Mathematics
Chair's introduction, Peter Constantin | Faculty Research Summaries
The research activities of the Department of Mathematics cover a wide variety of areas reflecting the broad expanse of modern mathematics. While mathematics has diversified, the interconnections between various directions have deepened and become much richer, so that one sees a remarkable unity and cohesion in mathematics even while there is constant evolution in new directions. The Mathematics Department is committed to this perspective both in terms of research and educating graduate students.
Mathematics has always reflected our perceptions of the universe and has often contributed to our understanding of it. The Department has members with joint appoint-ments in Physics, Statistics and Computer Science, and there is very significant and productive interdisciplinary activity in the relationship both to these areas and to mathematical biology and astrophysics as well.
The department has groups in algebra, analysis, geometry, topology, applied mathe-matics and logic, but the lines between these groups are not rigid, and often research in the department takes little notice of these boundaries. As discussed in more detail in the statements of individual faculty members below, some of the areas in which there is very significant research activity in the department include: algebraic geometry and commutative algebra, number theory, representation theory, finite groups, logic, Lie groups, algebraic topology, geometric topology, differential geometry, ergodic theory, complex geometry, harmonic analysis, partial differential equations, probability, numerical analysis, computational mathematics, and neural networks. A few specific areas of interest in the department currently include the geometric Langlands program relating representation theory and arithmetic, the theory of motives in algebraic geometry and number theory, p-adic Hodge theory, the large-scale geometry of discrete groups and spaces such as the Teichmuller space of a Riemann surface, inverse problems such as the determination of conductivity from surface measurements, blowup problems in partial differential equations, the effect of fluid flows on propagation and extinction of flames, the geometry of configuration spaces of points in complex manifold, propagation of waves in random media, the “fractal” nature of the moduli space of Riemannian metrics on a manifold, and the exploration of a “brave new algebra” in stable homotopy theory.
In addition to the regular members of the department, there are a large number of junior faculty, a very active visitor's program, and an excellent and diverse graduate student body.
Peter Constantin, Chair