Homogeneous Flows, Moduli Spaces and Arithmetic, Pisa, Italy

Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- Summer school School on Homogeneous Flows, Moduli • classical study by Nalini Anantharaman; Spaces and Arithmetic Equidistribution on homogeneous spaces The Centro di Ricerca Matematica Ennio De • and the analytic theory of L-functions by Giorgi in Pisa, Italy provided a wonderful setting Akshay Venkatesh; for the 2007 Clay Mathematics Institute Summer School. The school was designed to serve as a Counting and equidistribution on homoge- comprehensive introduction to the theory of flows • neous spaces, via mixing and unipotent on homogeneous spaces, moduli spaces and their flows by Hee Oh; many applications. These flows give concrete ex- amples of dynamical systems with highly interest- Informal introduction to unipotent flows by ing behavior and a rich and powerful theory. They • Gregory Margulis; are also a source of many interesting problems and conjectures. Furthermore, understanding the dy- Modular shadows by Yuri Manin; namics of such concrete system lends to numer- • ous applications in number theory and geometry On the regularity of solutions of the coho- regarding equidistributions, diophantine approxi- • mological equation for IET’s and transla- mations, rational billiards and automorphic forms. tion flows Modular Shadows by Giovanni Aerial view of Pisa. © Peter Adams. The program was built around three foundation Forni; courses: Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- The Distribution of free path lengths • • School on Homogeneous Flows, Moduli (1) Unipotentclassical study flowsby and Nalini applications Anantharaman;by Alex in the Periodic Lorentz Gas by J. Marklof; Spaces and Arithmetic Eskin and Dmitry Kleinbock; Equidistribution on homogeneous spaces Uniform spectral gap bounds and arithmetic • • The Centro di Ricerca Matematica Ennio De (2) Diagonalizableand the analytic actions theory and of L-functions arithmetic ap-by applications by A. Gamburd; Giorgi in Pisa, Italy provided a wonderful setting plicationsAkshay Venkatesh;by Manfred Einsiedler and Elon Multi-valued Hamiltonians and Birkhoff for the 2007 Clay Mathematics Institute Summer Lindenstrauss; • School. The school was designed to serve as a Counting and equidistribution on homoge- sums over rotations and IET by • comprehensive introduction to the theory of flows (3) Intervalneous spaces, exchange via mapsmixing and and translation unipotent sur- C. Ulcigrai; on homogeneous spaces, moduli spaces and their facesflows byby Jean-Christophe Hee Oh; Yoccoz. Random hyperbolic surfaces and measured many applications. These flows give concrete ex- • amples of dynamical systems with highly interest- Informal introduction to unipotent flows by laminations by M. Mirzakhani. These• were supplemented by various Short Courses ing behavior and a rich and powerful theory. They and AdvancedGregory Mini Margulis; Courses and Lectures: are also a source of many interesting problems and conjectures. Furthermore, understanding the dy- Modular shadows by Yuri Manin; namics of such concrete system lends to numer- • Equidistribution and L-Functions by Gerge- • ly Harcos; ous applications in number theory and geometry On the regularity of solutions of the coho- One way to orientate oneself within the formida- • regarding equidistributions, diophantine approxi- mological equation for IET’s and transla- ble mathematical landscape explored in the school Reveiw of Vatsal’s work on equidistribution mations, rational billiards and automorphic forms. tion flows Modular Shadows by Giovanni is to consider the familiar space SL(2, )/SL(2, ): • and non-vanishing L-functions by Nicolas R Z The program was built around three foundation Forni; to a hyperbolic geometer, it is the unit tangent Templier; courses: bundle of a hyperbolic surface; to a number the- The Distribution of free path lengths orist, it is the space of elliptic curves; to a low- • Homogeneous flows, buildings and tilings 22 CMI ANNUAL REPORT in the Periodic Lorentz Gas by J. Marklof; dimensional topologist it is the moduli space of flat (1) Unipotent