# Past and Future Talks

## Spring 2017

 Feb 2: The number of non-negative curvature triangulations of the sphere Phil Engel, Harvard University Feb 8: Flat surfaces and stability structures on categories Fabian Haiden, Harvard University Feb 15: Constructing pseudo-Anosov mapping classes with small stretch factor Eriko Hironaka, AMS Feb 22: Strata of abelian differentials and the effective cone of M_g,n Scott Mullane, Harvard University March 1: Global dynamics of multicurves in complex dynamics Russell Lodge, Stony Brook March 8: Extremal and rigid divisors on moduli spaces of curves Dawei Chen, Boston College March 22: Differentiating Blaschke products Oleg Ivrii, Caltech March 29: Smooth compactifications of strata of abelian differentials Matt Bainbridge, Indiana University April 5: Berkovich spaces and complex dynamics Yusheng Luo, Harvard University April 12: Rationality of canonical height Laura DeMarco, Northwestern University April 19: Affine surfaces part I Eduard Duryev, Harvard University April 25: Affine surfaces part II Jane Wang, MIT

## Fall 2016

 Sep 14, 21: Almost simple geodesics on the triply-punctured sphere C. McMullen , Harvard Sep 28: Introduction to Teichmueller curves in genus 2 C. McMullen , Harvard Oct 5, 12: Square-tiled surfaces of genus 2 E. Duryev , Harvard Oct 19: Moduli space, surface bundles, and the Atiyah-Kodaira examples B. Tshishiku , Harvard Oct 26: C != K on Teichmueller space, after Markovic Y. Luo , Harvard Nov 2,9: From Teichmueller curves to higher-dimensional invariant subvarieties S. Filip , Harvard Nov 16: Large balls in Teichmueller space are not convex, after Bourque and Rafi J. Wang , MIT

## Spring 2015

 Feb 4: Kronecker's congruence and Teichmüller curves in positive characteristic Ronen Mukamel, University of Chicago Feb 11: Transversality John H. Hubbard, Cornell University Feb 18: Rubber Bands and Rational Maps Dylan Thurston, Indiana University Feb 25: Equidistribution of Shears and Applications Alex Kontorovich, Rutgers University March 4: Circle actions on the boundary of Schottky space Alden Walker, University of Chicago March 11: Hyperbolic extensions of free groups Sam Taylor, Yale University March 18: Have a good Spring Break! March 25: Generalizing Douady's Magic Formula Adam Epstein, University of Warwick April 1: On the classification of critically fixed rational maps Kevin Pilgrim, Indiana University April 15: Families of K3 surfaces and their asymptotic properties Simion Filip, University of Chicago April 22: A case of the dynamical André-Oort conjecture Holly Krieger, MIT April 29: A polynomial endomorphism of C^2 with a wandering Fatou component Matthieu Astorg, Université Paul Sabatier May 1-3: Reading Group Reunion, Harvard University May 13: Complex dynamics and elliptic curves Laura DeMarco, Northwestern University

## Spring 2014

 C. Taubes. Limits of flat SL(2;C) connections and higher dimensional analogs of holomorphic quadratic differentials Date TBA C. McMullen. Fibered 3-manifolds and the simplest known surface dynamics 30 April C. McMullen. A brief tour of the mathematical plaster models 16 April C. McMullen. Constant curvature and Outer space 2 April T. Church. Moebius functions and growth rates of groups 26 March C. McMullen. Optimal metrics on graphs 12 Feb -- 12 Mar

## 2013

 G. Tiozzo. Galois conjugates of entropies of quadratic maps 4 Dec. M. Bridgeman. The pressure metric for convex Anosov representations. 13 Nov. C. McMullen. The complex hyperbolic geometry of moduli space. 4 Nov. E. Hironaka. Towards a fibered face theory for free group automorphisms. 30 Oct. C. McMullen. The clique polynomial of a graph. 14 Aug. D. Dumas. Convex polygons, complex polynomials, and hyperbolic affine spheres. 8 May. C. Smart. The Abelian sandpile and Apollonian circle packings. 1 May. G. Tiozzo. Entropy, dimension and combinatorial moduli for one-dimensional dynamical systems. 24 April. M. Bridgeman, M. Duchin, V. Gadre, D. Kleinbock, G. Walsh. PechaKucha: Dynamics & Geometry, 20x20. 17 April. D. Calegri. Random groups contain surface subgroups. 10 April. K. Rafi. Geometry of Teichmüller space. 3 April. A. Wright. SL(2,R)-orbit closures of translation surfaces. 27 March. E. Bedfod. Automorphisms of complex manifolds with positive entropy. March 13. B. Gross, O. Knill, S. Koch, B. Mazur, C. McMullen, E. Riehl. PechaKucha: Mathematics, 20x20. 6 March. J. Athreya. Gap Distributions and Homogeneous Dynamics. 20 Feb. L. Bartholdi. Computing with branched coverings. 13 Feb. D. Davis. Cutting sequences on Bouw-Möller surfaces 6 Feb. O. Ivrii. Completing the Cardioid. 30 Jan.

