Research Topics
Both research topics, while possessing a great deal in common in terms of
fundamental mathematics, seem to have completely different starting points.
For example, in the second set of problems, in contrast with the first, we will
begin by discussing biological modelling. Even the language will seem different
at first, but students will eventually see the common mathematical themes, and
consolidate deeper concepts in mathematics -- especially in analyis and linear
algebra -- which will be important for them later on and in graduate school.
(E.g., sequences, metrics, convergence, completeness, compactness, Banach
spaces, fixed points, asymptotic stability, difference equations, matrices
operators, diagonalization, spectral theory, convex combinations etc.)
This is a great way to contrast the unity of mathematics (with the many
common mathematical themes) with the breadth, diversity and applications of
mathematics.
Topic 1: Analysis, linear algebra, functional analysis
Let (s_1, s_2, ..., s_n) and (t_1, t_2, ..., t_n) be two sequences of
real numbers. (s_1, ..., s_n) is said to be majorized by (t_1, ..., t_n)
if
s_1 + ... + s_k <= t_1 + ... + t_k for all k, and
s_1 + ... + s_n = t_1 + ... + t_n.
Let A and B be two self-adjoint n by n matrices.
A is said to be majorized by B if the eigenvalues of A, arranged in decreasing
order and with repetitions allowed, is majorized by the corresponding
(similarly arranged) eigenvalues of A. Majorization is an important topic
studied throughout analysis and its applications.
A fundamental result in matrix analysis says that if A, B are two self-adjoint
n by n matrices, then A is majorized by B if and only if there exists a
unitary matrix U such that the diagonal of U B U^* is the list of eigenvalues
of A if and only if A is in the convex hull of the unitary orbit of B.
These ``Schur--Horn" type results, and their infinite dimensional
generalizations, have a long history and have still recently been of great
interest in operator theory, operator algebras and their applications.
Topic 2: Mathematical biology, modelling, dynamics and linear algebra
Amphibian decline has become a major focus in conservation biology with up to
one-third of amphibian species threatened with extinction. Some causes of this
decline are known to be habitat loss, exploitation, invasive species, climate
change, and pollutants. However, the high population decrease of amphibians
can not only be attributed to the above causes. Because of the decline, the
scientific community has recognized the need and importance to study amphibian
populations.
An infectious disease known as Chytridiomycosis, has emerged and researchers believe a major cause of the decline of amphibians can be attributed to this disease. For instance, Chytridiomycosis was the cause of death for 55.2% of free-living Australian frogs and for 58.4% of captive frogs. Chytridiomycosis is a disease that occurs when an amphibian is infected with Batrachochytrium dendrobatidis (Bd), a chytrid fungus that reproduces by waterborne zoospores. Bd is virulent to adults, does not kill tadpoles, and prefers cooler temperatures. Infection with Bd occurs inside the cells of the outer layer of the skin that contain large amounts of keratin. Keratin is the material that makes the outside of the skin tough and resistant to injury. With chytridiomycosis, the skin becomes very thick due to microscopic changes in the skin called hyperplasia and hyperkeratosis. The broad host range suggests Bd may infect all frog species that occur within suitable environment, and the low host specificity could be a factor facilitating the emergence of chytridiomycosis.