Instructor: Matilde Marcolli

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.

- 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

- 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

- 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"

- 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

- 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