## Fall 2012

 9/5: Fundamental domains and generators for lattice Veech groups R. Mukamel, Stanford University 9/12: Geodesic ray tracking for random walks G. Tiozzo, Harvard University 9/19: Winning sets of foliations for Schmidt games H. Masur, University of Chicago 9/26: Holomorphic families of Riemann surfaces and their degenerations H. Shiga, Tokyo Institute of Technology 10/3: Two extensions of Thurston's spectral theorem of surface diffeomorphisms A. Karlsson, University of Geneva 10/10: Pullback invariants of critically finite rational maps K. Pilgrim, Indiana University 10/17: Translation surfaces and Teichmüller dynamics J. Smillie, Cornell University Thursday, 10/25 (Department Colloquium, 4:30PM): Special curves and postcritically finite polynomials L. DeMarco, University of Illinois at Chicago 10/31: Convexity and counting A. Sambarino, Orsay 11/7: Effective Veech dichotomy M. Bainbridge, Indiana University 11/14: On the non-existence of super-Teichmüller disks S. Antonakoudis, Harvard University 11/21: no meeting - Happy Thanksgiving! Tuesday, 11/27 (we will meet in room 530): Trinitas for the Hènon map Y. Ishii, Kyushu University 12/5: On Hubbard trees, etc. J. H. Hubbard, Université de Provence/Cornell University 12/12 (Number Theory Seminar, 3PM room 507): The cohomology of SL(n,Z) and its congruence subgroups B. Farb, University of Chicago

## Spring 2012

 S. Antonkoudis. Infinitesimal shape of Teichmueller space. 11 Jan. O. Ivrii. Ghosts of the mapping class group. 11 Jan. G. Tiozzo. Entropy and dimension of sets of external rays. 11 Jan. C. McMullen. Navigating moduli space with complex twists. 11 Jan. 2/1: Mapping class groups, homology and finite covers of surfaces Thomas Koberda, Harvard University 2/15: Complex and p-adic Dynamics ICERM, Providence RI 2/22: Entropy and dimension of real sections of the Mandelbrot set Giulio Tiozzo, Harvard University 2/29: Matings with laminations Dzmitry Dudko, Jacobs University 3/7: Infinitesimal deformations of nodal stable curves Scott Wolpert, University of Maryland 3/14: no meeting, have a good spring break! 3/21: Global Arithmetic Dynamics ICERM, Providence RI 3/28: Invariant components of translation surfaces Kathryn Lindsey, Cornell University 4/4: Polynomial matings and rational maps with cluster cycles Tom Sharland, University of Warwick 4/11: Integer homology 3-spheres with large injectivity radius Nathan Dunfield, University of Illinois at Urbana-Champaign 4/18: Moduli Spaces Associated to Dynamical Systems ICERM, Providence RI 4/25: News about Newton: global dynamics and efficient root finding Dierk Schleicher, Jacobs University 5/2: Conservative diffeomorphisms from a C^1 generic perspective Amie Wilkinson, University of Chicago C. McMullen. Moduli spaces of isoperiodic forms on Riemann surfaces. 6 June. X. Buff. Antipode-preserving cubic rational maps. 6 June

