|
|
431 Malott Hall |
|
Research:
Differential equations are the main way in which mathematical models of real systems
are constructed, and understanding their behavior is the main contribution a mathematician can make to applications. I am
interested in understanding the behavior of differential equations and their close relatives: iterative systems. In
particular, I try to see how such systems behave in the complex domain, largely because complex analysis brings new and
powerful techniques to bear on the problems.
Books:
Selected Papers:
Multicorns are not path connected (with Dierk Schleicher).
An analytic construction of the Deligne-Mumford
compactification of the moduli space of curves (with Sarah Koch).
A first look at differential algebra (with Benjamin Lundell).
Newton's method applied to two quadratic equations in C^2 viewed as a global dynamical system (with Peter
Papadopol).
Exponential Thurston maps and limits of quadratic differentials
(with Dierk Schleicher and Mitsuhiro Shishikura).
Equidistribution of horocyclic flows on complete hyperbolic surfaces of finite
area (with Robyn Miller).
The KAM theorem
Andreev's theorem on hyperbolic polyhedra (with Roland Roeder and William Dunbar).
Parametrizing unstable and very unstable manifolds.
A proof of Kolmogorov's theorem
(with Yulij Ilyashenko).
A geometric view of rational Landen transformations
(with Victor Moll).
Farey curves
(with Xavier Buff and Christian Henriksen).
How to find all roots of complex polynomials by Newton's method
(with Dierk Schleicher and Scott Sutherland).
Linked solenoid mappings and the nontransversality locus invariant
(with Ralph Oberste-Vorth).
The convergence of an Euler approximation of an initial value problem is
not always obvious
(with Beverly West and Samer Habre).
A compactification of Hénon mappings in $\C^2$ as dynamical systems
(with Peter Papadopol and Vladimir Veselov).
Preface to the book The Mandelbrot Set, theme and variations
.
A Fatou-Bieberbach domain avoiding a neighborhood of a variety of
codimension 2
(with Greg Buzzard).
Hairs for the complex exponential family
(with Clara Bodelón, Bob Devaney, Michael Hayes, and Gareth Roberts).
The forced damped pendulum: chaos, complication and control.
Groups of automorphisms of tree and their limit sets
(with Sa'ar Hersonsky).
Hénon mappings in the complex domain II: projective and inductive limits of
polynomials (with Ralph
Oberste-Vorth).
The spider algorithm (with Dierk Schleicher).
Hénon mappings in the complex domain I: the global topology of dynamical
space (with Ralph
Oberste-Vorth).
Superattractive Fixed Points in $\C^n$ (with Peter Papadopol)
A proof of Thurston's topological
characterization of rational functions (with Adrien Douady).
Local Connectivity of Julia sets and bifurcation loci: three
theorems of J-C Yoccoz.
The iteration of cubic polynomials part II: patterns and
parapatterns (with Bodil
Branner).
The classification of critically preperiodic polynomials as dynamical
systems (with Benjamin Bielefeld and Yuval Fisher).
The classification of topologically expansive Lorenz maps (with Colin Sparrow).
A proof of the uniformization theorem for arbitrary plane
domains (with Benjamin Wittner and Yuval Fisher).
The iteration of cubic polynomials part I: the global topology of
parameter space (with Bodil
Branner).
The Hénon mapping in the complex domain.
On the dynamics of polynomial-like mappings (with Adrien Douady).
The Orsay Notes English, French (with Adrien Douady).
Itération des polynômes quadratiques complexes (with Adrien Douady).
The monodromy of projective structures.
On the convex hull genus of space curves.
The space of closed subgroups of $\R^2$ (with Ibrahim Pourezza).
Quadratic differentials and foliations (with Howie Masur).
On the existence and uniqueness of Strebel differentials
(with Howie Masur).
On the density of Strebel differentials (with Adrien Douady).
Transversalité.
Le theorem de M. Artin sur les solutions d'equations analytiques.
On the cohomology of Nash sheaves.
A note on nonrigid Nash structures.
Sur les sections analytiques de la courbe universelle de Teichmüller
(my thesis).
Selected Talks:
Marden Lecture, Forced Damped Pendulum, March 2008
A talk for the Undergraduate Math Club at Cornell, explaining the
Mandelbrot
Set, April 2008
In memory of Adrien Douady, Paris, May 2008
The Bott-Duffin synthesis: in memory of Raoul Bott, Montreal, June 2008
Monodromy and Henon maps, Oberwolfach, October 2008
Transversality according to Adam Epstein, Oberwolfach, October 2008
Monomial mappings and Hilbert modular varieties, October 2008
Some new approaches to Henon mappings, Stony Brook, June 2009
Parabolic Blow ups, Banff, February 21, 2011
The Price of Anarchy, Public Lecture, Cornell University, April 2011
Links:
Cornell math department
Centre de Mathématiques et Informatique
Oliver Club
Dynamics seminar
Dynamics page at Cornell
Matrix Editions
The PhD thesis of C.
Lipa.
The availability of powerful computers and computer graphics has changed
the way this sort of research is done. In the course of investigations of even such simple iterative systems as quadratic
polynomials, amazing pictures show up, rather as
artifacts to be investigated than as man-made objects. I anticipate that this sort of experimental mathematics will become a
major trend.
Most of my research over the last five years has been devoted to dynamics in several complex variables. I have co-authored
four foundational papers in the field. I am also writing three books on a very advanced level, one giving a treatment of
Teichmüller space and its applications in theorems of Bill Thurston, the second on dynamics of one complex variable, and
the third
on differential equations.
Vector Calculus, Linear Algebra, and Differential Forms, A Unified
Approach (with Barbara Burke Hubbard).
Teichmüller Theory and Applications to Geometry, Topology, and
Dynamics, Volume I: Teichmüller theory
Calcul Scientifique de la théorie à la pratique,
Volume I: Equations algébriques, traitement du signal et géométrie effective
(with Florence Hubert).
Calcul Scientifique de la théorie à la pratique,
Volume II: Equations différentielles et équations aux dérivées partielles
(with Florence Hubert).
Differential
Equations: A dynamical systems approach (with Beverly West).
