Organizing Committee
 David Fisher
Rice University  Dubi Kelmer
Boston College  Hee Oh
Yale university  Alan Reid
Rice University
Abstract
This workshop focuses on the interplay between dynamics, rigidity, and arithmetic in hyperbolic geometry and related areas. There have been many striking developments in recent years, particularly related to totally geodesic submanifolds in both finite and infinite volume hyperbolic and even complex hyperbolic manifolds.
One aim of this workshop is to expose young researchers to these breakthroughs providing them with the necessary background from dynamics, and geometry to allow them to appreciate some of these recent advances, and prepare them to make new original contributions. For this purpose, we will have minicourses on "Arithmeticity, Superrigidity and totally geodesic manifolds", and "Rigidity and geodesic planes in infinite volume hyperbolic manifolds". These courses will be preceded by an introductory minicourse on Hyperbolic geometry. We will also have a minicourse on "Understanding of geodesic planes in hyperbolic 3manifolds via computations and visualization". In addition, we wish to bring together experts in these fields to discuss the recent developments and open problems that lie at the crossroads of these different fields and to encourage more interaction among people working in these diverse areas.
This workshop is partially funded by NSF CAREER award DMS1651563
Confirmed Speakers & Participants
Talks will be presented virtually or inperson as indicated in the schedule below.
 Speaker
 Poster Presenter
 Attendee
 Virtual Attendee

Fernando Al Assal
Yale University

Alina Al Beaini
Brown University

Konstantin Andritsch
ETH Zurich

Anna Antal
Yale University

Juan Arosemena Serrato
Rice University

Juhun Baik
KAIST

Gregorio Baldi
Institut des Hautes Études Scientifiques

Erin Bevilacqua
University of Texas at Austin

Ian Biringer
Boston College

Christine Breiner
Brown University

Nic Brody
UC Santa Cruz

Tamunonye CheethamWest
Rice University

Hyein Choi
Rice university

Inhyeok Choi
Korea Institute for Advanced Study

Mikey Chow
Yale University

Ethan Cohen
Yale University

David Constantine
Wesleyan University

Emilio Corso
University of British Columbia

Gregory Cosac
Universidade de São Paulo (USP)