## Fall 2011

 9/7: Twisted Matings of Polynomials Sarah Koch, Harvard University 9/14: Holomorphic coordinates on Teichmüller space and compactified moduli space Clifford Earle, Cornell University 9/21: Polygonal outer billiards, polytope exchanges, and renormalization Rich Schwartz, Brown University 9/28: Solenoids over 3-manifolds Will Cavendish, Princeton University 10/5: A Weil-Petersson metric on Blaschke products Oleg Ivrii, Harvard University 10/12: A proof of Pilgrim's conjecture Nikita Selinger, SUNY Stony Brook Monday, 10/17: Geometric Limits and Renormalization Jeremy Kahn, Brown University 10/26: Intertwined Basins for a cylinder map arising from statistical physics Roland Roeder, Indiana University-Purdue University Indianapolis 11/2: Rescaling limits of rational maps Xavier Buff, Université Paul Sabatier 11/9: The critical locus for complex Hénon maps Tanya Firsova, SUNY Stony Brook 11/16: Orbit trees in the curve complex Johanna Mangahas, Brown University 11/23: no meeting, Happy Thanksgiving! 11/30: Rock-paper-scissors in the complex world Joshua Bowman, SUNY Stony Brook 12/7: Geometry of Hyperbolic 3-Manifolds with Constrained Rank Ian Biringer, Yale University 12/14: Badly approximable directions on flat surfaces Jon Chaika, University of Chicago

## Spring 2011

 C. McMullen. Periodic, Diophantine and ergodic foliations on surfaces. 9 Feb - 9 Mar. J. Hubbard. On the density of Strebel differentials. 23 March. S. Koch. On the Deligne-Mumford compactification of moduli space. 6, 13 April. N. Dunfield. The least spanning area of a knot and the Optimal Bounding Chain Problem. 20 April. M. Duchin. Flat structures as currents. 27 April. C. McMullen. Manifolds, topology and dynamics: Milnor's work and subsequent developments. 11 May. C. McMullen. What is a K3 surface? 27-30 June.

## Fall 2010

 R. Kirby. A new calculus for 4-manifolds. 8 Sep. L. DeMarco. Dynamics of cubic polynomials and enumeration of cusps. 15 Sep. C. McMullen The evolution of geometric structures on 3-manifolds. 20 Sep. (Monday) T. Koberda. A ping-pong theorem for the mapping-class group. 29 Sep. V. Gadre. Singularity of harmonic measures for random walks on the mapping class group. 6 Oct. J. Bourgain. Expansion in linear groups (4:30, Lecture Hall D). 13 Oct. R. Roeder. Blaschke products and renormalization on the diamond hierarchical lattice. 20 Oct. X. Buff. Transversality for Herman rings. 27 Oct. E. Uhre. The Mandelbrot set and its deformations. 3 Nov. G. Tiozzo. Continued fractions and kneading sequences of unimodal maps. 17 Nov. D. Meyer. Invariant Peano curves of expanding Thurston maps. 1 Dec.

## Spring 2010

 C. McMullen. Entropy. 10, 17 Feb. M. Baker. Complex dynamics and adelic potential theory. 24 Feb C. McMullen. Surface dynamics. 3, 10 March T. Koberda. Surface bundles of small volume. 24 March G. Tiozzo. Absolute continuity of invariant measures for random walks on Lie groups. 31 March R. Mukamel. Teichmueller dynamics in genus two, modular forms and zeta functions. 7 April S. Koch. Dynamics in superattracting basins. 14 April A. Putnam. The Picard group of the moduli space of curves with level structures. 21 April D. Margalit. Symplectic representations of braid groups. 28 April G. Walsh. The bumping set and the characteristic submanifold. 5 May

## Fall 2009

 E. Hironaka. Small dilatation pseudo-Anosov mapping classes coming from the simplest pseudo-Anosov braid. H. Oh. Equidistribution and counting for geometrically finite hyperbolic groups. S. Fenley. Pseudo-Anosov flows and large scale geometry of 3-manifolds. S. Koch. Dynamics of maps on moduli space. M. Bridgeman. The orthospectra of finite volume hyperbolic manifolds with totally geodesic boundary and associated volume identities. A. Cotton-Clay. Holomorphic Pants in R times a mapping torus. D. Margalit. Small dilatation pseudo-Anosovs and 3-manifold. G. Walsh. Commensurability of knot complements. G. Tiozzo. The entropy of alpha-continued fractions

## Spring 2009

 H. Oh. Apollonian circle packings and horosphercial flows on hyperbolic 3-manifolds. Wed, 18 Feb D. Margalit. Torelli groups and the complex of minimizing cycles. Wed, 25 Feb R. Devaney. Dynamics of zn + C/zn: Why n=2 is crazy. Wed, 4 March E. Hironaka. Pseudo-Anosov mapping classes with small dilatation constructed from graphs. Wed, 11 March A. Wilkinson. Absolute continuity, Lyapunov exponents and rigidity. Wed, 18 March D. Damjanovic. Perturbations of some higher rank unipotent actions Wed, 1 Apr N. Shah. Limits of expanding translates of shrinking curves on hyperbolic manifolds. Wed, 8 April C. Taubes. An introduction to geometric quantization. Wed, 15 Apr A. Bufetov. Suspension flows over Vershik's automorphisms. Wed, 22 Apr E. Ghys. Dynamics in Dimension 3. (Talk at MIT: room 34-101, 4:30 pm.) Wed, 29 Apr T. Koberda. Braids and Their Dilatations: The Burau, Gassner and Bigelow representations. Wed, 3 May

