Titles, Abstracts and Slides
Conference (July 2023):
Mahan Mj: Matings, holomorphic correspondences, and a Bers slice
There are two frameworks for mating Kleinian groups with rational maps on the Riemann sphere: the algebraic correspondence framework due to Bullett-Penrose-Lomonaco going back to the early 90s, and a more recent simultaneous uniformization mating framework. We shall first describe the simultaneous uniformization mating framework. We shall then proceed to extend this mating framework and show how to unify and generalize the earlier two frameworks in the case of principal hyperbolic components. To achieve this, we shall consider genus zero finite volume hyperbolic orbifolds with at most one orbifold point of order greater than two, and at most one orbifold point of order two. We give an explicit description of the resulting conformal matings in terms of uniformizing rational maps. Using these rational maps, we construct correspondences that are matings of such hyperbolic orbifold groups (including the modular group) with polynomials in principal hyperbolic components. We also define an algebraic parameter space of correspondences and construct an analog of a Bers slice of the above orbifolds in this parameter space.
Kohei Iwaki: Exact WKB analysis and topological recursion
Exact WKB analysis is a method to study singularly-perturbed ODEs such as the Schrodinger equations.
For second-order ODEs, this theory allows us to describe the monodromy and Stokes matrices of the ODE
in terms of an infinite sum of periods of the Riemann surface, which is obtained as the classical limit of the ODE.
On the other hand, topological recursion is a remarkable recursive algorithm; from a given Riemann surface
with additional data, it associates an analogue of the correlation function and free energy of the matrix model.
It is expected (and proved in many examples) that various enumerative invariants, which are related to
the moduli space of Riemann surfaces, are computed through the topological recursion.
In the first part of the talk, I’ll briefly summarize these two theories. In the second part, I’ll explain how they interact with each other,
and show several results (such as a construction of tau-function of the Painleve equations; arXiv:1902.06439) if time permits.
Here are references.
・Exact WKB analysis:
T. Kawai and Y. Takei, Algebraic Analysis of Singular Perturbation Theory,
Translations of Mathematical Monographs, Iwanami Series in Modern Mathematics, Volume: 227; 2005.
・Topological Recursion:
B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion,
Comm. Number Theory Phys. 1 (2007), 347–452; arXiv:math-ph/0702045.
Chikako Mese: Infinite energy harmonic maps
Harmonic maps have been a useful tool in various branches of mathematics. The focus of this talk is on infinite energy harmonic maps and their applications. In particular, we generalize a theorem by Gromov-Schoen by constructing an equivariant infinite energy harmonic map with some suitable asymptotic behavior into the Bruhat-Tits building and the Weil-Petersson completion of Teichmuller space.
Hideki Miyachi: A complex geometric aspect in the Teichmuller theory
In this talk, I will propose a direction to develop the complex geometry of Teichmuller space. The Teichmuller space is a complex manifold biholomorphic to a bounded domain in the complex Euclidean space. The holomorphic tangent and cotangent structures are well-developed fundamental objects in the Teichmuller theory. In this talk, I first recall facts from the complex analytical aspect of the Teichnuller theory. After then, I will discuss the double tangent and cotangent structures of the Teichmuller space, aiming for developing the (complex) Finsler geometry of the Teichmuller space.
Tsukasa Ishibashi: Teichmuller and lamination spaces with pinnings
The enhanced Teichmüller space of a punctured bordered surface has cross ratio coordinates associated with ideal triangulations by the work of Chekhov—Fock (1999). These coordinates lead to the theory of cluster varieties and quantum (higher) Teichmüller theory. While the Chekhov—Fock’s original construction does not involve coordinates assigned to the boundary intervals, later in the higher rank context, Goncharov—Shen (2019) introduced the moduli spaces of G-local systems with ``pinnings’’ which also have boundary coordinates and well-behaved under gluing of surfaces along them.
In this talk, we give an interpretation of the Goncharov—Shen’s construction purely in terms of the hyperbolic geometry, and introduce the ``Teichmüller space with pinnings’’. Time permitting, I will also mention its Thurston compactification.
Cyril Lecuire: Skinning hyperbolic 3-manifolds
The skinning map is defined by peeling off the boundary of a compact hyperbolic 3-manifold and recording the invariant (the conformal structure at infinity) of the inner side of this peeled skin. Proving that the skinning map has a bounded orbit (the Bounded Orbit Theorem) is one of the main steps in the proof of Thurston's Hyperbolization Theorem (of Haken 3-manifolds).