flows and applications by Alex • by Shahar Mozes; Eskin and Dmitry Kleinbock; metrics (with an associated vector field) on genus 1 Uniform spectral gap bounds and arithmetic surfaces; to those who study Diophantine approx- • Fuchsian groups, geodesic flows on surfaces applications by A. Gamburd; imation, it is the space of unimodular lattices in (2) Diagonalizable actions and arithmetic ap- • of constant negative curvature and symbolic 2; and to Lie theorists it is a motivating example plications by Manfred Einsiedler and Elon coding of geodesics by Svetlana Katok R Multi-valued Hamiltonians and Birkhoff of a finite volume homogeneous space G/Γ, that is, Lindenstrauss; • sums over rotations and IET by of a lattice Γ inside a Lie group G. ChaoticityC. Ulcigrai; of the Teichmuller flow by Ar- (3) Interval exchange maps and translation sur- • tur Avila; Two principal generalizations provided the set- faces by Jean-Christophe Yoccoz. Random hyperbolic surfaces and measured ting for much of the material presented at the school: • laminations by M. Mirzakhani. These were supplemented by various Short Courses and Advanced Mini Courses and Lectures:

Equidistribution and L-Functions by Gerge- • ly Harcos; One way to orientate oneself within the formida- ble mathematical landscape explored in the school Reveiw of Vatsal’s work on equidistribution is to consider the familiar space SL(2, )/SL(2, ): • and non-vanishing L-functions by Nicolas R Z to a hyperbolic geometer, it is the unit tangent Templier; bundle of a hyperbolic surface; to a number the- orist, it is the space of elliptic curves; to a low- Homogeneous flows, buildings and tilings dimensional topologist it is the moduli space of flat • by Shahar Mozes; metrics (with an associated vector field) on genus 1 surfaces; to those who study Diophantine approx- Fuchsian groups, geodesic flows on surfaces imation, it is the space of unimodular lattices in • of constant negative curvature and symbolic 2; and to Lie theorists it is a motivating example coding of geodesics by Svetlana Katok R of a finite volume homogeneous space G/Γ, that is, of a lattice Γ inside a Lie group G. Chaoticity of the Teichmuller flow by Ar- • tur Avila; Two principal generalizations provided the set- ting for much of the material presented at the school: Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- Clay Mathematics Institute 2007 Summer • Eigenfunctions of the laplacian: a semi- School on Homogeneous Flows, Moduli • classical study by Nalini Anantharaman; School onSpaces Homogeneous and Arithmetic Flows, Moduli classical study by Nalini Anantharaman; Spaces and Arithmetic Equidistribution on homogeneous spaces • Equidistribution on homogeneous spaces Summer school The Centro di Ricerca Matematica Ennio De • and the analytic theory of L-functions by GiorgiThe in Centro Pisa, Italydi Ricerca provided Matematica a wonderful Ennio setting De Akshayand the Venkatesh; analytic theory of L-functions by Organizers: David Ellwood (CMI), Manfred Einsiedler (Ohio State), Alex Eskin (Chicago), Dmitry Kleinbock (Brandeis), forGiorgi the in 2007 Pisa, Clay Italy Mathematics provided a Institute wonderful Summer setting Akshay Venkatesh; Elon Lindenstrauss, chair (Princeton), Gregory Margulis (Yale), Stefano Marmi (La Scuola Normale Superiore di Pisa), School.for the 2007 The Clay school Mathematics was designed Institute to serve Summer as a Counting and equidistribution on homoge- PeterSchool. Sarnak The (Princeton), school was Jean-Christophe designed to Yoccoz serve as(Collège a de France), Don• Counting Zagier (MIT) and equidistribution on homoge- comprehensive introduction to the theory of flows • neous spaces, via mixing and unipotent oncomprehensive homogeneous introduction spaces, moduli to the spaces theory and of flowstheir flowsneousby spaces, Hee Oh; via mixing and unipotent manyon homogeneous applications. spaces, These moduli flows give spaces concrete and their ex- flows by Hee Oh; amplesmany applications. of dynamical These systems flows with give highly concrete interest- ex- Informal introduction to unipotent flows by amples of dynamical systems with highly