## Fall 2008

 C. McMullen. Random walks on the hyperbolic plane and barycenter subdivision. Wed, 15 Oct C. McMullen. Braid groups and Hodge theory. Wed, 22 Oct R. Schwartz. Outer billiards and the modular group. Wed, 29 Oct D. Fisher. The space of discrete linear groups. Wed, 5 Nov J. Duncan. Fuchsian groups and affine Dynkin diagrams. Wed, 12 Nov N. Avni. Dynamics on representation varieties --- finite, compact and hyperbolic. Wed, 19 Nov S. Sheffield Hamburgers, cheeseburgers, and scaling limits of discrete random surfaces. Wed, 3 Dec R. Mukamel Surfaces with triangular veech broups Wed, 10 Dec

## Spring 2007

 G. Mondello. Hyperbolic surfaces and systems of arcs. Wed, 14 March D. Kleinbok. Expanding translates of horospheres and applications to number theory. Wed, 21 March J. Behrstock. Asymptotic geometry of the mapping-class group. Wed, 4 April L. Silberman. Measure rigidity for Cartan actions: the "low-entropy" method. Wed, 11 April D. Damjanovic. Littlewood's conjecture and SL_3(R)/SL_3(Z). Wed, 25 April S. Brooks. Halfway to quantum unique ergodicity. Wed, 2 May C. McMullen. Complex dynamics on the unit disk: measures, multipliers and R-trees. Wed, 9 May

## 19--21 June 2006

 D. Dumas. Grafting and shearing hyperbolic surfaces M. Bainbridge. Volumes of Hilbert modular surfaces M. Mirzakhani. Ergodic properties of the space of measured laminations C. McMullen. Thermodynamics, dimension and the Weil-Petersson metric R. Schwartz. Nearly isosceles billiards, Veech points, and universal power series expansions J. Brock. Heegaard splittings, handlebodies and the geometry of hyperbolic 3-manifolds

## May 2005

 Mon, May 9. Prym varieties and Teichmüller space. Curt McMullen Wed, May 11. Euler characteristics of Teichmüller curves. Matt Bainbridge Mon, May 16. Coble sextics and holomorphic actions of lattices on P1. Izzet Coskun Wed, May 18. Polynomial dynamics and trees. Laura DeMarco Wed, May 25. Hausdorff dimension and bendings of Fuchsian groups. Martin Bridgeman Fri, May 27. Train tracks. Maryam Mirzakhani

## Fall 2003

 Oct. 22. Markov maps and discreteness tests for punctures torus groups. David Dumas Oct. 29. The boundary of the space of rational maps. Laura DeMarco. Nov. 5. Is there a polynomial Julia set of positive measure? Matt Bainbridge. Nov. 20. Crystals and algebraic curves. Colloquium, Andrei Okounkov (Princeton) Dec. 12. Billiards and dynamics over moduli space. Gauge theory seminar, Curt McMullen

## Fall 2002

 Sept. 25. Billiards and Riemann surfaces of infinite complexity. Curt McMullen. Oct. 2. Galois flux and measured foliations. Curt McMullen. Oct. 9. What is a random 3-manifold and what does it look like? Nathan Dunfield. Oct. 16. Does a random 3-manifold fiber over the circle? Nathan Dunfield. Oct. 23. No meeting. Go to Rich Schwartz's colloquium on Thursday instead. Oct. 30. Physical measures and periodic orbits of quadratic polynomials. Matt Bainbridge. Nov. 6. Moduli of Polyhedra. Laura DeMarco. Nov. 13. Ergodic theory of horocycles and the earthquake flow on moduli space. Maryam Mirzakhani. Nov. 20. Is the Jones polynomial the same size as the Alexander polynomial? Jacob Rasmussen. Nov. 27. No meeting Dec. 4. Abelian differentials and dynamics Izzet Coskun. Dec. 11. Projective Riemann surfaces with Fuchsian holonomy David Dumas. Dec. 18. The Patterson-Sullivan theory of discrete quasiconformal groups Ed Taylor.

