Applied Mathematics

To get a feel for what I was thinking about in terms of applied mathematics when working at the Max Planck Institute for Cell Biology & Genetics, Dresden and Yale University, I will send you to my biology page. A few things that really piqued my interest were
  • scaling principles in biology: for example, Kleiber's Law states that the basal metabolic rate (i.e., at rest) of an organism goes like \(M^{3/4}\), where \(M\) is the mass of the organism; another great example is Murray's Law which tells us how the radii of daughter branches should scale as we proceed down the vascular system from aorta to capillaries; I am interested in how organelles scales with cellular volume and am particularly fascinated by how this works during development, where cells run a whole gamut of sizes over a very short period of time;
  • Pattern formation and utilizing the language of both deterministic and stochastic partial differential equations to describe how patterns form in biological systems: the most commonplace examples are Turing-type mechanisms and Gierer-Meinhardt systems; I am currently curious about how the structure of mitochondrial networks can form utilizing what we know about pattern formation;
  • The role of noise & fluctuations in biological systems: I am currently thinking about the role of stochastic resonance in biological oscillations.
Pure Mathematics

My mathematical research background is firmly grounded in the Pure Mathematics. With the gift of hindsight, I can see that my interests lay in the study of algebraic objects (groups, fields, rings, algebras) as manifested in topological and geometric structures. So it happened that my honours thesis was in Algebraic Topology and my PhD thesis in Algebraic Geometry.

Generally speaking, my honours work revolved around looking at invariant structures associated with topological objects such as spheres, tori and knots.

My PhD research requires a few more words: the general idea was to use classical techniques from 20th century Algebraic Geometry to help us to think about Noncommutative Algebraic Geometry. More specifically, under the tutelage of Daniel Chan at the University of New South Wales, I was interested in a class of noncommutative algebro-geometric objects known as orders. Prior to my graduate work, there was a substantial amount known about orders and yet there were relatively few concrete examples known. My project was to construct a zoo of such objects in order to 1) provide insight into the geometry of the related algebraic structures and 2) develop an intuition into the algebraic properties of the geometric structures at hand. To say anything more about this here would be almost futile. If you are interested in particulars, you should definitely check out my PhD thesisThe Explicit Construction of Orders on Surfaces, and/or the paper I published in Journal of Algebra that stems from the work, The construction of numerically Calabi–Yau orders on projective surfaces.