1. Dennis Sullivan

Title: To be announced

2. Alberto Verjovsky

Title: Twistor Spaces of Hyperbolic Four-Manifolds and Ergodic Foliations by Riemann Surfaces

Abstract: We will construct new examples of foliations by Riemann surfaces on a $P^1(C)$ bundle over the twistor space $Z(M)$ of a hyperbolic 4-manifold $M$. By a celebrated theorem of Atiyah-Hitchin-Singer $Z(M)$ is a complex 3-fold. Using Eells and Salamon's theorems, we construct foliations by Riemann surfaces on a $\mathbb{P}^1(\mathbb{C})$ bundle over $Z(M)$, which is a complex 4-fold. Using the results of Ratner and Shah one proves that these foliations are ergodic and therefore have a set of full measure of dense leaves.

This is joint work with Helena Lizarraga Collí.

3. Anna Benini

Title: Universality in Transcendental Dynamics.

Abstract: The Mandelbrot set is a fractal object encoding the dynamical behaviour of the family of quadratic polynomials z^2+c, where c is a parameter varying over the complex plane. It surprisingly appears also in the parameter spaces of all (reasonable) rational maps and in such sense, it also encodes the dynamical behaviour of this much larger class. The explanation is intricate and relies on the concept of renormalization: essentially, renormalization isolates and extrapolates the behaviour of a rational functions near  its critical values, and brings it back to analogous behaviours for quadratic polynomials.  In this work we present an analogous object for transcendental maps, which arises from a model family  and yet encodes the dynamical behaviour of all  (reasonable) families of transcendental meromorphic maps.

This is joint work with M. Astorg and N. Fagella.

4. Steve Kerckhoff

Title: Hyperbolic 3-Manifolds that Fiber over the Circle

Abstract: Thurston showed that the mapping torus of a pseudo-Anosov homeomorphism has a hyperbolic structure. It also has a singular Anosov structure obtained by suspending the invariant foliations of the homeomorphism. This lecture will describe an approach to connecting these two structures through local deformations and representation varieties.

5. Ángel Cano Cordero

Title: Limit Sets, Flag Degenerations, and Ordinary Domains in $\mathbb{P}^2_{\mathbb{C}}$

Abstract: I will present a unified asymptotic description of limit sets for discrete subgroups of $\mathrm{PSL}(3,\mathbb{C})$. The main tool is the pseudo-projective compactification, which records divergent sequences through kernel and image data and yields a trichotomy in terms of flag degenerations. This asymptotic boundary recovers the Myrberg, extended Conze-Guivarc'h, Cartan, and Kulkarni limit sets by natural projections. Under a general-position hypothesis, the Kulkarni limit set is a dynamically defined union of projective lines, and the ordinary set is the complement of this configuration. As a consequence, its connected components are complete Kobayashi hyperbolic and Kobayashi hyperbolically embedded in $\mathbb{P}^2_{\mathbb{C}}$. I will also discuss a genericity result for strongly irreducible Schottky-type groups, placing these phenomena within a projective analogue of Sullivan's dictionary.

6. Linda Keen

Title: Characterizing the dynamical behavior of Meromorphic functions with finitely many singular values

Abstract: Meromorphic functions are infinite degree maps of the complex plane to Riemann sphere. They can be iterated to form a dynamical system although points that land on infinity will have finite orbits. Understanding whether these systems behave ergodically is an interesting and complicated question and is characterized by these points with finite orbits. In this lecture, I will describe the dynamics of a special class of meromorphic functions, those whose singular set consists of finitely many asymptotic values. In particular, I will characterize the properties of the asymptotic values that control whether the function acts ergodically on its Julia set. Much of this work is joint with others, including Tao Chen, Bob Devaney, Nuria Fagella, Yunping Jiang and Janina Kotus.

7. Lasse Rempe

Title: Thurston's theorem for transcendental Belyi.

Abstract: Thurston's characterisation of post-critically finite rational maps is one of the cornerstones of the dynamical study of rational maps of one complex variable. It is a long-standing question whether an analogue of this theorem holds for classes of transcendental dynamical systems, in particular for post-singularly finite transcendental entire functions. A number of partial results are known for classes of functions where either the geometry of the functions or the combinatorics of the post-singular set is restricted in some way, but in general the question remains open.

