈C×R(z,\lambda) \in \mathbb{C} \times \mathbb{R} such that λ\lambda  is the growth rate of a critically periodic unimodal self-map of an interval and zz is a Galois conjugate of λ\lambda.  I will present a new characterization of which points are in this set.  This characterization gives a way to think of each horizontal slice of the Master Teapot as an analogy of the Mandelbrot set for a ``restricted iterated function system.''  An application of this characterization is that the Master Teapot is not invariant under the map (z,λ)↦(−z,λ)(z,\lambda) \mapsto (-z,\lambda).  This presentation is based on joint work with Chenxi Wu. 


Tuesday 10-11 am, Giulio Tiozzo: Statistical properties of coarse expanding dynamical systems

Abstract: Expanding Thurston maps are a class of branched covers of the sphere which extend the notion of hyperbolic rational map to a general metric context, introduced by Bonk-Meyer. Moreover, Haissinsky and Pilgrim defined the class of coarse expanding conformal maps of metric spaces,  which in particular generalize expanding Thurston maps to the postcritically infinite case.

We establish the thermodynamic formalism for these systems. In particular, we prove existence and uniqueness of equilibrium states for a wide class of potentials, as well as statistical laws such as a central limit theorem, law of iterated logarithm, exponential decay of correlations and a large deviation principle.

       This is joint work with T. Das, F. Przytycki, M. Urbanski and A. Zdunik.


Tuesday 11am-12pm, Volodia Nekrashevych: Conformal dimension of self-similar groups,

Abstract: We will talk about conformal and topological dimensions of contracting self-similar groups, and their connection to algebraic properties such as word problem, amenability, growth, presentations, etc..


Wednesday 7-8 am, Alex Eremenko: Moduli spaces for Lamé functions and Abelian differentials of the second kind

Abstract: This talk is based on a joint work with Andrei Gabrielov, Gabriele Mondello and Dmitri Panov.

The moduli space for Lamé functions of degree m is isomorphic to the space of Abelian  differentials on elliptic curves with single zero of order 2m at the origin, and m double poles with vanishing residues. We describe the topology of this space: it a Riemann surface of finite type; we find the number of components and genera and the numbers of punctures for each component.

Two applications will be discussed: the proof of a conjecture of Robert Maier on degrees of Cohn's polynomials, and a description of the degeneration locus of spherical metrics on tori with one conic singularity.


Wednesday 8-9, Kirill Lazebnik: Bers Slices in Families of Univalent Maps

Abstract: We will look at a few motivating examples of how two different conformal dynamical systems can be "combined" to yield a single conformal dynamical system. Then we will explain how "combining" (1) a class of groups generated by reflections in Euclidean circles with (2) the anti-holomorphic polynomial z−>¯zdz->\overline{z}^d yields (3) a certain class of dynamical systems arising from a classical family of univalent maps. This talk is based on joint projects with Nikolai Makarov and Sabyasachi Mukherjee.


Abstract: We introduce a quantity called Loewner potential, associated to a collection of n disjoint simple chords (multichords) joining 2n boundary points of a simply connected domain in the complex plane. It is first motivated by the large deviations of multiple SLE0+, a probabilistic model of interfaces in 2D statistical mechanics configurations.

On the analytic side, we show that finite potential multichords are quasiconformal images of analytic curves, and potential minimizers are real locus of a rational function. Moreover, the potential can be expressed in terms of determinants of Laplacians, and the minimal potential satisfies the semiclassical limit of PDEs arising from the Belavin-Polyakov-Zamolodchikov equations in conformal field theory.

This is joint work with Eveliina Peltola.


Abstract: Thurston proved that a branched cover of the plane that satisfies certain finiteness conditions is either equivalent to a polynomial (that is: conjugate via a mapping class) or it has a topological obstruction.  We use topological techniques – adapting tools used to study mapping class groups – to produce an algorithm that determines when a branched cover is equivalent to a polynomial, and if it is, determines which polynomial a branched cover is equivalent to.  Our approach gives a new, topological solution to Hubbard's twisted rabbit problem, as well as generalizations of this problem.  This is joint work with Jim Belk, Justin Lanier, and Dan Margalit.