## Spring 2002

 Jan 30. Billiards and Curves of Genus 2. Curt McMullen. Feb 6. Laminations and groups of homeomorphisms of the circle. Nathan Dunfield. Feb 13. The configuration space of points on the projective line and the moduli of polygons. Haruko Nishi. Feb 20. Simple geodesics and the Weil-Petersson volume of the moduli space of Riemann surfaces. Maryam Mirzakhani. Feb 27. Experiments with Quasifuchsian Groups and Projective Structures. David Dumas. Mar 6 (starts 4:30). Growth of the number of periodic points for generic diffeomorphisms. Vadim Kaloshin. Mar 13. Some algebraic questions related to the Poincaré Conjecture. Andrew Casson. Mar 20. A Variational Study of Curvature and Potential Theory. Laura DeMarco. Mar 27. Spring Break. Apr 3. Complex hyperbolic reflection groups: a primer. Daniel Allcock. Apr 10. Rigidity and non-rigidity of hyperbolic 3-manifolds. Kevin Scannell. Apr 17. Shadow pictures: pair-of-pants decompositions of 3-manifolds. Dylan Thurston. Apr 24. Weil-Petersson and Hodge volumes of moduli space, and the Schottky problem. Samuel Grushevsky. May 1. Mostow Rigidity and Lattice Classification. Kathy Paur.

## Fall 2000

 1. Local connectivity -- conformal mapping, dynamics and combinatorics, C. McMullen Quotients of the circle and the sphere Laminations The Riemann mapping Julia sets of polynomials 2. The boundary of an abstract group and simple loops on the torus, C. McMullen The boundary of a group Limit sets of Kleinian groups 3-manifolds that fiber over the circle Simple factorization on the torus References: Local connectivity, Kleinian groups and geodesics on the blowup of the torus 3. Surfaces in finite covers of 3-manifolds: the virtual Haken conjecture, N. Dunfield A closed hyperbolic 3-manifold is Haken if it contains an incompressible surface. If a hyperbolic manifold M=H3/G is not Haken, then G can be taken to lie in SL2(O) for the ring of integers O in a number field. Conjecture: a closed, irreducible 3-manifold M with infinite fundamental group has a finite cover N such that (i) N contains an incompressible surface; or even (ii) b1(N) > 0; or even (iii) N fibers over the circle. Evidence: among the first 10,000 hyperbolic manifolds, 9,999 admit a finite cover with b1(N) > 0. 4. Circles and hyperbolic geometry I, D. Calegari Milnor-Wood: a circle bundle over a surface, E-> S, admits a foliation transverse to the fibers iff |Euler class of E| <= |Euler class of TS|. Automatic structures on G= pi1(S) produce PL actions of G on S1 5. Circles and hyperbolic geometry II, D. Calegari References: Notes by Calegari 6. Hyperbolic volume and the Jones polynomial, D. Thurston Reference: Murakami, The asymptotic behavior of the colored Jones function of a knot and its volume 7. Lyapunov exponents and complex dynamics, L. DeMarco Reference: Complex dynamics, domains of holomorphy, and a positive current on the bifurcation locus 8. Julia sets, harmonic measure and rotation, I. Binder Reference: Harmonic measure and rotation of simply connected planar domains 9. The topology of the space of hyperbolic structures on a 3-manifold, J. Holt Reference: Some new behaviour in the deformation theory of Kleinian groups 10. Complex projective structures on Riemann surfaces, D. Dumas 11. Simple closed curves on surfaces and the volume of moduli space, M. Mirzakhani 12. Dynamics on K3 surfaces, C. McMullen 13. Exotic complex projective surfaces and the boundaries of quasifuchsian and Schottky spaces, H. Tanigawa