In joint work with Prochorov, we establish an analogue of Thurston's theorem for a large class of (non-entire) transcendental meromorphic functions: _Belyi functions_ defined on non-empty open proper subsets of the Riemann sphere, and with bounded criticality. Here a Belyi function is a branched covering of the sphere, branched over only three points; it has bounded criticality if the degree of branched points is uniformly bounded. In particular, such maps have no asymptotic values, and hence cannot be entire.

8. Janina Kotus

Title: Metric Attractors, Lyapunov Exponents, and Invariant Measures

Abstract: In the first part of the lecture, we will recall several interesting examples of transcendental functions for which the point at infinity or the postsingular set serves as a metric attractor in the Julia set. In the second part, for these functions, we will analyze the relationship between Lyapunov exponents and the existence of absolutely continuous invariant measures with respect to the Lebesgue measure $l_2$ on the Julia set.

9. Richard Canary

Title: Bending, entropy and proper affine actions of surface groups

Abstract: We study the variation of complex length with respect to bending deformations of quasifuchsian groups. As one application, we exhibit an explicit open neighborhood of the Fuchsian locus in quasifuchsian space so that the only critical points of the entropy function in this neighborhood lie on the Fuchsian locus. As a second application, we exhibit an explicit open neighborhood of the Fuchsian locus so that the adjoint of every representation which is not Fuchsian is the linear part of a proper affine action on the Lie algebra of SL(2,C).

10. Israel García

Title: Volume growth on discontinuity regions of split-solvable kleinian groups

Abstract:A Heintze group is a semidirect product $H = \mathbb{R} \rtimes N$, where $N$ is nilpotent and $H$ admits a left-invariant metric of negative curvature. The group $H$ is called abelian when $N$ is abelian. Given two Heintze groups $H_1$ and $H_2$, one can define their horospherical product $G = H_1 \bowtie H_2$.

In this talk, I will present a family of split-solvable Lie groups of the form $ H_1 \bowtie H_2$ for abelian $H_i$ that act on complex projective space $\mathbb{P}^n_{\mathbb{C}}$ and admit lattices arising as complex Kleinian groups. These groups generalize Thurston's SOL geometry to higher-dimensional settings.

A particularly interesting feature of these examples is the interplay between geometry and dynamics. Exploiting this relationship, one can compute the volume entropy governing the growth of geodesic balls in the discontinuity region and derive asymptotic formulas for volume growth in the Heintze factors $H_i$. I will discuss these results and highlight the rich geometric structure of the discontinuity region.

11. Rodrigo Robles

Title: Pinching deformations and Baker domains in holomorphic dynamics

Abstract: We deform a transcendental entire function with a Baker domain in order to construct a transcendental entire function with a wandering domain that remains a positive Euclidean distance from its post-singular set.

Joint work with G. Sienra and P. Domínguez.

Poster Session (Coffee Break)

12. John Parker

Title: Complex hyperbolic triangle groups

Abstract: In this talk we consider subgroups of SU(2,1) generated by three involutions, each fixing a complex line, with the property that the product of each pair is elliptic or parabolic. We can think of these as being reflections in the sides of a complex triangle in complex hyperbolic space. Unlike for the case of constant curvature, the space of such triangles has real dimension four. Fixing (up to conjugation) the pairwise products of the generators we still have one free parameter. We ask for which parameter values the resulting group is discrete. Much of the work in this topic has been inspired by the conjectural picture presented by Rich Schwartz in his talk at the 2001 ICM. I will discuss the progress that has been made and I will present new conjectures that refine those of Schwartz.

13. Krishnendu Gongopadhay

Title: Amenable extension of Anosov subgroups

Abstract: We study entropy and periodic-orbit growth for normal covers arising from projective Anosov surface groups in $\mathrm{SL}_d(\mathbb{R})$. A projective Anosov representation naturally gives rise to a metric Anosov flow, providing a dynamical framework in which tools from thermodynamic formalism can be applied. Given a normal subgroup $\Gamma_0 \triangleleft \Gamma$, we analyze the lifted flow on the associated normal cover and compare the growth rate of its periodic orbits with the entropy of the base flow. Our main theorem shows that equality of these two quantities is equivalent to amenability of the quotient group $\Gamma/\Gamma_0$. Thus, we obtain a higher-rank analogue of the classical relation between amenability and critical-exponent equality, connecting Anosov dynamics, thermodynamics formalism, periodic-orbit growth, and amenability.