Ozkan Demir
University of Illinois Chicago

Subhadip Dey
Yale University

Ethan Dlugie
University of California, Berkeley

Sami Douba
Institut des Hautes Études Scientifiques

Elias Dubno
University of Zurich

Sara EdelmanMunoz
Rice University

Leonardo Ferrari
Université de Neuchâtel

David Fisher
Rice University

Mikolaj Fraczyk
University of Chicago

Sam Freedman
Brown University

Alex Furman
University of Illinois at Chicago  UIC

Milana Golich
Purdue University

Yanlong Hao
University of Illinois at Chicago

Paige Hillen
University of California, Santa Barbara

Junzhi Huang
Yale University

Sebastian Hurtado
Yale University

Seung uk Jang
The University of Chicago

Yushan Jiang
City University of New York, the Graduate Center

Junehyuk Jung
Brown University

Dubi Kelmer
Boston College

Wooyeon Kim
ETH Zurich

Dongryul Kim
Yale University

Dmitry Kleinbock
Brandeis University

Or Landsberg
Yale University

Minju Lee
University of chicago

Ricky Lee
University of California, Santa Barbara

Homin Lee
Northwestern University

Joaquin Lema
Boston College

Zuo Lin
University of California San Diego

Beibei Liu
University of California, Davis

Trent Lucas
Brown University

Simon Machado
Institute for Advanced Study

Alexandre Maldague
Rice University

Ari Markowitz
The University of Auckland

Curtis McMullen
Harvard University

Jonah Mendel
Brown University

Katherine Merkl
UC Santa Barbara

Nicholas Miller
University of Oklahoma

Shahriar Mirzadeh
UNIVERSITY OF CINCINNATI

Sayantika Mondal
Graduate school and university center, CUNY

Casandra Monroe
University of Texas  Austin

Hamid Naderiyan
University of North Texas

Hee Oh
Yale university

Tariq Osman
Brandeis University

Michael Pandazis
CUNY Graduate Center

Sungjin Park
Yale University

Insung Park
ICERM

Julien Paupert
Arizona State University

Mark Pengitore
University of Virginia

Carsten Peterson
University of Michigan

Lam Pham
Brandeis University

Plinio Pino Murillo
Fluminense Federal University

Amelia Pompilio
University of Illinois at Chicago

Alan Reid
Rice University

Megan Roda
University of chicago

Rafael Saavedra
Harvard University

Anthony Sanchez
University of California San Diego

Geoffrey Sangston
University of Maryland

Pratyush Sarkar
UC San Diego

Connor Sell
Rice University

Juno Seong
University of CaliforniaSan Diego

Ekaterina Shchetka
University of Michigan

Aleksander Skenderi
University of WisconsinMadison

Raz Slutsky
Weizmann Institute of Science

Miri Son
Rice University

Matthew Stover
Temple University

Nattalie Tamam
University of Michigan

Jacob Tolman
Wesleyan University

Tina Torkaman
Harvard University

Bena Tshishiku
Brown University

Hunter Vallejos
University of Texas at Austin

Franco Vargas Pallete
Yale University

Itamar Vigdorovich
Weizmann Institute of Science

Thi Hanh Vo
Arizona State University

Mujie Wang
Boston College

Amy Wang
Yale University

Jane Wang
University of Maine

Zhiren Wang
Penn State University

Vicky Wen
University of Wisconsin, Madison

Anna Wienhard
Heidelberg University

Amanda Wilkens
University of Texas at Austin

Becca Winarski
MSRI/College of the Holy Cross

Karl Winsor
Fields Institute

Christian Wolf
The City College of New York

Leyla Yardimci
Wesleyan University

Matthew Zevenbergen
Boston College

Yongquan Zhang
Stony Brook University

Michael Zshornack
UC Santa Barbara

Jonathan Zung
Princeton University
Workshop Schedule
Monday, May 15, 2023

9:30  9:50 am EDTCheck In / AM Coffee BreakCheck In  11th Floor Collaborative Space

9:50  10:00 am EDTWelcome11th Floor Lecture Hall

10:00  10:45 am EDTHyperbolic geometry  An Introduction11th Floor Lecture Hall
 Speaker
 Ian Biringer, Boston College
 Session Chair
 David Fisher, Rice University
Abstract
The aim of this minicourse is to introduce fundamental concepts in hyperbolic geometry, such as limit sets, geometric finiteness, and critical exponent, and Mostow rigidity. We will discuss examples of arithmetic hyperbolic manifolds, and illustrate flexible geometric constructions like Dehn filling, quasiconformal deformation, and the gluing constructions of nonarithmetic lattices by Gromov and PiatetskiShapiro.

11:00  11:45 am EDTTotally geodesic subvarieties via Hodge theory11th Floor Lecture Hall
 Speaker
 Gregorio Baldi, Institut des Hautes Études Scientifiques
 Session Chair
 David Fisher, Rice University
Abstract
In this Mini Course, following a joint work with E.Ullmo, I will explain how (integral) Hodge theory naturally comes up in the study of totally geodesic subvarieties of a complex hyperbolic ball quotient S. From such a point of view, the finiteness of the maximal totally geodesics of S becomes a consequence of a very general conjecture about 'unlikely intersections'. The two lectures will give some motivations and an introduction to such techniques (no prior knowledge in Hodge theory will be assumed).

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTGeometric, dynamical and arithmetic properties of Anosov representations11th Floor Lecture Hall
 Speaker
 Anna Wienhard, Heidelberg University
 Session Chair
 Nattalie Tamam, University of Michigan
Abstract
Anosov representations provides a rich class of discrete embeddings of hyperbolic groups into semisimple Lie groups, which generalizes the classes of convex cocompact subgroups to the setting of higher rank Lie groups. In this talk I will discuss some results (old and new) regarding geometric, dynamical and arithmetic properties.

2:30  3:15 pm EDTHyperbolic geometry  An Introduction11th Floor Lecture Hall
 Speaker
 Ian Biringer, Boston College
 Session Chair
 Nattalie Tamam, University of Michigan
Abstract
The aim of this minicourse is to introduce fundamental concepts in hyperbolic geometry, such as limit sets, geometric finiteness, and critical exponent, and Mostow rigidity. We will discuss examples of arithmetic hyperbolic manifolds, and illustrate flexible geometric constructions like Dehn filling, quasiconformal deformation, and the gluing constructions of nonarithmetic lattices by Gromov and PiatetskiShapiro.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTRigidity and geodesic planes in infinite volume hyperbolic manifolds11th Floor Lecture Hall
 Speaker
 Hee Oh, Yale university
 Session Chair
 Nattalie Tamam, University of Michigan
Abstract
The aim of this mini course is to discuss some basic theory of homogeneous dynamics on the quotient of SO(n,1) by a discrete subgroup and to explain the topological rigidity of geodesic planes in hyperbolic nmanifolds of Fuchsian ends which was proved by McMullenMohammadiO. (for n=3) and by Minju LeeO. (for n>3).