In this talk, I will present a proof of the Bounded Image Theorem, a generalization, announced without proof by Thurston, of the Bounded Orbit Theorem. The goal is to show that an iterate of the skinning map has a bounded image.
This is a joint work with Ken'ichi Ohshika.
Inkang Kim: Convexity of energy functions of harmonic maps homotopic to covering maps of the product of surfaces
In this talk, we consider the strict convexity of energy functions of harmonic maps at its critical points from Riemann surfaces to product of negatively curved surfaces, such that each projection of the harmonic map is homotopic to a covering map. In general, the result is not true for the product of three surfaces.
Joint work with X. Wan and G. Zhang.
Sang-hyun Kim: First order rigidity of the groups Homeo(M) and Homeo(M,vol)
Two groups are said to be elementarily equivalent if they have the same sets of true first order group theoretic sentences. We prove that if the homeomorphism groups of two compact connected manifolds are elementarily equivalent, then the manifolds are homeomorphic. The same conclusion holds with the additional volume--preserving hypotheses on the groups. This generalizes Whittaker’s theorem on isomorphic homeomorphism groups (1963) without relying on it. Our proof utilizes classical and modern results of geometric topology, such as those of M. Brown, Edwards--Kirby, Cheeger--Kister, Oxtoby--Ulam and Fathi. Joint work with Thomas Koberda (UVa) and Javier de la Nuez-Gonzalez (KIAS).
Hiroaki Karuo: Skein algebras and quantum Teichmuller spaces
The skein algebra of an oriented surface is an algebra with a quantum parameter. For a punctured surface with an ideal triangulation, Bonahon--Wong constructed an embedding of its skein algebra into its quantum Teichm\"uller space, called a quantum trace map. Moreover, one can construct quantum trace maps for some generalizations of skein algebras of marked surfaces in similar way. In this talk, I explain how to construct quantum trace maps and related topics based on a joint work with W. Bloomquist and T. T. Q. Le (Georgia Tech).
Dounnu Sasaki : Counting subgroups and subset currents
Given a closed hyperbolic surface S of genus g, Mirzakhani proved that the number of closed geodesics of length at most L and of a given type is asymptotic to cL^{6g-6} for some c>0.
Since a closed geodesic is corresponding to a conjugacy class of the fundamental group G of S, as a generalization, we consider the counting problem of conjugacy classes of finitely generated subgroups of G.
In this case, by using "half the sum of the lengths of the boundaries of the convex cores of a subgroup" instead of the length of a closed geodesic, we find that the number is also asymptotic to cL^{6g-6} for some c>0. In addition, we see that such a length for a subgroup is natural from the view point of subset currents, which is a completion of weighted conjugacy classes of finitely generated subgroups of G.
Richard Canary: Transverse groups
Transverse groups are discrete subgroups of semi-simple Lie groups whose discretenese is registered
by a specific collection of simple roots and whose limit set is a collection of mutually transverse flags in the associated
partial flag variety. Discrete subgroups of rank one Lie groups and Anosov and relatively Anosov subgroups of higher
rank Lie groups provide examples of transverse groups.
We describe recent work on transverse groups with Tengren Zhang and Andy ZImmer. Results include a common
generalization of results of Bishop-Jones and Pozzetti-Sambarino-Wienhard on Hausdorff dimension of limit sets, entropy
rigidity results for Hitchin representations of Fuchsian group, a Hopf-Tsuji-Sullivan dichotomy for Patterson-Sullivan measures
and entropy gap results for subgroups.
Ara Basmajian: Identities: Old and New
Let X be a compact n dimensional hyperbolic manifold with totally geodesic boundary. A homotopy class (rel the boundary) of a non-trivial arc from the boundary to itself can be realized by an orthogeodesic- a geodesic segment perpendicular to the boundary at its initial and terminal points. This talk will be an introduction to the study of such arcs, their properties, and the identities they satisfy. For example, it is a consequence of the orthospectrum identity that the set of all orthogeodesic lengths determine the area of the boundary of X. In fact, such lengths are also related to the topological entropy of the manifold. In dimension two, there are special subclasses of orthogeodesics called prime orthogeodesics.