Spring 2020

March 31st: Mario Bonk, UCLA, Uniformly branching trees


Abstract: A quasiconformal tree T is a (compact) metric tree that is doubling and of bounded turning. We call T trivalent if every branch point of  T has exactly three branches. If the set of branch points is uniformly relatively separated and uniformly relatively dense, we say that  T is uniformly branching. In this talk I will explain that a metric space T is quasisymmetrically equivalent to the continuum self-similar tree (I’ll define what this is) if and only if it is a trivalent quasiconformal tree that is uniformly branching. In particular, any two trees of this type are quasisymmetrically equivalent. 


Slides available here. 


April 7th: Sergiy Merenkov, CUNY, Quasi-connections between groups and rational maps 

Abstract: I plan to discuss the (non-)existence of quasisymmetric maps between limit sets of certain Kleinian groups and Julia sets of post-critically finite rational maps. The main emphasis will be on reflection groups associated to circle packings and critically finite anti-rational maps with gasket-like Julia sets. Even though it is easy to see that there is no quasisymmetry between the corresponding limit and Julia sets, there is a trans-quasiconformal (or David) map from the Julia set to the limit set in the Apollonian gasket case. The talk is based on several joint works with Mario Bonk, Russell Lodge, Misha Lyubich, and Sabya Mukherjee.  


Slides available here. 

Recording link sent on email list or available upon request from organizers.


April 14th: Sylvester Eriksson-Bique, UCLA, Isoperimetry via iteration and Poincarè inequalities

Abstract: A number of conditions, such as the Loewner condition, Muckenhoupt and the quasisymmetry condition, have "weak’’ analogues. That is, if they hold at all scales with sufficiently good parameters, then they can be iterated. The goal of this talk is to bring out a similar behavior for isoperimetric inequalities via a new condition from my joint work with Jasun Gong. This is a new perspective that has shown application in the regime of Poincaré inequalities for exponents greater than one - in particular the Keith-Zhong self-improvement. Here, we show that despite significant differences the p=1 still has some iterations that can be used and which give easier conditions to verify. This has some applications of interest, such as constructing new examples. Additionally we will discuss time permitting recent work on modulus, duality and the applications that these have to the structure and characterization of functions of bounded variation. 


Slides available here. 

Recording link available in email list or upon request from organizers.


April 21st: Guy C. David, Ball State University, Quantitative decompositions of Lipschitz mappings

Abstract: Given a Lipschitz map, it is often useful to chop the domain into pieces on which the map has simple behavior. For example, depending on the dimensions of source and target, one may ask for pieces on which the map behaves like a bi-Lipschitz embedding or like a linear projection. For many issues, it is even more useful if this decomposition is quantitative, i.e., with bounds independent of the particular map or spaces involved. After surveying the question of bi-Lipschitz decomposition, we will discuss the more complicated case in which dimension decreases, e.g., for maps from R^3 to R^2. This is recent joint work with Raanan Schul, improving a previous result of Azzam-Schul.


Slides available here. 

Video to be made available on Friday on the email list. If you wish to receive it afterwards, contact the organizers.


April 28th: Istvan Prause, University of Eastern Finland, Integrability of limit shapes

Abstract: Many models in statistical mechanics exhibit limit shape formation: in the macroscopic limit the random system settles into a fixed deterministic limit. Notable examples include the dimer model and the six-vertex model. These geometric limit surfaces often (known or conjectured to) display arctic boundaries, sharp transitions from ordered (frozen) to disordered (liquid) phases. From an analysis point of view, limit shapes are minimizers of (singular) gradient variational problems. I’ll show that techniques from the quasiworld reveal an intrinsic conformal structure in which the problem simplifies dramatically and allow for explicit parametrizations even beyond free fermionic settings. The talk is based on joint work with Rick Kenyon.

Slides available here. 

Video to be made available on Friday on the email list. If you wish to receive it afterwards, contact the organizers.


