Tom Duchamp, Jerry Folland, Robin Graham, Jack Lee, James Morrow, Dan Pollack, Hart Smith, John Sylvester, Tatiana Toro, Gunther Uhlmann, Yu Yuan
| First Year | |||
|---|---|---|---|
| 544 | 545 | 546 | (Topology and Geometry of Manifolds) |
| 554 | 555 | 556 | (Linear Analysis) |
| and one of the following: | |||
| 504 | 505 | 506 | (Modern Algebra)* |
| 524 | 525 | 526 | (Real Analysis)* |
| 534 | 535 | 536 | (Complex Analysis)* |
| Second Year | |||
| Three of the following (to be chosen in consultation with the advisor, depending on student's interest): | |||
| 504 | 505 | 506 | (Modern Algebra) |
| 524 | 525 | 526 | (Real Analysis) |
| 534 | 535 | 536 | (Complex Analysis) |
| 527 | 528 | 529 | (Functional Analysis) |
| 537 | 538 | 539 | (Several Complex Variables) |
| 547 | 548 | 549 | (Geometric Structures) |
| 557 | 558 | 559 | (Introduction to Partial Differential Equations) |
| 564 | 565 | 566 | (Algebraic Topology) |
| 577 | 578 | 579 | (Lie Groups and Lie Algebras) |
| 594 | 595 | 596 | (Special Topics in Numerical Analysis) |
*Modern Algebra is most appropriate for students interested in more topological, less analytic approaches to differential geometry. Real Analysis is most appropriate for students with a particular interest in Fourier analysis, geometric measure theory, or minimal surface theory. Complex Analysis is most appropriate for students interested in PDE theory, complex manifolds, or the geometric theory of several complex variables.
The research interests of the large group of faculty members who have submitted this recommended program are quite varied. To more clearly identify what they do, short descriptions of research interests for a few of them follow.
- Jack Lee
- Differential geometry of Riemannian
manifolds, conformal manifolds, complex manifolds, and CR manifolds
(which are the abstract models of real submanifolds of complex
n-space, and arise naturally as boundaries of domains in several
complex variables). I mostly use tools from the theory of elliptic
partial differential equations to study metrics with prescribed
curvature properties, and to approach questions like existence,
uniqueness, boundary regularity, and other qualitative geometric
properties of such metrics.
- Gunther Uhlmann
- Most of my recent work has centered in the mathematical theory of inverse problems.Typical questions in which I have been interested are the following: determining the density of a medium by X-rays, CAT scans or similar measurements. The problem of determining the electrical conductivity of a body by making voltage and current measurements at the boundary. The problem of determining the sound speed and density of a medium by scattering information.Most of these problems can be formulated in mathematical terms as determining coefficients of a partial differential equation by knowing some property of its solutions. Therefore I mostly use tools from the theory of partial differential equations.