14. Moira Chass

Title: Computing the intersection of curves on surfaces via the Goldman Lie algebra

Abstract: In the eighties, Goldman discovered two Lie algebra structures on two vector spaces generated by free homotopy classes of closed curves on a surface. In one case, the basis is given by the classes of oriented curves, and in the other, by the classes of unoriented curves.

These Lie brackets, by definition, combine transversal intersection structure with loop product of curves.

We will describe how the algebraic structure then captures minimal intersection structure of curves on surfaces, in particular counting minimal intersections of a general curve with simple curves and showing the central elements are parallel to the boundary.

The proof uses both hyperbolic geodesic geometry and the effect of Thurston earthquakes on angles at intersection points.

These results are joint work with Arpan Kabiraj.

15. William Goldman

Title: Moduli of geometric structures and cubic surfaces

Abstract: The classification of locally homogeneous geometric structures on manifolds leads, in dimension two, to dynamical systems on deformation spaces of surface group representations. The building blocks for these spaces are the relative character varieties for the one-holed torus and four-holed sphere. In these cases these correspond to affine cubic surfaces, including the Markoff family defined by the polynomial k = x^2 + y^2 + z^2 - x y z - 2. We describe how the dynamics bifurcates at the level k = 18, which intimately relates to Clebsch's projective cubic surface and how the geometry of the cubic surface interacts with the geometric structures it parametrizes.

16. Sabyasachi Mukherjee

Title: Fatou-Sullivan dictionary: Where rational dynamics meets Kleinian groups

Abstract:The iteration of rational maps and the action of Kleinian groups on the Riemann sphere represent two central themes in conformal dynamics. The striking conceptual parallels between these theories are classically formalized as Sullivan's Dictionary. As early as the 1920s, Fatou envisioned unifying these dynamical systems through the framework of iterated algebraic correspondences.

We will present concrete realizations of Fatou's vision, constructing algebraic correspondences that intertwine the dynamics of complex polynomials and Fuchsian groups within a single dynamical plane. We will also explore rational Julia set realizations of Kleinian limit sets, including classical Apollonian-like gaskets. Finally, we will outline the main analytic and algebraic ideas underlying this program and highlight their applications to problems arising in analysis and statistical physics.

17. Peter Makienko

Title: Automorphic Measures, Lyapunov Exponents, and Instability of Rational Maps

Abstract: To construct obstructions to the stability of rational maps with non-summable critical points in their Julia sets, we introduce and study automorphic measures with complex eigenvalues for rational maps of the Riemann sphere. These measures extend the classical notions of quasi-invariant and conformal measures by allowing a complex Radon--Nikodym derivative proportional to the multiplicative cocycle Js, t(R)=∣R′∣s(R′∣R′∣)t, which plays the role of an automorphy factor in the sense of group actions. The existence of such measures reveals a close connection between the geometric and dynamical properties of rational maps.

We prove that the existence of certain automorphic measures, particularly unimodular measures and their associated vector fields, implies instability of the corresponding rational map. More precisely, if a weakly dissipative rational map admits a (-1,1)-unimodular measure, then there exists an integer q ≥ 1 such that the map is q-unstable. This result extends previous instability criteria obtained in the summable case and involving pseudoconformal measures, and establishes a direct link between automorphic measures and the failure of structural stability.

18. Adolfo Guillot Santiago

Title: Deformations of quotients of non-classical flag domains

Abstract:We will describe some compact complex non-Kähler threefolds that arise as quotients of domains in the flag manifold of the complex projective plane. Their locally homogeneous geometry is both rigid and maximal(they cannot be deformed by either deforming the group, or considering a more general geometry), but they can be deformed as complex manifolds. We will describe some deformations that arise through aconstruction involving projective structures along the leaves of their tautological foliations, reminiscent of Bers' simultaneous uniformization. This is joint work with Bertrand Deroin.