May 5th: Bruce Kleiner, Courant Institute, NYU,  Rigidity and regularity of mappings between Carnot groups.

Abstract: I will discuss mappings between Carnot groups under various regularity assumptions -- smooth, bilipschitz, quasiconformal, and Sobolev, giving new regularity and rigidity results.  This is joint work with Stefan Muller and Xiangdong Xie.


Talk based on the paper: https://arxiv.org/abs/2004.09271 .

Video available on email list or upon request.

May 12th: Sarah Koch, University of Michigan, Irreducibility in complex dynamics

Abstract: A major goal in complex dynamics is to understand dynamical moduli spaces; that is, conjugacy classes of holomorphic dynamical systems. One of the great successes in this regard is the study of the moduli space of quadratic polynomials; it is isomorphic to the complex plane. This moduli space contains the famous Mandelbrot set, which has been extensively studied over the past 40 years. Understanding other dynamical moduli spaces to the same extent tends to be more challenging as they are often higher-dimensional. In this talk, we consider the moduli space of quadratic rational maps, which is isomorphic to C^2. We will focus on special algebraic curves, called "Milnor curves" in this space. In general, it is unknown if Milnor curves are irreducible over C. Because these curves are smooth, this is equivalent to asking whether they are connected. We will exhibit the first infinite collection of Milnor curves that are connected. This is joint work with X. Buff and A. Epstein. 

Video available on email list.

May 19th: Dragomir Saric, Queens College, The ergodicity of the geodesic flow and Fenchel-Nielsen coordinates

Abstract: Fix a topological pants decomposition P of an infinite surface X. The set of conformal hyperbolic metrics on X is parametrized by the Fenchel-Nielsen coordinates on the pants decomposition P. We give sufficient conditions on the Fenchel-Nielsen coordinates such that the harmonic type of X is parabolic, i.e. X in O_G. This is a joint work with Ara Basmajian and Hrant Hakobyan. 


Slides for the talk here. 

Video available in the Friday emal for the email list.


May 26th: John Mackay, University of Bristol, Conformal dimension and decompositions of hyperbolic groups


Abstract:  The conformal dimension of a metric space is the infimum of the possible values of its Hausdorff dimension under quasisymmetric homeomorphisms.  The conformal dimension of the boundary at infinity of a Gromov hyperbolic group is a fundamental quasi-isometric invariant.  I will discuss how this invariant behaves when the group splits over two-ended subgroups (i.e. when the boundary has local cut points), and applications.  Joint work with Matias Carrasco.

June 2nd: Mikhail Hlushchanka, UCLA, Invariant graphs and curves and their applications in complex dynamics.


Abstract: Invariant graphs and curves appear naturally in various aspects of complex dynamics. For instance, Hubbard trees are used to classify all postcritically-finite polynomials, while Thurston’s obstructions, which are certain invariant (multi)curves, appear in the celebrated Thurston characterization of rational maps. 

Concentrating on specific families of postcritically-finite rational maps, I will discuss how invariant graphs help to approach several open questions. In particular, we will talk about dynamics of simple closed curves under the pullback by a rational map, as well as algebraic properties (such as growth and amenability) of the associated iterated monodromy groups


Slides for the talk available here. 



June 9th, Toby Hall, University of Liverpool, Inverse limits of tent maps and sphere homeomorphisms


The inverse limit construction allows the study of a non-invertible dynamical system to be converted into the study of a self-homeomorphism of the inverse limit. The inverse limit is usually much more complicated than the original space: for example, inverse limits of tent maps - simple piecewise affine self-maps of the interval - typically contain indecomposable continua.

In this talk I'll describe how mild identifications on the inverse limits of tent maps turn them into 2-spheres, and make it possible to construct parameterized family of sphere homeomorphisms corresponding to the parameterized family of tent maps. I'll discuss the dynamics of these homeomorphisms, which include examples of Thurston's pseudo-Anosov maps, so-called generalized pseudo-Anosovs, and the (further generalized) measurable pseudo-Anosovs.

This is joint work with Philip Boyland (University of Florida) and André de Carvalho (University of Sao Paulo). Next week André will give a related talk describing similar constructions in higher dimension.


