Recommended program: Differential geometry and PDE
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.
|