interest- • Informal introduction to unipotent flows by ing behavior and a rich and powerful theory. They • Gregory Margulis; areing alsobehavior a source and of a rich many and interesting powerful problems theory. They and Gregory Margulis; conjectures.are also a source Furthermore, of many interesting understanding problems the and dy- Modular shadows by Yuri Manin; conjectures. Furthermore, understanding the dy- • Modular shadows by Yuri Manin; namics of such concrete system lends to numer- • ousnamics applications of such concrete in number system theory lends and to geometry numer- On the regularity of solutions of the coho- ous applications in number theory and geometry • On the regularity of solutions of the coho- regarding equidistributions, diophantine approxi- • mological equation for IET’s and transla- mations,regarding rational equidistributions, billiards and diophantine automorphic approxi- forms. tionmological flows equation Modular for Shadows IET’sby and Giovanni transla- Themations, program rational was billiards built around and automorphic three foundation forms. Forni;tion flows Modular Shadows by Giovanni courses:The program was built around three foundation Forni; courses: The Distribution of free path lengths • The Distribution of free path lengths (1) Unipotent flows and applications by Alex in the Periodic Lorentz Gas by J. Marklof; • in the Periodic Lorentz Gas by J. Marklof; (1) EskinUnipotent and Dmitryflows and Kleinbock; applications by Alex Eskin and Dmitry Kleinbock; Uniform spectral gap bounds and arithmetic • Uniform spectral gap bounds and arithmetic (2) Diagonalizable actions and arithmetic ap- applications by A. Gamburd; • applications by A. Gamburd; (2) plicationsDiagonalizableby Manfred actions Einsiedler and arithmetic and Elon ap- plications by Manfred Einsiedler and Elon Multi-valued Hamiltonians and Birkhoff Lindenstrauss; • Lindenstrauss; sumsMulti-valued over rotations Hamiltonians and IET andby Birkhoff • sums over rotations and IET by (3) Interval exchange maps and translation sur- C. Ulcigrai; C. Ulcigrai; (3) facesIntervalby exchange Jean-Christophe maps and Yoccoz. translation sur- faces by Jean-Christophe Yoccoz. Random hyperbolic surfaces and measured • laminationsRandom hyperbolicby M. Mirzakhani. surfaces and measured These were supplemented by various Short Courses • laminations by M. Mirzakhani. andThese Advanced were supplemented Mini Courses by variousand Lectures: Short Courses and Advanced Mini Courses and Lectures: Equidistribution and L-Functions by Gerge- • lyEquidistribution Harcos; and L-Functions by Gerge- • ly Harcos; One way to orientate oneself within the formida- bleOne mathematical way to orientate landscape oneself explored within in the the formida- school Reveiw of Vatsal’s work on equidistribution ble mathematical landscape explored in the school • Reveiw of Vatsal’s work on equidistribution is to consider the familiar space SL(2, R)/SL(2, Z): • and non-vanishing L-functions by Nicolas is to consider the familiar space SL(2, )/SL(2, ): Photo courtesyand Centro non-vanishing di Ricerca Matematica L-functions Ennio De Giorgi.by Nicolas to a hyperbolic geometer, it is theR unit tangentZ Templier; to a hyperbolic geometer, it is the unit tangent Templier; bundle of a hyperbolic surface; to a number the- orist,bundle it of is a the hyperbolic space of surface; elliptic tocurves; a number to a low- the- Homogeneous flows, buildings and tilings orist, it is the space of elliptic curves; to a low- • Homogeneous flows, buildings and tilings dimensional topologist it is the moduli space of flat by Shahar Mozes; dimensional topologist it is the moduli space of flat • by Shahar Mozes; metrics (with an associated vector field) on genus 1 surfaces;metrics (with to those an associated who study vector Diophantine field) on approx- genus 1 Fuchsian groups, geodesic flows on surfaces surfaces; to those who study Diophantine approx- • Fuchsian groups, geodesic flows on surfaces imation, it is the space of unimodular lattices in of constant negative curvature and symbolic imation,2 it is the space of unimodular lattices in • of