Past and Future Talks

Spring 2011

Fall 2010

Spring 2010

Fall 2009

Spring 2009

Fall 2008

Spring 2007

19--21 June 2006

May 2005

Fall 2003

Fall 2002

Spring 2002

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