About

Where and when

photo
Dates: 29 May 2023 - 2 June 2023
Venue: Congressi Stefano Franscini - Monte Verità
Strada Collina 84
CH-​6612 Ascona
Switzerland

Organizers

Hyungryul Baik
Mladen Bestvina
Sebastian Hensel
Alessandra Iozzi
Howard Masur
Bram Petri
Beatrice Pozzetti

Funded by

Speakers

Paul Apisa (University of Wisconsin Madison)
Uri Bader (Weizman Institute)
Yves Benoist (Paris - Sud)
Jonas Beyrer (Bonn)
Lei Chen (Caltech)
Benson Farb (Chicago)
Simion Filip (Chicago)
Elia Fioravanti (Bonn)
Koji Fujiwara (Kyoto)
Jingyin Huang (Ohio State)
Dawid Kielak (Oxford)
Bruce Kleiner (NYU)
Chris Leininger (Rice University)
Seonhee Lim (Seoul National)
Kathryn Mann (Cornell)
Yair Minsky (Yale)
Juan Souto (Rennes)
Karen Vogtmann (Warwick)
Amie Wilkinson (Chicago)

Schedule

Monday
9:30 - 10:00     Coffee and registration
10:00 - 10:15CSF Welcome address
10:15 - 11:15Talk 1: Juan Souto
11:20 - 12:20Talk 2: Paul Apisa
12:30 - 15:00Lunch
15:00 - 16:00Talk 3: Jingyin Huang
16:00 - 16:30Coffee
16:30 - 17:30Talk 4: Simion Filip
17:45 - 18:45Talk 5: Amie Wilkinson
     
Tuesday
9:30 - 10:30Talk 6: Lei Chen
10:30 - 11:15Coffee
11:15 - 12:15Talk 7: Uri Bader
12:30 - 15:00Lunch
15:00 - 16:00Talk 8: Chris Leininger
16:00 - 16:30Coffee
16:30 - 17:30Talk 9: Seonhee Lim
17:45 - 18:45Talk 10: Dawid Kielak
     
Wednesday
9:30 - 10:30Talk 11: Benson Farb
10:30 - 11:15Coffee
11:15 - 12:15Talk 12: Karen Vogtmann
12:30 - 14:00Lunch
     
Thursday
9:30 - 10:30Talk 13: Bruce Kleiner
10:30 - 11:15Coffee
11:15 - 12:15Talk 14: Jonas Beyrer
12:30 - 15:00Lunch
15:00 - 16:00Talk 15: Yves Benoist
16:00 - 16:30Coffee
16:30 - 17:30Talk 16: Koji Fujiwara
17:45 - 18:45Talk 17: Kathryn Mann
     
Friday
9:30 - 10:30     Talk 18: Elia Fioravanti
10:30 - 11:15Coffee
11:15 - 12:15Talk 19: Yair Minsky
12:30 - 14:00Lunch

Travel information

General travel information: The closest airports are Milano Malpensa and Zurich, click here for further information.

Shuttle bus: On Sunday 28 May there will be a shuttle bus: the Monte Verità minibus, 8 seats (first come-first served basis) from Locarno main station at the following times:

15:20 / 16:00 / 16:40 / 17:20 / 18:00 / 18:40


Shuttles will also be organised on Friday after lunch

Registration

Registration is closed

Titles and abstracts

Paul Apisa

Title: Hurwitz Spaces, Hecke Actions, and Orbit Closures in Moduli Space

Abstract: The moduli space of Riemann surfaces is a space whose points correspond to the ways to endow a surface with a hyperbolic metric or, equivalently, complex structure. Geodesic flow on moduli space can be used to generate an action of \(\mathrm{GL}(2, \mathbb{R})\) on its cotangent bundle. While work of Eskin, Mirzakhani, Mohammadi, and Filip implies that \(\mathrm{GL}(2, \mathbb{R})\) orbit closures are varieties, the question of which ones occur is wide open. Aside from two well-understood constructions (taking loci of branched covers and subloci of rank two orbit closures) there are only 3 known families of orbit closures: the Bouw-Moller curves, the Eskin-McMullen-Mukamel-Wright (EMMW) examples, and 2 sporadic examples. Building on ideas of Delecroix-Rueth-Wright, I will describe work showing that the Bouw-Moller and EMMW examples can be constructed using just the representation theory of finite groups. The main idea is to connect these examples to Hurwitz spaces of G-regular covers of the sphere for an appropriate finite group \(G\). In the end, I will describe a construction that inputs a finite group \(G\) and a set of generators satisfying a combinatorial condition and outputs a \(\mathrm{GL}(2, \mathbb{R})\) orbit closure in moduli space.

Uri Bader

Title: Higher property T of arithmetic lattices, applications and conjectures

Abstract: We will discuss a higher-degree version of property T and a recent proof that arithmetic lattices in a semisimple Lie group G satisfy it below the rank of G. The proof relies on functional analysis and the polynomiality of higher Dehn functions of arithmetic lattices below the rank.