19. Geoffrey Sangston

Title: Toward a Classification of Closed Conformally Flat Lorentzian 3-Manifolds with Nilpotent Holonomy

Abstract: A conformally flat Lorentzian 3-manifold determines a (G, X)-structure, where X is the 3-dimensional Einstein universe (the conformal compactification of Minkowski space) and G = PO(3,2) is its conformal group. The classification of closed (G, X) manifolds with nilpotent holonomy splits into cases corresponding to nine conjugacy classes of maximal nilpotent subgroups of G; the unipotent case is due to Lee. I will describe progress on another case, in which the holonomy lies in the direct product of the pointwise stabilizer of two disjoint photons with a unipotent group stabilizing a third photon meeting both. I will present partial classification results in this case and discuss the open subcases.

20. Erin Jingxuan Son

Title: Algebraic and Geometric Convergence of Discrete and Non-Faithful Representations

Abstract: Let $M$ be a compact, orientable, hyperbolizable $3$-manifold with boundary, and let $\rho_i$ be a sequence of discrete representations of $\pi_1(M)$ with torsion-free image. We say that $\rho_i$ converges strongly if its algebraic limit agrees with its geometric limit. When $\rho_i$ are faithful, Anderson and Canary showed that strong convergence is guaranteed if the algebraic limit $\rho(\pi_1(M))$ contains no parabolic elements. In the non-faithful case, however, Biringer and Souto identified an additional obstruction to strong convergence: the non-embedded degenerate ends.

In this talk, I will review the results of Anderson-Canary and Biringer-Souto, and explain the non-embedded degenerate ends through Biringer-Souto’s example for surface groups. I will then demonstrate that this obstruction also exists for multiple families of 3-manifold groups and discuss in which case we can prevent this obstruction from happening.

Poster Session (Coffee break)

21. Tengren Zhan

Title: A rigidity theorem for complex Kleinian groups

Abstract: The notion of hyperconvexity for representations into PGL(d,R) plays a central role in the study of positive representations in Higher Teichmuller theory; for instance, this property is responsible for ensuring good regularity properties of the limit sets of positive representations. On the other hand, hyperconvex representations into PGL(d,C) are much more rigid. In this talk, I will explain a rigidity results about hyperconvex representations into PGL(d,C). This is joint work with Richard Canary and Andrew Zimmer.

22. Mahan Mj

Title: The Fatou-Sullivan dictionary and Thurston's questions

Abstract: Nearly hundred-year-old question by Fatou asks for a synthesis of the following two kinds of holomorphic dynamical systems under a common framework of holomorphic correspondences on the Riemann sphere: (a) Kleinian groups acting on the Riemann sphere (b) iteration of complex polynomials on the Riemann sphere. Sullivan's dictionary gave us a way of translating techniques from one of these fields to give results in the other. In a relatively recent development, building on Sullivan's dictionary, a bridge has been built between these two classes in the spirit of Bers' simultaneous uniformization theorem. New holomorphic dynamical systems on the Riemann sphere have thus been discovered that arise as combinations or matings of Kleinian groups and polynomials. In some cases, these single valued matings give rise to multi-valued algebraic correspondences on the Riemann sphere, partially fulfilling Fatou's dream. A particular consequence of these constructions is an analog of the compactness theorem for Bers slices of punctured sphere groups. In 1982, Thurston posed a number of questions that guided the development of the theory of Kleinian groups for the next 3 decades. With the above analog of Bers compactness in place, many of these questions reincarnate themselves in this new context. We will survey some of these developments and questions. This is joint work with Yusheng Luo and Sabyasachi Mukherjee.

23. Ara Basmajian

Title: Curves, designer metrics, and the inf spectrum of Moduli

Abstract: In this talk, we consider the relationship, in various settings, between length and self-intersection of a fixed closed curve on a surface. One such setting involves the inf invariant associated to a curve γ and its so-called designer metric. By considering the inf invariant over each homotopy class of curve we are naturally led to a discrete spectrum of positive real numbers we call the inf spectrum of the surface. This spectrum is independent of metric and is associated to the moduli space of hyperbolic structures on the surface. This is a joint project with Sayantika Mondal and Hugo Parlier.