Slides here. 



June 16th, André de Carvalho, University of São Paolo, Title: Inverse limits of expanding Thurston maps and pseudo-Anosov homeomorphisms of the 3-sphere

The inverse limit and natural extension constructions extend a non-invertible dynamical system "naturally" to an invertible one. Inverse limits were probably first used in dynamics by Bob Williams, in the late 1960's, in his study of expanding 1-dimensional attractors. Since then, several researchers have studied inverse limits in dynamics. In complex dynamics, and rational maps in particular, this started to be done systematically in the 1990's, with Dennis Sullivan's work on renormalization and with the work of Meiyu Su and Minsky-Lyubich on Riemann surface laminations associated to rational maps. 

In this talk we discuss inverse limits of expanding Thurston maps with a certain symmetry. Typical examples are real rational maps (i.e., holomorphic maps that can be written with real coefficients). Inverse limit spaces are almost always very complicated topological spaces. We show how a mild collapse on the inverse limit spaces under consideration yields the 3-sphere and the natural extension then becomes a "nice" 3-sphere homeomorphism. More precisely, the statement we discuss is that, starting from our expanding Thurston map of the 2-sphere, there is a mild collapse which turns its inverse limit space into the 3-sphere and its natural extension into a "pseudo-Anosov" homeomorphism. This means that on the 3-sphere there are two singular invariant foliations; these foliations are transverse to each other, are transversely measured and the holonomies preserve these measures; the homeomorphism preserves the foliations and contracts one and expands the other transverse measure.

This talk is closely related to the talk "Inverse limits of tent maps and sphere homeomorphisms" given by Toby Hall on June 9th. There generalized pseudo-Anosov maps of the 2-sphere were constructed from inverse limits of interval endomorphisms. Thus our construction can be viewed as a higher dimensional analog. 

This is joint work (in progress) with Daniel Meyer.

Here are notes from André's talks.   


June 23rd, Lukas Geyer, Montana State University, Title: Classification of critically fixed anti-rational maps

The study of anti-holomorphic dynamics shares many similarities with holomorphic dynamics, but there are a few subtle and substantial differences. Apart from studying the dynamics of anti-rational maps as a subject of independent interest, there are intriguing connections with Kleinian groups and problems in gravitational lensing. In this talk, I will give a complete description of critically fixed anti-rational maps through simple topological models associated with certain planar graphs. I will compare and contrast this result with Mikhail Hlushchanka's recent classification of critically fixed rational maps, sketch the main ideas involved in the proof, and illustrate the theorem with several examples and applications to critically fixed anti-polynomials.


Here  are Lukas's slides. The talk is based on the speakers paper. 



June 30th, Dylan Thurston,  Indiana University Bloomington, Ahlfors-regular conformal dimension from graph energies

We identify the Ahlfors-regular conformal dimension of the Julia set of a rational map as the critical exponent of energies associated to a graph virtual endomorphism. This lets us give numerical upper and lower bounds for the conformal dimension. We illustrate the techniques by giving two families of quartic rational maps, one with conformal dimension approaching 1 and one with conformal dimension approaching 2.


This is joint work with Kevin Pilgrim.


The talk was in a blackboard style, but the images of the "slides" are availble here. 


July 7th, Sabyasachi Mukherjee,  Tata Institute of Fundamental Research, Mumbai, Reflection groups, anti-rational maps, and univalent functions

Abstract: The goal of this talk is to explain novel connections between Kleinian reflection groups, anti-holomorphic rational maps, and Schwarz reflection maps arising from univalent functions. We will describe a natural correspondence between a class of Kleinian reflection groups and anti-rational maps such that the group dynamics on the limit set is equivalent to the anti-rational dynamics on the Julia set. We will also discuss how conformal matings of reflection groups and anti-polynomials are realized as Schwarz reflection maps associated with univalent functions. Time permitting, we will outline applications of this mating theory to extremal problems for schlicht functions, and embedding of Teichmüller spaces of reflection groups in families of schlicht functions.

             Slides available here.