## Fall 1999

 1. Hausdorff dimension and the Laplacian on Riemann surfaces, C. McMullen Reflection through 3 circles Linear Cantor sets The bottom on the spectrum on Hd+1 From conformal densities to eigenfunctions 2. Cusps of hyperbolic manifolds and the boundary of the Mandelbrot set, C. McMullen The critical exponent of the Poincaré series A cusp of rank r gives dimension > r/2 Geometric limits and rank 2 cusps A bound of 2q/(q+1) for Julia sets near a p/q rotation References: Hausdorff dimension and conformal dynamics, I, II, III 3. 3-manifold groups, surfaces and actions on trees, N. Dunfield. Reference: P. Shalen, Representations of 3-manifold groups 4. Surgery on 3-manifolds and volume rigidity, N. Dunfield. Reference: N. Dunfield, "Cyclic surgery, ..." 5. Teichmüller space, quadratic differentials and rational billards, H. Masur. 6. Counting periodic points for billiards and flat structures, H. Masur. Reference: A. Eskin, H. Masur, "Pointwise asymptotic formulas on flat surfaces". 7. Salem numbers in geometry, Eriko Hironaka References: E. Hironaka, "The Lehmer polynomial and pretzel knots"; "The arithmetic and geometry of Salem numbers" (with E. Ghate) 8. Average bending of convex hull boundaries, M. Bridgeman Reference: M. Bridgeman, "Average Bending of Convex Pleated Planes in H^3" 9. Currents and instability in complex dynamics, Laura DeMarco Reference: L. DeMarco, "Domains of holomorphy in complex dynamics" 10. Foliated Teichmüller theory, C. McMullen Reference: "Polynomial invariants for fibered 3-manifolds..." 11. Diophantine numbers, quadratic differentials and failure of ergodicity, Yitwah Cheung Reference: Thesis draft 12. An Approach to the Low-Volume Question for Hyperbolic 3-Manifolds, Robert Meyerhoff References: Work in progress

## Spring 1999

 Course Notes 1. Riemann surfaces and their Jacobians The modulus of an annulus Holomorphic 1-form = flat metric + oriented line field The area of the image of X under a 1-form The Bergman metric and the Poincare metric (Kazhdan) Calculating the area from periods The Jacobian and the period matrix Mordell's conjecture: can a finitely generated subgroup in Jac(X) meet X in an infinite set? 2. Lipschitz maps and nets in Euclidean space Field trip to MIT Most separated nets Y in R^n, n>1, are not bilipschitz to Z^n. Most L^\infty functions on R^n cannot be realized as the divergence of a Lipschitz vector field. Reference: GAFA 8 (1998), 304--314. 3. Periods; dynamics The adjunction formula Hyperelliptic surfaces; billiards Introduction to dynamics of f(z)=z^2+c 4. The Mandelbrot set is connected Escape rates A natural metric on the basin of infinity Invariants of z^2+c for c not in M Insulated, planar Riemann surfaces Constructing a polynomial with given invariants 5. Schottky groups and Teichmüller space Schottky groups Teichmüller space The modular group Hyperbolic isometries Pairs of pants Fenchel-Nielsen coordinates Teichmüller space is a cell 6. Earthquakes and hyperbolic geometry Grisha Mikhalkin. Reference: W.P. Thurston, Earthquakes in two-dimensional hyperbolic geometry, in Low-dimensional Topology and Kleinian Groups, Cambridge Univ. Press, 1987, 91-112. 7. Complex dynamics, measures and foliations Laura DeMarco. Reference: J.H. Hubbard and P. Papadopol, Superattractive fixed points in C^n, Indiana Univ. Math. J. 43 (1994), 321--365. 8. Compactness in moduli space Thick-thin decomposition Finiteness of automorphisms Discreteness of the length spectrum Discreteness of action of Mod(S) Loops of length O(log g) Mumford's compactness theorem 9. Simple closed curves The hairy torus: pants with cuffs O(sqrt g) Pants, trivalent graphs and Whitehead moves Polar FN coordinates Deligne-Mumford compactification Divisors at infinity: [g/2]+1 #S(L) = O(L^6g-6) Simple curves via incidence with pants Train tracks Foliations and laminations 10. Geodesic currents Jeff Brock (Stanford University) Reference: F. Bonahon, The geometry of Teichmüller spacee via geodesic currents', Invent. math. 92, 139-162 (1988). 11. The Nielsen problem Binding curves define a compact set Convexity of length Twist formulas The Nielsen realization problem Reference: S. Kerckhoff, The Nielsen realization problem', Annals of Math. 117 (1983), 235-265 12. Teichmüller space is a domain of holomorphy Daniel Allcock Bers embedding via Schwarzian derivatives Kobayashi and Carathéodory metrics Completeness and pseudoconvexity 13. Extremal length, univalent maps and Teichmüller space Bers embedding and quasifuchsian groups L_g(Q(X,Y)) <= 2 L_g(Y) Univalent maps and the area theorem z + sum a_n/z^n univalent => sum n|a_n|^2 <= 1. |Sf| < 3/2 for univalent maps Cor: Teichmüller space is a bounded domain Ahlfors-Weill extension |Sf| < 1/2 => Im(f) is a quasidisk