Along the way we will discuss rigidity results regarding unitary representations of semisimple groups and their lattices and present some relevant conjectures. If time permits, we will discuss some applications to dynamics, to character rigidity and to stability.

Based on a joint work with Roman Sauer.

Yves Benoist

Title: Bounded harmonic maps

Abstract: One hundred years ago, Fatou and Herglotz proved that bounded harmonic functions on the hyperbolic disk are in one to one correspondence with bounded measurable functions on the boundary circle. With D. Hulin, we extend this theorem to harmonic maps when the source and the target spaces are pinched Hadamard manifolds.

Jonas Beyrer

Title: Convex cocompactness in pseudo-Riemannian hyperbolic space and higher higher rank Teichmueller spaces

Abstract: In the last two decades there has been extensive research on 'higher rank Teichmueller spaces' - those are connected components, filled with discrete and faithful representations, of the space of surface groups representations into a simple Lie group. As it turns out there are also two classes of examples of such components for other (higher dimensional) groups than surface groups. In this talk we want to show that there are discrete and faithful representations of hyperbolic groups, different from surface groups, into \(O(p,q)\) constituting connected components of the representation space. Those representations are not only discrete and faithful but have convex cocompact actions on pseudo-Riemannian hyperbolic space. This generalizes results of T. Barbot in the context of Anti-de Sitter space.
Based on joint work with F. Kassel.

Lei Chen

Title: Mapping class groups of circle bundles over a surface

Abstract: In this talk, we study the algebraic structure of mapping class group \(Mod(M)\) of 3-manifolds M that fiber as a circle bundle over a surface \(S^1 \to M \to S_g\). We prove an exact sequence \(1 \to H_1(S_g) \to Mod(M) \to Mod(S_g) \to 1\), relate this to the Birman exact sequence, and determine when this sequence splits. This is joint work with Bena Tshishiku.

Benson Farb

Title: Mapping class groups of K3 surfaces from a Thurstonian point of view

Abstract: In many ways the state of our understanding of homeomorphisms of 4-manifolds in 2022 is essentially that of our understanding of homeomorphisms of 2-manifolds in 1973, before Thurston changed everything. In this talk I will report on some first steps in a project (joint with Eduard Looijenga) whose ultimate goal is to change this.

Simion Filip

Title: Some applications of hyperbolic geometry to algebraic geometry

Abstract: Quadratic forms of signature \((1,n)\) appear naturally in some of the cohomology groups of complex algebraic manifolds. They also appear frequently in spaces that do not, a priori, admit such quadratic forms, and seem to be closely related to the group of (pseudo-)automorphisms of the algebraic manifold. I will discuss joint work with John Lesieutre and Valentino Tosatti in which we consider some situations of the latter kind where the geometry and dynamics of Calabi-Yau manifolds is controlled by certain hyperbolic reflection groups. The necessary background will be explained.

Elia Fioravanti

Title: Global estimates for the pressure metric on quasi-Fuchsian spaces

Abstract: Thermodynamic formalism, as initially developed by Bowen and Ruelle, can be used to quantify the difference between two Anosov flows in terms of a "pressure semi-norm" on suitable spaces of Hölder functions. McMullen showed that the Weil-Petersson metric on Teichmüller space can be interpreted in these terms, after which pressure metrics were defined on broader representation varieties, for instance on quasi-Fuchsian spaces by Bridgeman and on Hitchin components by Bridgeman, Canary, Labourie and Sambarino.

While the infinitesimal structure at the Fuchsian locus is often well-understood in these varieties, almost nothing is known on the global behaviour of pressure metrics. For instance, it remains an open problem whether quasi-Fuchsian spaces and Hitchin components admit representations arbitrarily far from the Fuchsian locus.

In this direction we show that, for every \(\varepsilon>0\), the set of \(\varepsilon\)-thick quasi-Fuchsian manifolds has finite diameter for the pressure metric. A curious consequence is that the Fuchsian locus is highly distorted inside quasi-Fuchsian spaces. Joint work with Ursula Hamenstädt, Frieder Jäckel and Yongquan Zhang.

Koji Fujiwara

Title: The rates of growth in hyperbolic groups.

Abstract: For a finitely generated group of exponential growth, we study the set of exponential growth rates for all possible finite generating sets. Let G be a hyperbolic group. In a joint work with Sela, we proved that the set of growth rates is well-ordered. Also, given a number, there are only finitely many generating sets that have this number as the growth rate. I will also discuss variations and generalization of this result other than a hyperbolic group.

Jingyin Huang

Title: Labeled four wheels and the \(K(\pi,1)\) problem for reflection arrangement complements

Abstract: The \(K(\pi,1)\)-conjecture for reflection arrangement complements, due to Arnold, Brieskorn, Pham, and Thom, predicts that certain complexified hyperplane complements associated to infinite reflection groups are Eilenberg MacLane spaces. We establish a close connection between a very simple property in metric graph theory about 4-cycles and the \(K(\pi,1)\)-conjecture, via elements of non-positively curvature geometry. We also propose a new approach for studying the \(K(\pi,1)\)-conjecture. As a consequence, we deduce a large number of new cases of Artin groups which satisfies the \(K(\pi,1)\)-conjecture.

