Remarks: This is a first semester of a two semester course.

One significant difference between the Texas (the method developed by the UT Professor R. L. Moore) style, and what we refer to as the Red Burton style, is that rather than to be led to the rediscovery of known results, the participants will work exclusively on conjectures of selected versions of Graffiti, without getting any hints whether these conjectures are true or false. Another difference is that unlike in traditional Texas style courses the participants will be allowed, to read textbooks and even solutions of previous conjectures of Graffiti, because the problems they will encounter are unlikely to be found in textbooks anyway. This will create a more realistic setting for acquisition of research experience. Active participants will have an opportunity to discover new original results.

That does not mean that the course will be more difficult than other math classes. The only prerequisites are graduate standing in the College of Natural Sciences and Mathematics or consent of the instructor. One advantage of running Graffiti individually, is that the difficulty of conjectures can be tailored to a preferred level of users, presumably making the class actually easier. The course will be conducted by email and a discussion list.

• A review of linear algebra.
• Autonomous first order linear systems, steady states and stability.
• An introduction to function spaces and the contraction mapping theorem.
• Well posedness for general first order nonlinear systems.
• Continuous dependence on initial data and parameters.
• Stability theory, and linearized stability.
• The implicit function theorem.
• The stable manifold theorem.
• Elementary bifurcation theory.

The final grade is aproximately based on a total score of 400 points consisting of homework (100 points), a semester test (100 points), and a final exam (200 points). The instructor may change this at his discretion.

• 1. Functional Spaces with a particular emphasis on Hilbert spaces and the projector theorem. Weak convergence.
• 2. Minimization of functional in Hilbert spaces.
• 3. Iterative solution of linear and nonlinear problems in Hilbert spaces.
• 4. The Lax-Milgram theorem and Galerkin methods in Hilbert spaces.
• 5. Some notions on the Theory of Distributions.
• 6. Application to the solution of variational problems from Mechanics and Physics.
• 7. Time dependent problems and operator-splitting.
• 8. Constructive methods for linear and nonlinear eigenvalue problems.
• 9. Boundary value problems and their approximation.

Reference texts:

• The Symmetry Perspective (Golubitsky and Stewart, Birkhauser)
• Singularities and Groups in Bifurcation Theory Vol. II (Golubitsky, Stewart, and Schaeffer, Springer)

(Texbooks that will be used: W. Strauss's: "Partial Differential Equations" , R. Glowinski: "Numerical Methods for Fluids (Part 3)", Chorin and Marsden: "Mathematical Introduction to Fluid Mechanics", Y.C. Fung: "Circulation", Y.C. Fung: "Biomechanics: Mechanical properties of living tissues." R. LeVeques: "Conservation Laws", Research Papers)

• Review of basic linear PDEs.
• Analysis of quasilinear PDEs with concentration to hyperbolic conservation laws.
• Introduction to fundamentals of fluid mechanics (basic equations of motion: continuity, momentum, energy, vorticity).
• Incompressible/compressible flow examples (derivation of the incompressible, viscous Navier-Stokes equations).
• A brief introduction to Sobolev spaces. Fluid-structure interaction arising in blood flow modeling (effective models).
• Energy estimates.
• Special topics related to the study of blood flow through compliant blood vessels.

Also notes will be handed out in class.

We will consider local and global Newton and Gauss-Newton methods and variants thereof. Emphasis will be put on a thorough affine invariant convergence analysis as well as on appropriate damping strategies and monotonicity tests for convergence monitoring. Compared to traditional approaches, the distinguishing affine invariance concept leads to shorter and more transparent proofs and permits the construction of adaptive algorithms. We will also address parameter dependent nonlinear problems and focus on path-following continuation methods for their numerical solution.