5:00  6:30 pm EDTReception11th Floor Collaborative Space
Tuesday, May 16, 2023

9:30  10:00 am EDTCoffee Break11th Floor Collaborative Space

10:00  10:45 am EDTArithmeticity, superrigidity, and totally geodesic manifolds11th Floor Lecture Hall
 Speaker
 Nicholas Miller, University of Oklahoma
 Session Chair
 Alan Reid, Rice University
Abstract
In the 1970s seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov–PiatetskiShapiro and Deligne–Mostow have shown that a similar superrigidity theorem cannot hold for all lattices in the isometry group of real or complex hyperbolic space, i.e., in the rank 1 setting. However in recent work joint with Bader, Fisher, and Stover we show that one can prove certain superrigidity/arithmeticity theorems provided the associated manifold satisfies the geometric condition that it contains infinitely many (maximal) totally geodesic submanifolds. Specifically, we show that this latter criteria forces a hyperbolic manifold to be arithmetic.
The goal of this minicourse will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting. 
11:00  11:45 am EDTRigidity and geodesic planes in infinite volume hyperbolic manifolds11th Floor Lecture Hall
 Speaker
 Hee Oh, Yale university
 Session Chair
 Alan Reid, Rice University
Abstract
The aim of this mini course is to discuss some basic theory of homogeneous dynamics on the quotient of SO(n,1) by a discrete subgroup and to explain the topological rigidity of geodesic planes in hyperbolic nmanifolds of Fuchsian ends which was proved by McMullenMohammadiO. (for n=3) and by Minju LeeO. (for n>3).

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTTotally geodesic subvarieties via Hodge theory11th Floor Lecture Hall
 Speaker
 Gregorio Baldi, Institut des Hautes Études Scientifiques
 Session Chair
 Dubi Kelmer, Boston College
Abstract
In this Mini Course, following a joint work with E.Ullmo, I will explain how (integral) Hodge theory naturally comes up in the study of totally geodesic subvarieties of a complex hyperbolic ball quotient S. From such a point of view, the finiteness of the maximal totally geodesics of S becomes a consequence of a very general conjecture about 'unlikely intersections'. The two lectures will give some motivations and an introduction to such techniques (no prior knowledge in Hodge theory will be assumed).

2:30  3:15 pm EDTExponential mixing of frame flows for geometrically finite hyperbolic manifolds11th Floor Lecture Hall
 Speaker
 Pratyush Sarkar, UC San Diego
 Session Chair
 Dubi Kelmer, Boston College
Abstract
The frame bundle of an ndimensional hyperbolic manifold X is the homogeneous space Γ\SO(n, 1)° for some discrete subgroup Γ and the frame flow is given by the right translation action by a oneparameter diagonalizable subgroup. We assume that Γ is Zariski dense and X is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. We endow the frame bundle with the unique probability measure of maximal entropy called the BowenMargulisSullivan measure. In a joint work with Jialun Li and Wenyu Pan, we prove that the frame flow is exponentially mixing. The proof uses a countably infinite coding and the latest version of Dolgopyat's method. To overcome the difficulty in applying Dolgopyat's method due to the cusps of nonmaximal rank, we prove a large deviation property for symbolic recurrence to certain large subsets of the limit set of Γ.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTLength functions in Lie groups and lattices.11th Floor Lecture Hall
 Speaker
 Sebastian Hurtado, Yale University
 Session Chair
 Dubi Kelmer, Boston College
Abstract
We will discuss the notion of length functions on groups, focusing on lattices in Lie groups of higher rank, and discuss how dynamics and ergodic theory can help to understand some questions about them.
Wednesday, May 17, 2023

9:30  10:00 am EDTCoffee Break11th Floor Collaborative Space

10:00  10:45 am EDTRigidity and geodesic planes in infinite volume hyperbolic manifolds11th Floor Lecture Hall
 Speaker
 Hee Oh, Yale university
 Session Chair
 David Fisher, Rice University
Abstract
The aim of this mini course is to discuss some basic theory of homogeneous dynamics on the quotient of SO(n,1) by a discrete subgroup and to explain the topological rigidity of geodesic planes in hyperbolic nmanifolds of Fuchsian ends which was proved by McMullenMohammadiO. (for n=3) and by Minju LeeO. (for n>3).

