Past and Future Talks

Spring 2024

Jan 31. Stretch maps, after Thurston
Rafael Saavedra, Harvard

Feb 14. The Weil-Petersson gradient flow of renormalized volume uniformizes relatively acylindrical manifolds
Martin Bridgeman, Boston College

Feb 28. Isoperiodic forms and invariant subvarieties of moduli space
Karl Winsor, SUNY Stony Brook

Mar 20. Clusters algebras and Teichmueller theory
Amanda Burcroff, Harvard

Apr 3. A new proof of Mordell's conjecture (for Riemann surfaces; after Xie - Yuan)
Jit Wu Yap, Harvard

Apr 17. Are fully intelligent robots coming soon?
David Mumford, Harvard

May 1. Counting minimal surfaces in hyperbolic 3-manifolds
Ruojing Jiang, MIT

Fall 2023

Oct 4 Automorphism of complex surfaces: What is the simplest interesting dynamical system?
Curtis McMullen, Harvard

Oct 11 Islands on K3 surfaces
Curtis McMullen, Harvard

Oct 18. Connectedness of moduli spaces: From plane curves to rational maps
Eriko Hironaka, NSF and FSU

Oct 25 Dynamics on blowups of P2
Curtis McMullen, Harvard

Nov 1. The 3D dimer model
Catherine Wolfram, MIT

Nov 15. Multipliers of rational maps
Laura DeMarco, Harvard

Nov 29. Monodromy over Riemann surfaces
Salim Tayou, Harvard


Feb 8. Galois orbits, equidistribution, and the moduli space of all triangles
Curtis McMullen, Harvard University

Feb 15. Entropy and geodesic currents on surfaces
Tina Torkaman, Harvard University

Feb 22. Earthquakes, triangulations and circle homeomorphisms
Catherine Wolfram, MIT

Sep. 7, 14: Exotic homeomorphisms and flows (after Sullivan and Freedman)
Curtis McMullen, Harvard University

Sep. 21, 28, Oct 5, 12, 19: Billiards and Teichmuller theory
Curtis McMullen, Harvard University

Oct 26: Unramified correspondences (Bogomolov, Tschinkel)
Aaron Landsman, Harvard University

Nov 2: A norm for the homology of 3-manifolds (Thurston)
Rafael Saavedra, Harvard University

Nov 9, 16: Bers, Hénon, Painlevé and Schrödinger (Cantat)
Max Weinreich, Harvard University. NOTES

Nov 30: Triangles groups and Hilbert modular varieties.
Curtis McMullen, Harvard University.

2019 - 2020

Sep. 11: Absolute periods of holomorphic 1-forms on Riemann surfaces
Karl Winsor, Harvard University

Sep. 18: On the Loewner energy of simple planar curves
Yilin Wang, MIT

Oct. 2: Elementary surfaces in the Apollonian manifold
Yongquan Zhang, Harvard University

Oct. 9: From veering triangulations to pseudo-Anosov flows (and back again)
Saul Schleimer, University of Warwick

Oct. 23: The Weil-Petersson gradient flow for renormalized volume
Martin Bridgeman, Boston College

Oct. 30: A walk from knots to exotic smooth Stein 4-manifolds
Eylem Yildiz, Harvard University

Nov. 6: Level sets of length function in moduli space
Tina Torkaman, Harvard University

Nov. 13: Experimental study of a meromorphic analogue of Teichmüller space
David Dumas, University of Illinois at Chicago

Nov. 20: Volumes and intersection theory on moduli spaces of differentials
Dawei Chen, Boston College

2017 - 2018

25 April: Andreev's theorem on hyperbolic polyhedra and Kleinian reflection groups
Yongquan Zhang, Harvard University

18 April: Galois conjugates of PCF generalized beta-transformations
Kathryn Lindsey, Boston College

11 April: Kaehler groups and surface bundles
Corey Bregman, Brandeis University

4 April: What to do when you cannot confine your planes? Teichmüller spaces of non-compact Riemann surfaces
Francis Bonahon, University of Southern California

28 March: Planes immersed in hyperbolic 3-manifolds
Curtis McMullen, Harvard University

21 March: Bounds on renormalized volume and the volume of the convex core
Ken Bromberg, University of Utah

14 March:
Enjoy the Spring Break!!!

7 March: The geometry of Hilbert's 13th problem
Jesse Wolfson, University of California, Irvine
Benson Farb, University of Chicago

*7 March: The theory of resolvent degree, after Hamilton, Klein, Hilbert, and Brauer
(This talk will be closely related to the one given in the informal seminar)
Number Thoery Seminar 3-4 Room 507
Jesse Wolfson, University of California, Irvine

28 Feb: 5 points on the sphere
Richard Schwartz, Brown University

21 Feb: Convex cores of thick hyperbolic 3-manifolds with bounded rank
Ian Biringer, Boston College

14 Feb: Hypergeometric equations and Lyapunov exponents (via Hodge theory)
Simion Filip, Harvard University

7 Feb: Penrose Tilings of Riemann Surfaces
Phil Engel, Harvard University

31 Jan: The behavior of planes in confinement
Curtis McMullen, Harvard University

29 Nov: Finitely many algebraically primitive Teichmüller curves in genus g at least 3
Jane Wang, MIT

8 Nov: Square-tilings of modular curves X(d)
Eduard Duryev, Harvard University

1 Nov: Cohomology of arithmetic groups and characteristic classes of manifold bundles
Bena Tshishiku, Harvard University

25 Oct: The metastable homology of mapping class groups
Sander Kupers, Harvard University

18 Oct: Algebraic dynamics from topological and holomorphic dynamics
Rohini Ramadas, Harvard University

4 Oct: Divergent trajectories in arithmetic homogeneous spaces of rational rank two
Nattalie Tamam, Tel-Aviv University

27 Sep: Trees in degenerating families of rational maps and isometric G-actions
Yusheng Luo, Harvard University

20 Sep: Plane curves, (higher) spin structures, and mapping class groups
Nick Salter, Harvard University

13 Sep: Tilings and Hurwitz theory
Phil Engel, Harvard University

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

Fall 2014


Normalized entropy versus volume (Kojima-McShane)
The renormalized volume and the volume of the convex core of quasifuchsian manifolds (Schlenker)
Geometric inflexibility and 3-manifolds that fiber over the circle (Brock-Bromberg)
Zippers and univalent functions (Thurston)
The Schlaefli formula in Einstein manifolds with boundary (Rivin and Schlenker)
The hyperbolic Gauss map and quasiconformal reflections (Epstein)


Lecture 1 (McMullen)
Lecture 2 (Bridgeman)
Lecture 3 (Yarmola)
Lecture 4 (Bridgeman)
Lecture 5 (Brock)
Lecture 6 (McMullen)

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


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
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

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