Mathematics has, since ancient times, been associated with many disparate areas of human inquiry, ranging from art and philosophy to science and engineering. This fruitful interchange has led to important mathematical discoveries, and has also stimulated the development of whole new fields of study. This tradition continues today in the Mathematics Department, where a wide variety of interdisciplinary research is conducted, both singly and in collaboration with researchers in other departments and universities.
Chris Beasley studies formal aspects of quantum field theory and string theory. Maxim Braverman has conducted research in superconductivity and hydrodynamics. Chris King has worked on problems in lattice gauge field theory and exactly solvable models in statistical mechanics. Mikhail Shubin has been working on several problems related to miscellaneous questions in quantum physics, in particular relations between completeness properties of classical and quantum problems, spectral behavior of Schrodinger operators, applications of non-commutative geometry to the quantum Hall effect. He also studied singular perturbations of oscillating systems by non-standard analysis methods. Valerio Toledano Laredo’s research draws inspiration from, and has consequences for Conformal Field Theory and String Theory. He has worked on the representation theory of loop groups and the rigorous construction of the corresponding interacting, 2-dimensional quantum field theories. He has also defined, in collaboration with B. Feigin and E. Frenkel a new class of quantum integrable systems which generalise the ones defined by Gaudin in the 1970’s. In recent work with Tom Bridgeland, he has shown how the change of vacuum, or wall-crossing in String Theory is described by Stokes phenomena for differential equations on the complex plane with irregular singularities. Peter Topalov has been working on questions arising in hydrodynamics and classical mechanics. More specifically, his work is related to fluid dynamics described by non-linear evolution equations such as the Korteweg-de Vries and Camassa-Holm equation. His work on Riemannian geometry has been applied to classical mechanics (the motion of rigid bodies) and, more recently, to astrophysics (the problem of studying light rays moving on black hole backgrounds in the presence of a cosmological constant).
Biostatistics and Bioinformatics
Adam Ding has worked on several problems in biostatistics, including the statistical analysis of data from HIV clinical trials, analysis of multiple correlated survival times and current status data in epidemiology, and the analysis of genetic micro-array data. Michael Maliuotov has developed probabilistic models of protein dynamics, and used them to analyse molecular diffusion in the human brain.
Quantum Information Theory
Chris King’s work in quantum information theory is directed toward quantum channel capacity problems and the role of entanglement in noisy quantum systems.
In other areas of interdisciplinary research, Jayant Shah has been working on problems related to Computer Vision. His current work involves applications of Differential Geometry to shape representation and a study of variations in brain structures either due to disease or due to population differences.
Chris King has worked on data compression for IP routing tables, stability and fairness of high-speed TCP variants, and analysis of jamming in 802.11 wireless protocols.