11:00  11:45 am EDTArithmeticity, superrigidity, and totally geodesic manifolds11th Floor Lecture Hall
 Speaker
 Nicholas Miller, University of Oklahoma
 Session Chair
 David Fisher, Rice University
Abstract
"In the 1970s seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov–PiatetskiShapiro and Deligne–Mostow have shown that a similar superrigidity theorem cannot hold for all lattices in the isometry group of real or complex hyperbolic space, i.e., in the rank 1 setting. However in recent work joint with Bader, Fisher, and Stover we show that one can prove certain superrigidity/arithmeticity theorems provided the associated manifold satisfies the geometric condition that it contains infinitely many (maximal) totally geodesic submanifolds. Specifically, we show that this latter criteria forces a hyperbolic manifold to be arithmetic.
The goal of this minicourse will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting." 
11:55 am  12:00 pm EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTDiscrete subgroups with finite BowenMargulisSullivan measure in higher rank11th Floor Lecture Hall
 Speaker
 Minju Lee, University of chicago
 Session Chair
 Julien Paupert, Arizona State University
Abstract
Let G be a connected semisimple real algebraic group and D be its Zariski dense discrete subgroup. We prove that if D\G admits any finite BowenMargulisSullivan measure, then D is virtually a product of higher rank lattices and discrete subgroups of rank one factors of G. This may be viewed as a measuretheoretic analogue of classification of convex cocompact actions by KleinerLeeb and Quint, which was conjectured by Corlette in 1994. This is joint work with Mikolaj Fraczyk. We will then discuss its application on the bottom of the L^2 spectrum, in joint work with Samuel Edwards, Mikolaj Fraczyk and Hee Oh.

2:30  3:15 pm EDTBilliards and the arithmetic of nonarithmetic groups11th Floor Lecture Hall
 Speaker
 Curtis McMullen, Harvard University
 Session Chair
 Julien Paupert, Arizona State University
Abstract
We will survey new results and open problems on triangle groups in SL_2(R), and their connections to Abelian varieties, Teichmueller curves and billiards in polygons.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTUnderstanding geodesic planes in hyperbolic 3manifolds via computations and visualization11th Floor Lecture Hall
 Speaker
 Yongquan Zhang, Stony Brook University
 Session Chair
 Julien Paupert, Arizona State University
Abstract
As a companion and complement to the minicourse on rigidity of geodesic planes in infinite volume hyperbolic manifolds, I will discuss several examples of rigidity and nonrigidity of geodesic planes in hyperbolic 3manifolds. These results are orthogonal to the ones obtained by McMullen, Mohammadi and Oh, and illustrate what can happen if some of their assumptions are relaxed. Some of these examples were produced by explicit computations and can be visualized very nicely, and I will discuss these aspects as well.
Thursday, May 18, 2023

9:30  10:00 am EDTCoffee Break11th Floor Collaborative Space

10:00  10:45 am EDTArithmeticity, superrigidity, and totally geodesic manifolds11th Floor Lecture Hall
 Speaker
 Nicholas Miller, University of Oklahoma
 Session Chair
 Jane Wang, University of Maine
Abstract
In the 1970s seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov–PiatetskiShapiro and Deligne–Mostow have shown that a similar superrigidity theorem cannot hold for all lattices in the isometry group of real or complex hyperbolic space, i.e., in the rank 1 setting. However in recent work joint with Bader, Fisher, and Stover we show that one can prove certain superrigidity/arithmeticity theorems provided the associated manifold satisfies the geometric condition that it contains infinitely many (maximal) totally geodesic submanifolds. Specifically, we show that this latter criteria forces a hyperbolic manifold to be arithmetic.
The goal of this minicourse will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting. 
11:00  11:45 am EDTUniform spectral gap and orthogeodesic counting for Kleinian groups11th Floor Lecture Hall
 Speaker
 Beibei Liu, University of California, Davis
 Session Chair
 Jane Wang, University of Maine
Abstract
Strongly convergent sequences of hyperbolic manifolds arise naturally in the study of Kleinian group representations, for example, the Dehn surgeries on hyperbolic knots. It turns out that such sequences usually have uniform control on the geometry and dynamics, such as the uniform convergence of small eigenvalues of the Laplacian, and the PattersonSullivan measures. We will talk about the uniform convergence results in this talk and apply them to count uniformly along the sequence the number of simple closed geodesics and orthogeodesics. This is joint work with Franco Vargas Pallete.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTLyapunov spectrum: simplicity and continuity beyond random products11th Floor Lecture Hall
 Speaker
 Alex Furman, University of Illinois at Chicago  UIC
 Session Chair
 Plinio Pino Murillo, Fluminense Federal University

