Ma142 Winter 2024: Ordinary and Partial Differential Equations
Winter 2024: TR 1:00-2:30 pm in 187 Linde Hall
Instructor:
Matilde Marcolli
Brief Course Description
This course will cover aspects of ordinary and partial
differential equations, including:
linear and nonlinear ordinary differential equations: existence and uniqueness, equilibria, bifurcation, orbits, limit sets, etc (additional topics may include Sturm-Liouville theory, Riemann-Hilbert problem, Painleve transcendents)
partial differential equations: heat equation, wave equation, Laplace equation, Schrodinger equation; Fourier transform, variational methods; distributions, weak solutions, Gelfand triples; geometric aspects (additional topics may include D-modules, geometric methods for soliton equations)
The class is graded P/F, the grade is assigned on the basis of attendance/participation and completion of an assigned reading/presentation or project individually assigned by the instructor.
Notes of Classes
scanned notes of classes will be posted here.
- pdf
slides with examples
- pdf
introduction to ODE/PDE, equilibrium points of linear ODEs, planar case,
Kronecker foliation on the torus
- pdf
examples of nonlinear ODEs: bifurcations (saddle-node, Hopf), nullclines,
nonlinear pendulum, gradient flows, Hamiltonian systems, closed orbits
and limit sets, Poincare return map, Poincare-Bendixon
- pdf
supplementary notes on nonlinear ODEs, further examples, Lyapunov functions
- pdf existence and uniqueness theorem as fixed point theorem
- pdf Sturm-Liouville
- pdf Sturm-Liouville and PDEs
- pdf
Painleve equations and nonlinear PDEs
- pdf Painleve equations
- pdf The Riemann-Hilbert problem
- pdf linear PDEs, wave & heat equations
- pdf Gelfand Triples and Schrodinger equation
- pdf D-modules
- pdf Solitons Part 1
- pdf Solitons Part 2
Summary of Lectures
- Thursday January 4: introduction to ODE/PDE, equilibrium points of linear ODEs,
classification of equilibria for planar systems
- Tuesday January 9: interaction of geometry and differential equations:
the case of the Kronecker foliation on the torus
- Thursday January 11: behavior of nonlinear ODEs: saddle-node and
Hopf bifurcations, nullclines
- Tuesday January 16: behavior of nonlinear equations: Poincare Bendixson theorem
- Thursday January 18: existence and uniqueness as a fixed point theorem
- Tuesday January 23: Sturm-Liouville problems
- Thursday January 25: Sturm-Liouville problems and spectral theory
- Tuesday January 30: Sturm-Liouville problems and PDEs
- Thursday February 1: Painleve equations and nonlinear PDEs
- Tuesday February 6: Painleve equations
- Thursday February 8: Riemann-Hilbert problem Fuchsian and regular sigular, Riemann-Hilbert boundary value problem
- Tuesday February 13: Riemann-Hilbert problem, isomonodromic deformations and Painleve equations, from Riemann-Hilbert problem to Riemann-Hilbert correspondence
- Thursday February 15: linear PDEs, differential operators, principal symbol, ellipticity, Minkowski geometry and hyperbolicity,
- Tuesday February 20: linear PDEs, wave equation, heat equation; distributions
- Thursday February 22: linear PDEs, heat equation, Laplace equation; Gelfand Triples and Schroedinger equation
- Tuesday February 27: D-modules and linear PDEs, maximally overdetermined systems, holonomic D-modules
- Thursday February 29: nonlinear PDEs, solitons, Burger eq, KdV eq, Weierstrass p-function, n-soliton solutions, Lax form of KdV
- Tuesday March 5: Lax equation and KdV hierarchy, other Lax pairs (Boussinesq, sine-Gordon), Lax equation and Jacobian of curves, KP equation and tau function
- Thursday March 7: KP equation and Grassmannian, pseudodifferential operators, KP hierarchy, soliton solutions from algebraic geometry
Some book references
There is no official textbook for this class, but the following books
will be useful references (available in the Caltech library) [Note: more
references will be added]
- Ferdinand Verhulst, "Nonlinear differential equations and dynamical systems"
- Vladimir Arnold, "Ordinary differential equations"
- Vladimir Arnold, "Lectures on partial differential equations"
- S.C. Coutinho, "A Primer of algebraic D-modules"
Other Reading Material
Suggested reading material including both material covered in the
lectures and possible suggestions for student presentations (more will
be added as the class progresses)
General reading material
- pdf Partial differential equations
- pdf Introduction
to partial differential equations
- pdf Introduction
to partial differential equations
- pdf Stability and
Hopf Bifurcation Analysis of the Delay Logistic Equation
- pdf
Foliations on complex manifolds
- pdf
Quasi periodic motions from Hipparcus to Kolmogorov
- pdf Poincare Bendixson theorem
- pdf Banach fixed point theorem and applications
- pdf existence and uniqueness theorem
- pdf On the origins of the Riemann-Hilbert problems in mathematics
- pdf The Riemann-Hilbert Problem
- pdf Some explicit solutions to the Riemann-Hilbert problem
- pdf Differential forms and their application to Maxwell's equations
- pdf One cannot hear the shape of a drum
- pdf Distribution theory and Fourier analysis
- pdf Geometrical structure of Laplacian eigenfunctions
- pdf Algebraic aspects of nonlinear differential equations
- pdf Introduction to the theory of D-modules
- pdf Painleve equations - nonlinear special functions
- pdf
Painleve V and time-dependent Jacobi polynomials
- pdf Kronecker foliation, D1 branes and Morita equivalence of noncommutative two-tori
- pdf Combinatorics of KP solitons from the real Grassmannian
- pdf Analysis of Riemann-Hilbert problems
- pdf Maxwell equations, Hodge Theory and Gravitation
- pdf Spectrum of the Laplace operator on closed surfaces
- pdf Selberg's trace formula: an introduction
- pdf Introduction to Otto Calculus
- pdf Hyperbolic equations in the 20th century
- pdf Lacunas for hyperbolic differential operators with constant coefficients
- pdf Introduction to Fixed Point Methods
- pdf Non-autonomous equations
- pdf Banach-Gelfand triples and their applications
- pdf D-module techniques for solving differential equations in the context of Feynman integrals
- pdf overdetermined PDEs
- pdf On the Numerical Analysis of Overdetermined Linear Partial Differential Systems
Schedule of Presentations:
- Tuesday March 12: Jihoon, Propagation of ultrasonic Love waves in nonhomogeneous elastic functionally graded materials
- Tuesday March 12: Reza, Introduction to Fixed Point Methods
- Tuesday March 12: Jonghyeon, PDEs in mathematical biology
- Thursday March 14: Jennifer, Otto calculus
- Thursday March 14: Chelsea, Bearing pressures and cracks
- Thursday March 14: Erik, Overdetermined PDEs