constant negative curvature and symbolic R ; and to Lie theorists it is a motivating example coding of geodesics by Svetlana Katok 2; and to Lie theorists it is a motivating example coding of geodesics by Svetlana Katok ofR a finite volume homogeneous space G/Γ, that is, of a finitelattice volumeΓ inside homogeneous a Lie group spaceG. G/Γ, that is, Chaoticity of the Teichmuller flow by Ar- of a lattice Γ inside a Lie group G. • Chaoticity of the Teichmuller flow by Ar- • tur Avila; Two principal generalizations provided the set- tur Avila; tingTwo for much principal of the generalizations material presented provided at the the school: set- ting for much of the material presented at the school: Clay Mathematics Institute 2007 Summer Eigenfunctions of the laplacian: a semi- Dynamics on the space of lattices: Let X = Dynamics on the moduli space of flat sur- School on Homogeneous Flows, Moduli • classical study by Nalini Anantharaman; n SL(n, )/SL(n, ). This is the space of unimodu- faces: Instead of considering the space X of lat- Spaces and Arithmetic R Z n lar lattices in n, as well as a homogeneous space tices in higher dimensions, we can consider the Equidistribution on homogeneous spaces R G/Γ. On our motivating example X there are two moduli spaces of flat metrics (with associated The Centro di Ricerca Matematica Ennio De • and the analytic theory of L-functions by 2 important dynamical systems arising from the left vector fields) onH surfaces of higher genus g 2. Giorgi in Pisa, Italy provided a wonderful setting Akshay Venkatesh; action of one-parameter subgroups: the geodesic There is a natural SL(2, ) action on this space,≥ for the 2007 Clay Mathematics Institute Summer R flow, given by action of the diagonal subgroup coming from the linear action on 2. The action School. The school was designed to serve as a Counting and equidistribution on homoge- R of the subgroup A is called Teichm¨uller geodesic comprehensive introduction to the theory of flows • neous spaces, via mixing and unipotent et 0 flow, and the orbit of a point x under this on homogeneous spaces, moduli spaces and their flows by Hee Oh; A = g = ; ∈ H t 0 e t flow yields information about the ergodic proper- many applications. These flows give concrete ex- − t   ∈R ties of the associated vector field. amples of dynamical systems with highly interest- Informal introduction to unipotent flows by • 2007 23 ing behavior and a rich and powerful theory. They Gregory Margulis; and the horocycle flow, given by action of the unipo- A seemingly unrelated family of dynamical sys- are also a source of many interesting problems and tent subgroup tems are interval exchange maps: given a partition conjectures. Furthermore, understanding the dy- Modular shadows by Yuri Manin; of the unit interval into n labeled subintervals, re- • namics of such concrete system lends to numer- 1 t arrange them according to a permutation π S . U = ht = . ∈ n ous applications in number theory and geometry On the regularity of solutions of the coho- 01 t However, if we take a first return map for the flow •   ∈R regarding equidistributions, diophantine approxi- mological equation for IET’s and transla- associated to a vector field x to a transverse mations, rational billiards and automorphic forms. tion flows Modular Shadows by Giovanni ∈ H Two sets of lectures focused primarily on gen- interval, we obtain exactly one of these exchange The program was built around three foundation Forni; eralizations of each of these actions: Alex Eskin maps. courses: and Dmitry Kleinbock (with a two lecture prequel In the last foundational course, Jean-Chrstophe The Distribution of free path lengths by Grigorii Margulis) delivered a lecture series on Yoccoz explored this connection from the perspec- • in the Periodic Lorentz Gas by J. Marklof; (1) Unipotent flows and applications by Alex the action of unipotent subgroups H on homoge- tive of combinatorics and dynamics of interval ex- Eskin and Dmitry Kleinbock; neous spaces G/Γ, exploring the results of Dani, changes. In particular, he showed how to use the Uniform spectral gap bounds and arithmetic Margulis, and