2:30  3:15 pm EDTIntersection number and intersection points of closed geodesics on hyperbolic surfaces11th Floor Lecture Hall
 Speaker
 Tina Torkaman, Harvard University
 Session Chair
 Plinio Pino Murillo, Fluminense Federal University
Abstract
In this talk, I will discuss the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I talk about the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTUnderstanding geodesic planes in hyperbolic 3manifolds via computations and visualization11th Floor Lecture Hall
 Speaker
 Yongquan Zhang, Stony Brook University
 Session Chair
 Plinio Pino Murillo, Fluminense Federal University
Abstract
As a companion and complement to the minicourse on rigidity of geodesic planes in infinite volume hyperbolic manifolds, I will discuss several examples of rigidity and nonrigidity of geodesic planes in hyperbolic 3manifolds. These results are orthogonal to the ones obtained by McMullen, Mohammadi and Oh, and illustrate what can happen if some of their assumptions are relaxed. Some of these examples were produced by explicit computations and can be visualized very nicely, and I will discuss these aspects as well.
Friday, May 19, 2023

9:30  10:00 am EDTCoffee Break11th Floor Collaborative Space

10:00  10:45 am EDTFourier decay of selfconformal measures for nonlinear IFS’s11th Floor Lecture Hall
 Speaker
 Zhiren Wang, Penn State University
 Session Chair
 Bena Tshishiku, Brown University
Abstract
We show that for a C^2 IFS on R, either up to smooth conjugacy the IFS has vanishing second derivative on its attractor, or the selfconformal measure has polynomial decay of Fourier coefficients. A key argument is a cocycle version of Dolgopyat's method and resulting spectral gaptype estimates and renewal theorem. This is a joint work with Amir Algom and Federico Rodriguez Hertz.

11:00  11:45 am EDTCentral extensions of real and complex hyperbolic lattices11th Floor Lecture Hall
 Speaker
 Matthew Stover, Temple University
 Session Chair
 Bena Tshishiku, Brown University
Abstract
I will describe joint work with Domingo Toledo on residual finiteness for cyclic central extensions of fundamental groups of aspherical manifolds, its application to central extensions of (arithmetic) real and complex hyperbolic lattices, and connections to several open problems.

12:00  1:30 pm EDTLunch/Free Time

1:30  2:15 pm EDTPoissonVoronoi tessellations in higher rank and the fixed price conjecture11th Floor Lecture Hall
 Speaker
 Mikolaj Fraczyk, University of Chicago
 Session Chair
 Dubi Kelmer, Boston College
Abstract
The cost of a probability measure preserving action of a countable group G on X is an invariant that generalizes the rank (minimal number of generators) of G and measures the “minimal average number of maps” needed to connect every pair of points of X in the same G orbit. The fixed price conjecture predicts that any two essentially free p.m.p. actions of the group G have the same cost. In my talk I will report on a joint work with Sam Mellick and Amanda Wilkens in which we prove fixed price one for higher ranks lattices in semisimple real or padic groups. As a corollary we obtain that the number of generators of index n subgroup of such a group grows like o(n) which implies new state of the art results on the growth of modp homology groups. The proof is based on certain miraculous properties of the PoissonVoronoi tessellation of higher rank symmetric spaces (not present in rank 1) that might be of independent interest.

2:30  3:15 pm EDTThe arithmetic of totally geodesic surfaces on Bianchi orbifolds11th Floor Lecture Hall
 Speaker
 Junehyuk Jung, Brown University
 Session Chair
 Dubi Kelmer, Boston College
Abstract
Bianchi subgroups are cofinite noncocompact lattices in PSL_2(C), defined by \Gamma_d = PSL_2(O_d), where O_d is the ring of integers of the imaginary quadratic field of discriminant d. The Bianchi orbifold \Omega_d = \Gamma_d\H^3 is known to contain infinitely many immersed totally geodesic surfaces, which can be identified with integral binary hermitian forms over O_d. In this talk, I will show that some of these immersed totally geodesic surfaces are in fact orientable embedded closed totally geodesic surfaces using number theoretic ideas. I then present some numerical data concerning these surfaces. This talk is based on a joint work with Alan Reid and on an ongoing project with Sam Kim and James Rickards.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space
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