A PDF copy of the syllabus can be found here: APMA_2200.pdf

April 19: We used the homotopy invariance of the Leray-Schauder degree to prove the Rabinowitz global bifurcation theorem. Paul Rabinowitz's original paper can be found here:

P. Rabinowitz, Some global results for nonlinear eigenvalue problems Journal of Functional Analysis, 7 (1971), 487-513.

April 12: We proved the Krasnoselski bifurcation theorem. A brief history of the early impact of degree theoretic methods to bifurcation theory can be found in section 2 of:

J.E. Marsden, Qualitative methods in bifurcation theory Bull. Amer. Math. Soc. 84 (1978), no. 6, 1125-1148.

April 10: We used the Leray-Schauder degree to obtain the existence of weak solutions to semi-linear elliptic equations when the nonlinearity exhibits sublinear growth. Along the way, we discussed the properties of the Nemitski operator.

The second homework assignment has been posted.

April 3 - April 5: We reviewed the basics of Sobolev spaces and weak solutions in preparation for the lectures on degree-theoretic methods and PDEs. We also proved that the inverse of the Laplacian over $H_{0}^{1} (Ω)$ is a compact operator.

The fourth presentation assignment has been posted.

March 22: We revisited the classical Riesz-Schauder theory and proved that any compact perturbation of the identity is a Fredholm operator of index zero.

March 15: We discussed the subtleties of the approximation problem in Banach spaces, and defined the Leray-Schauder degree for compact perturbations of the identity operator.

March 8: We focused on the homotopy invariance of the Brouwer degree, and we proved the Brouwer fixed point theorem.

March 6: We defined the Brouwer degree for continuous maps in finite-dimensional spaces, and discussed its basic properties (see section II.1 in Kielhöfer's book).

The third presentation assignment has been posted.

Feb 27 - Mar 1: We discussed the spectral theory of closed linear operators, and derived the principle of exchange of stability for fold, transcritical, and pitchfork bifurcations in Banach spaces. See also:

M. Crandall and P. Rabinowitz, Bifurcation, perturbation of simple eigenvalues, and linearized stability Archive for Rational Mechanics and Analysis, 56, 161-180, 1973.

February 22: We derived explicit expressions for the tangent vector of the bifurcation curve in the Crandall-Rabinowitz theorem. In turn, this computation allowed us to classify the bifurcation as transcritical, supercritical, or subcritical.

The descriptions of the first and second presentation assignments have been posted.

February 15: We proved the Crandall−Rabinowitz theorem on bifurcations from simple eigenvalues. The original paper by Crandall and Rabinowitz can be found here:

Michael Crandall and Paul Rabinowitz, Bifurcation from simple eigenvalues Journal of Functional Analysis, 8(2), 321-340, 1971.

The first homework assignment has been posted.

February 13: We used the Lyapunov-Schmidt reduction to derive conditions on the bifurcation equation that ensure the existence of a fold (saddle-node) bifurcation. See section I.4 in Kielhöfer's book.

February 8: We proved the implicit function theorem in Banach spaces and introduced the Lyapunov-Schmidt reduction method.

February 6: We discussed the notions of Fréchet and Gâteaux differentiability, along with several elements of differential calculus in Banach spaces.

February 1: We continued our discussion of bounded linear operators, and we paid special attention to the properties of Fredholm operators and the Fredholm alternative theorem.

January 25: We reviewed some basic material about bounded linear operators.