others on non-divergence of orbits; ergodicity of Teichm¨ullerflow and associated renor- • applications by A. Gamburd; (2) Diagonalizable actions and arithmetic ap- Margulis’s use of unipotent dynamics to prove the malization procedures on the space of interval ex- plications by Manfred Einsiedler and Elon Oppenheim conjecture on the values of quadratic changes to show the resolution (by Masur & Veech) Multi-valued Hamiltonians and Birkhoff Lindenstrauss; forms; and Ratner’s classification of orbit closures of the Keane conjecture that almost every interval • sums over rotations and IET by and invariant measures. Here, even in the case exchange map is uniquely ergodic. C. Ulcigrai; (3) Interval exchange maps and translation sur- n = 2, the situation is quite rigid: for example, all faces by Jean-Christophe Yoccoz. orbits of the group U are either periodic or dense. Following Yoccoz’s lectures, Artur Avila, Gio- Random hyperbolic surfaces and measured vanni Forni, and Maryam Mirzakhani gave further • laminations by M. Mirzakhani. In contrast, the action of A for n = 2 is re- talks on Teichm¨ullerdynamics, exploring applica- These were supplemented by various Short Courses markably chaotic: given any 1 α 3, one can tions to the ergodic theory of polygonal billiards, and Advanced Mini Courses and Lectures: ≤ ≤ produce an orbit whose closure has Hausdorff di- and studying closely related spaces of foliations and mension α. However, for n 3, there are conjec- laminations on surfaces. Equidistribution and L-Functions by Gerge- tures of Margulis on the rigidity≥ of the action of • ly Harcos; the diagonal subgroup. Two of the main contrib- All the lecturers made a special effort to ensure One way to orientate oneself within the formida- utors to this field, Manfred Einsiedler and Elon their presentations would be accessible to all the ble mathematical landscape explored in the school Reveiw of Vatsal’s work on equidistribution Lindenstrauss, gave lectures on the progress made participants in the summer school, from beginning • is to consider the familiar space SL(2, R)/SL(2, Z): and non-vanishing L-functions by Nicolas toward these conjectures, focusing on the theory of graduate students on upwards. Ninety-seven young to a hyperbolic geometer, it is the unit tangent Templier; entropy; the applications to the Littlewood conjec- mathematicians participated, from Europe, Asia, bundle of a hyperbolic surface; to a number the- ture on simultaneous diophantine approximation; the Middle East, and the Americas. In addition orist, it is the space of elliptic curves; to a low- Homogeneous flows, buildings and tilings and the theory of quantum unique ergodicity. to bringing many of the participants in these fields • dimensional topologist it is the moduli space of flat by Shahar Mozes; together to exchange ideas, the summer school has metrics (with an associated vector field) on genus 1 Building on these lectures, Nalini Ananthara- hopefully helped spark the interest of a new gener- surfaces; to those who study Diophantine approx- man, Gergely Harcos, Hee Oh, and Akshay Venka- Fuchsian groups, geodesic flows on surfaces ation of mathematicians in these beautiful areas. imation, it is the space of unimodular lattices in tesh gave shorter series of more advanced lectures, • of constant negative curvature and symbolic 2; and to Lie theorists it is a motivating example indicating applications to quantum chaos, auto- coding of geodesics by Svetlana Katok R of a finite volume homogeneous space G/Γ, that is, morphic forms, and counting points on varieties. In of a lattice Γ inside a Lie group G. addition, a special session on Diophantine approxi- Chaoticity of the Teichmuller flow by Ar- mation was organized by Dmitry Kleinbock, allow- • tur Avila; Two principal generalizations provided the set- ing many of the younger participants to present ting for much of the material presented at the school: their recent results in the subject. Dynamics on the space of lattices: Let Xn = Dynamics on the moduli space of flat sur- SL(n, R)/SL(n, Z). This is the space of unimodu- faces: Instead of considering the space Xn of