In work with Hugo Parlier and Ser Peow Tan we show that the prime orthogeodesics arise naturally in the study of maximal immersed pairs of pants in X and are intimately connected to regions of X in the complement of the natural collars. These considerations lead to far reaching generalizations of the orthospectrum identity in the form of continuous families of equations (so called identities) that remain constant on the deformation space of hyperbolic structures.
Hidetoshi Masai: A distance on Teichmueller space via renormalized volume
In this talk, we consider volumes of hyperbolic 3-manifolds and construct a new distance on the Teichmueller space of a closed surface of genus >1. We will compare the new distance with other known distances: Teichmueller distance, and Weil-Petersson distance. This talk is based on the preprint arxiv:2108.06059.
Hiroki Ishikura: Permutation stability of free metabelian groups
A countable group is called permutation stable if every asymptotic homomorphism into finite symmetric groups can be approximated by actual homomorphisms. For example, free abelian groups have this property. Recently, permutation stability of amenable groups is found to be related to invariant random subgroups, which are studied in measured group theory. In this talk, I will explain a result showing that free metabelian groups are permutation stable.
Ser Peow Tan: Identities on the thrice-punctured sphere and their index sets
Many interesting identities have been discovered/obtained by McShane, Basmajian, Bridgeman, Mirzakhani, Tan-Wong-Zhang, Luo-Tan (and many other authors) for hyperbolic surfaces and have found interesting applications. The infinite sums in the identities are indexed by various geometrically defined sets like the set of (simple) closed geodesics, or orthogeodesic. However, these reduce to trivial identities when applied to the hyperbolic thrice-punctured sphere.
By introducing grades on the cusps, which govern the tubular neighborhoods of the cusps to be removed, Basmajian, Parlier and the author were able to obtain families of interesting and non-trivial identities on the thrice-punctured sphere with these tubular neighborhoods removed (the so called concave core). We will discuss these identities and their index sets (the set of prime orthogeodesics on the concave core) including a combinatorial criteria for prime orthogeodesics, an efficient algorithm for enumerating these prime orthogeodesics and questions about multiplicities (with respect to length). Applications to other surfaces, and connections to other questions, like enumerating reciprocal geodesics on the modular surface may be discussed, time permitting. This is based on joint work with various collaborators including Ara Basmajian, Hugo Parlier, Nhat Minh Doan, Boon Wei Chow and Yichen Tao.
Lecture Series (July 2022):
Jean-Marc Schlenker: The renormalized volume of hyperbolic 3-manifolds
Abstract: Quasifuchsian manifolds, and more generally convex co-compact hyperbolic 3-manifolds, have infinite volume. However they have a well-defined "renormalized" volume, first defined and studied by theoretical physicists in a wider setting. After recalling some key properties of quasifuchsian and convex co-compact hyperbolic 3-manifolds, we will give a definition of the renormalized volume and prove some of its key properties. We will then outline some of its recent mathematical applications to Teichmüller theory and to the geometry of hyperbolic 3-manifolds. We will also try to provide some hints about the physical motivations for its definition. [Slides 1] [Slides 2] [Slides 3] [Slides 4] [Slides 5]
Athanase Papadopoulos: Distorsion of maps between surfaces: From geography to Teichmüller spaces
Abstract: I will explain the notion of the distorsion of maps between surfaces, from the early works of Euler and Lagrange on geographical maps from the sphere or spheroid onto the plane, to the more recent work of Thurston on maps between hyperbolic surfaces. I will present recent results in Thurston's theory of stretch maps that were done in collaboration with Yi Huang, Hideki Miyachi, Ken'ichi Ohshika and I. Saglam. [Slides 1] [Slides 2]
Joan Porti: Anosov representations and symmetric spaces
Abstract: Anosov representations were introduced by Labourie for surface groups and by Guichard and Wienhard for word hyperbolic groups. I plan to discuss the different characterizations of being Anosov in terms of the action on symmetric spaces of non compact type, as a joint work with Kapovich and Leeb. I'll make the analogy with the characterizations of convex-cocompact groups of isometries of hyperbolic space. [Slides 1] [Slides 2] [Slides 3] [Slides 4] [Slides 5] [Slides 6] [Slides 7] [Slides 8]
To be announced.