Dawid Kielak

Title: Kazhdan constants for Chevalley groups over \(\mathbb{Z}\)

Abstract: I will report on computations yielding lower bounds for Kazhdan constants of all elementary Chevalley groups over \(\mathbb{Z}\) associated to irreducible root systems of rank at least 2. For groups other than \(\mathrm{SL}_n(\mathbb{Z})\), these are the first such bounds. This is joint work with Marek Kaluba.

Bruce Kleiner

Title: Rigidity and flexibility of mappings between Carnot groups.

Abstract: After covering background and motivation, the lecture will discuss some recent progress on geometric mapping theory in Carnot groups. This is based on joint work with Stefan Muller, Laszlo Szekelyhidi, and Xiangdong Xie.

Chris Leininger

Title: End-periodic mapping tori and hyperbolic volume

Abstract: The mapping torus of an end-periodic homeomorphism is the interior of a compact manifold. Under an appropriate "irreducibility" assumption, the manifold admits a unique, complete hyperbolic structure with totally geodesic boundary. We prove that the volume of this manifold is comparable to the (asymptotic) translation length of the homeomorphism acting on the pants graph, by analogy with Brock's Theorem for mapping tori of pseudo-Anosov homeomorphisms. This represents joint work with Elizabeth Field, Autumn Kent, Heejoung Kim, and Marissa Loving.

Seonhee Lim

Title: Inhomogeneous Diophantine Approximation in the function field

Abstract: Diophantine approximation around a target vector has been studied in the real case over several years. In this talk, we will first review some results on the Hausdorff dimension of the target vectors with a fixed approximating matrix which is epsilon-bad, and also the set of epsilon-badly approximable matrices for a fixed target vector in the real case. We will then talk about the function field analog. The Hausdorff dimension of the former set depends on the behavior of the orbit of a lattice point under certain diagonal flow. Joint work with Frédéric Paulin and Taehyeong Kim.

Kathryn Mann

Title: Anosov flows, foliations and 3-manifolds

Abstract: The study of Anosov (and pseudo-Anosov) flows on 3-manifolds is a beautiful interface of dynamics, geometry and topology: each flow gives rise to two invariant transverse 2-dimensional foliations, as well as the 1-dimensional foliation of the manifold by orbits, and the study of these foliations can help us classify flows. Starting with geodesic flow on the unit tangent bundle of a surface, I will describe some of the ways we construct many examples on manifolds with given geometry/topology, and recent work towards the classification problem. This involves joint work (in various iterations) with Barthelmé, Bowden, and Frankel.

Yair Minsky

Title: Horospheres, Laminations and Lipschitz maps

Abstract: Horospherical orbit closures in hyperbolic manifolds are not very well understood in the infinite-volume setting. In the case of \(\mathbb{Z}\)-covers of compact manifolds there is an interesting connection between such closures and the geometry of "tight" circle valued functions on the compact manifolds and their maximal-stretch laminations. In dimension 2, this makes it possible to describe the orbit closures in many cases. In particular these closures are quite sensitive to the dynamics of the stretch laminations, and vary discontinuously under changes of the metric. Joint work with James Farre and Or Landesberg.

Juan Souto

Title: Counting geodesics of given commutator length

Abstract: Let \(S\) be a closed hyperbolic surface. The commutator length of a homologically trivial curve in \(S\) is the minimal number of commutators one needs to multiply to represent the associated element in the fundamental group. In this talk I will discuss for fixed \(k\) the asymptotic behaviour, when \(L\) tends to infinity, of the number of closed geodesics in \(S\) of length at most \(L\) and commutator length equal to \(k\). This is joint work with Viveka Erlandsson.

Karen Vogtmann

Title: The Euler characteristic of the moduli space of graphs

Abstract: The moduli space of n-loop metric graphs, the outer automorphism group of the free group of rank n and Kontsevich's Lie graph complex in degree n all have the same rational cohomology. We determine the asymptotic behavior of the associated Euler characteristic, and thereby prove that the total dimension of this cohomology grows rapidly with n. This is joint work with Michael Borinsky.

Amie Wilkinson

Title: Centralizer rigidity

Abstract: The centralizer \(Z(f)\) of a diffeomorphism \(f\colon M\to M\) of a closed manifold \(M\) is the group of all diffeomorphisms commuting with \(f\); it is the collection of dynamical symmetries of \(f\). The centralizer of \(f\) always contains the group \(\langle f\rangle \) generated by \(f\) as a normal subgroup, and conjecturally the two typically coincide. In this talk, I will describe some results and conjectures in an ongoing project with Danijela Damjanovic and Disheng Xu that addresses the question: what happens when \(Z(f)\neq \langle f\rangle\)?

Activities

Here is a website with activities in and around Ascona.

Here are two suggestions for walks:
- A map for a walk through the hills with panoramic views over the lake (10.6 km, 500m of ascent, ~3.5 hours walking) can be found here.
- A walk along the lake (7km, flat, ~1.5 hours of walking) can be found here.

Conference photo

The conference photo can be downloaded here.