lat- lar lattices in Rn, as well as a homogeneous space tices in higher dimensions, we can consider the G/Γ. On our motivating example X2 there are two moduli spaces of flat metrics (with associated important dynamical systems arising from the left vector fields) onH surfaces of higher genus g 2. ≥ action of one-parameter subgroups: the geodesic There is a natural SL(2, R) action on this space, Homogeneousflow, given by action of the diagonal Flows, subgroup Modulicoming Spaces from the linear and action on R2. The action of the subgroup A is called Teichm¨uller geodesic et 0 flow, and the orbit of a point x under this Arithmetic,A = g = Pisa, Italy; t t flow yields information about the∈ ergodicH proper- 0 e− t   ∈R ties of the associated vector field. and the horocycle flow, given by action of the unipo- A seemingly unrelated family of dynamical sys- tent subgroup tems are interval exchange maps: given a partition of the unit interval into n labeled subintervals, re- 1 t arrange them according to a permutation π S . U = ht = . ∈ n 01 t However, if we take a first return map for the flow   ∈R associated to a vector field x to a transverse interval, we obtain exactly one∈ ofH these exchange DynamicsTwo sets on of lectures the space focused of lattices: primarilyLet onX gen-= Dynamics on the moduli space of flat sur- n maps. eralizationsSL(n, R)/SL of(n, eachZ). Thisof these is the actions: space of Alex unimodu- Eskin faces: Instead of considering the space Xn of lat- andlar lattices Dmitry in KleinbockRn, as well (with as a a homogeneous two lecture prequel space ticesIn thein higher last foundational dimensions, course, we can Jean-Chrstophe consider the byG/Γ Grigorii. On our Margulis) motivating delivered example aX lecture2 there series are two on Yoccozmoduli explored spaces thisof flatconnection metrics from (with the associated perspec- Summer school theimportant action dynamical of unipotent systems subgroups arisingH fromon homoge- the left tivevector of combinatoricsfields) onH surfaces and dynamics of higher ofgenus intervalg ex-2. ≥ neousaction spaces of one-parameterG/Γ, exploring subgroups: the results the ofgeodesic Dani, changes.There is a In natural particular,SL(2 he, R showed) action how on this to use space, the Margulis,flow, given and by action others of on the non-divergence diagonal subgroup of orbits; ergodicitycoming from of Teichm¨ullerflow the linear action and on associatedR2. The actionrenor- Margulis’s use of unipotent dynamics to prove the malizationof the subgroup proceduresA is calledon theTeichm¨uller space of interval geodesic ex- Oppenheim conjecture onet the0 values of quadratic changesflow, and to the show orbit the resolution of a point (byx Masurunder & Veech) this A = g = ; forms; and Ratner’st classificationt of orbit closures offlow the yields Keane information conjecture about that almost the∈ ergodic everyH interval proper- 0 e− t and invariant measures. Here, even∈R in the case exchangeElonties Lindenstrauss of the map associated conducting is uniquely a session vector in ergodic. Pisa. field. nand= the2, thehorocycle situation flow is, quitegiven rigid:by action for ofexample, the unipo- all orbits of the group U are either periodic or dense. FollowingA seemingly Yoccoz’s unrelated lectures, family Artur of dynamical Avila, Gio- sys- tent subgroup vannitems are Forni,interval andMaryam exchange Mirzakhani maps: given gave a partition further In contrast, the action of A for n = 2 is re- talksof the on unit Teichm¨ullerdynamics, interval into n labeled exploring subintervals, applica- re- markably chaotic: given any1 t 1 α 3, one can tionsarrange to them the ergodic according theory to a of permutation polygonal billiards,π S . U = ht = ≤ ≤. ∈ n produce an orbit whose closure01 hast Hausdorff di- andHowever, studying if we closely take arelated first return spaces map of foliations for the flow and   ∈R mension α. However, for n 3, there are conjec- laminationsassociated to on a surfaces. vector field x to a transverse tures of Margulis on the rigidity≥ of the action of ∈ H Two sets of lectures focused primarily on gen- interval, we obtain