About this course
MATH4401 - Advanced Analysis
COURSE CONTENT:
After a review of some background material in analysis, this course will cover three main areas: degree theory in finite dimensions (Brouwer degree), degree theory in infinite dimensions (Schauder degree) and the theory of nonlinear eigenvalues (bifurcation theory).
Degree theory is a argument to count topologically the number of solutions of an operator equation. From this, the famous Brouwer fixed point theorem can be derived. There are other important topological results also, like the hairy ball problem which says that a 'hairy' sphere in odd dimensions (for example a tennis ball) cannot be completely smoothed all over - there is always a point wheree the hair sticks up.
In infinite dimensions, we work in Banach spaces. The Schauder degree can be constructed with compact mappings because these behave like finite dimensional maps. An analogy to the Brouwer fixed point theorem is attained.
Bifurcation theory is able to describe the buckling of a longitudinally loaded beam. At a critical loading, the beam fails, resulting in catastrophic failure. This critical loading corresponds with an eigenvalue of a nonlinear operator. The bifurcation result by Krasnoselskii is proven which confirms that nonlinear operator equations can be linearised and a bifurcation branch is attained when the linear system has eigenvalues of odd multiplicity.
We lead up to the global bifurcation theorem of Rabinowitz. This theorem says that the bifurcation of a compact operator is a global phenomenon. A bifurcation branch must either become unbounded or reconnect with another bifurcation point.
WHERE IS IT USED:
Degree theory and bifurcation theory is a field of pure and applied mathematics studied by researchers in nonlinear analysis. Bifurcation theory is an important tool used by engineers and applied mathematicians in fields as diverse as combustion theory, dynamical systems and chaos.
Researchers are currently working on bifurcation theories for noncompact operators. Research into fixed point theory is a popular topic, especially with multivalued maps.
The material in this course is complemented by the topics in MATH4404 (Functional Analysis) which continues covers other important topics in analysis.
WHO IS INTERESTED:
Students with interests in the application of pure mathematics and the theory behind topics in applied mathematics will be interested. A research career in analysis requires good understanding of the topics in this courses such as operator equations, the topological degree, the Ascoli-Arzela theorem, the contraction mapping principle, and the Schauder fixed point theorem. The topics in bifurcation will be useful for applied mathematicians who intend pursuing theory behind physical phenomena such as combustion, phase transitions or stability analysis. The subject is suitable for both third and fourth year students.
WHAT DO I NEED:
This course is based on some founding analysis. Students should know about metric spaces, multivariate calculus and the inverse function theorem. Some knowledge about Hilbert spaces preferred and the Riesz representation theorem will be used extensively. In particular, Advanced Analysis assumes that the student knows the material in MATH2400.
WHEN IS IT AVAILABLE:
Semester 1 odd numbered years.