exactly one of these exchange the diagonal subgroup. Two of the main contrib- All the lecturers made a special effort to ensure eralizations of each of these actions: Alex Eskin maps. utors to this field, Manfred Einsiedler and Elon their presentations would be accessible to all the and Dmitry Kleinbock (with a two lecture prequel Lindenstrauss, gave lectures on the progress made participantsIn the last in foundational the summer course, school, Jean-Chrstophe from beginning by Grigorii Margulis) delivered a lecture series on toward these conjectures, focusing on the theory of graduateYoccoz explored students this on upwards.connection Ninety-seven from the perspec- young the action of unipotent subgroups H on homoge- entropy; the applications to the Littlewood conjec- mathematicianstive of combinatorics participated, and dynamics from Europe,of interval Asia, ex- neous spaces G/Γ, exploring the results of Dani, ture on simultaneous diophantine approximation; thechanges. Middle In East, particular, and the he Americas.showed how In to addition use the Margulis, and others on non-divergence of orbits; and the theory of quantum unique ergodicity. toergodicity bringing of many Teichm¨ullerflow of the participants and associated in these renor- fields Margulis’s use of unipotent dynamics to prove the togethermalization to procedures exchange ideas, on the the space summer of interval school has ex- OppenheimBuilding onconjecture these lectures, on the values Nalini of Ananthara- quadratic hopefullychanges to helped show the spark resolution the interest (by Masur of a new & Veech) gener- man,forms; Gergely and Ratner’s Harcos, classification Hee Oh, and of Akshay orbit closures Venka- ationof the of Keane mathematicians conjecture that in these almost beautiful every interval areas. teshand gaveinvariant shorter measures. series of more Here, advanced even in thelectures, case exchange map is uniquely ergodic. indicatingn = 2, thesituation applications is quite to quantum rigid: for chaos, example, auto- all morphicorbits of forms, the group andU countingare either points periodic on varieties. or dense. In Following Yoccoz’s lectures, Artur Avila, Gio- addition, a special session on Diophantine approxi- vanni Forni, and Maryam Mirzakhani gave further In contrast, the action of A for n = 2 is re- mation was organized by Dmitry Kleinbock, allow- talks on Teichm¨ullerdynamics, exploring applica- markably chaotic: given any 1 α 3, one can ing many of the younger participants to present tions to the ergodic theory of polygonal billiards, produce an orbit whose closure≤ has Hausdor≤ ff di- their recent results in the subject. and studying closely related spaces of foliations and mension α. However, for n 3, there are conjec- laminations on surfaces. ≥ Dynamics on the space of lattices: Let Xn = Dynamicstures of Margulis on the on moduli the rigidity space of the of flat action sur- of SL(n, R)/SL(n, Z). This is the space of unimodu- faces:the diagonalInstead subgroup. of considering Two of the the space mainXn contrib-of lat- All the lecturers made a special effort to ensure lar lattices in Rn, as well as a homogeneous space ticesutors in to higher this field, dimensions, Manfred we Einsiedler can consider and Elon the their presentations would be accessible to all the G/Γ. On our motivating example X2 there are two moduliLindenstrauss, spaces gaveof lectures flat metrics on the (with progress associated made participants in the summer school, from beginning important dynamical systems arising from the left vectortoward fields) these conjectures, onH surfaces focusing of higher ongenus the theoryg 2. of graduate students on upwards. Ninety-seven young ≥ action of one-parameter subgroups: the geodesic Thereentropy; is the a natural applicationsSL(2, toR) the action Littlewood on this conjec- space, mathematicians participated, from Europe, Asia, flow, given by action of the diagonal subgroup comingture on from simultaneous the linear diophantine action on R approximation;2. The action the Middle East, and the Americas. In addition ofand the the subgroup theory ofA quantumis called uniqueTeichm¨uller ergodicity. geodesic to bringing many of the participants in these fields et 0 flow, and the orbit of a point x under this together to exchange ideas, the summer school has A = g = ; Building on these lectures, Nalini∈ H Ananthara- t 0 e t flow yields information about the ergodic proper- hopefully helped spark the interest of a new gener- − t man, Gergely Harcos, Hee Oh, and Akshay Venka-   ∈R ties of the associated vector field. ation of mathematicians in these beautiful areas. tesh gave shorter series of more advanced lectures, and the horocycle flow, given by action of the unipo- 24 CMI indicatingANNUALA seemingly REPORT applications unrelated to family quantum of dynamical chaos, auto- sys- tent subgroup temsmorphic are forms,interval and exchange counting maps points: given on varieties. a partition In ofaddition, the unit a specialinterval session into n onlabeled Diophantine subintervals, approxi- re- 1 t arrangemation was them organized according by to Dmitry a permutation Kleinbock,π allow-S . U = ht = . ∈ n 01 t However,ing many if of we the take younger a firstreturn participants map for to the present flow   ∈R associatedtheir recent to results a vector in the field subject.x to a transverse ∈ H Two sets of lectures focused primarily on gen- interval, we obtain exactly one of these exchange eralizations of each of these actions: Alex Eskin maps. and Dmitry Kleinbock (with a two lecture prequel In the last foundational course, Jean-Chrstophe by Grigorii Margulis) delivered a lecture series on Yoccoz explored this connection from the perspec- the action of unipotent subgroups H on homoge- tive of combinatorics and dynamics of interval ex- neous spaces G/Γ, exploring the results of Dani, changes. In particular, he showed how to use the Margulis, and others on non-divergence of orbits; ergodicity of Teichm¨ullerflow and associated renor- Margulis’s use of unipotent dynamics to prove the malization procedures on the space of interval ex- Oppenheim conjecture on the values of quadratic changes to show the resolution (by Masur & Veech) forms; and Ratner’s classification of orbit closures of the Keane conjecture that almost every interval and invariant measures. Here, even in the case exchange map is uniquely ergodic. n = 2, the situation is quite rigid: for example, all orbits of the group U are either periodic or dense. Following Yoccoz’s lectures, Artur Avila, Gio- vanni Forni, and Maryam Mirzakhani gave further In contrast, the action of A for n = 2 is re- talks on Teichm¨ullerdynamics, exploring applica- markably chaotic: given any 1 α 3, one can tions to the ergodic theory of polygonal billiards, ≤ ≤ produce an orbit whose closure has Hausdorff di- and studying closely related spaces of foliations and mension α. However, for n 3, there are conjec- laminations on surfaces. tures of Margulis on the rigidity≥ of the action of the diagonal subgroup. Two of the main contrib- All the lecturers made a special effort to ensure utors to this field, Manfred Einsiedler and Elon their presentations would be accessible to all the Lindenstrauss, gave lectures on the progress made participants in the summer school, from beginning toward these conjectures, focusing on the theory of graduate students on upwards. Ninety-seven young entropy; the applications to the Littlewood conjec- mathematicians participated, from Europe, Asia, ture on simultaneous diophantine approximation; the Middle East, and the Americas. In addition and the theory of quantum unique ergodicity. to bringing many of the participants in these fields together to exchange ideas, the summer school has Building on these lectures, Nalini Ananthara- hopefully helped spark the interest of a new gener- man, Gergely Harcos, Hee Oh, and Akshay Venka- ation of mathematicians in these beautiful areas. tesh gave shorter series of more advanced lectures, indicating applications to quantum chaos, auto- morphic forms, and counting points on varieties. In addition, a special session on Diophantine approxi- mation was organized by Dmitry Kleinbock, allow- ing many of the younger participants to present their recent results in the subject.