Friday, October 20, 2006
| Speaker |
Boris Shekhtman |
| Topic |
Algebra of Interpolation |
| Time |
3:00-4:00 p.m. |
| Place |
ENG 004 |
Abstract
I will couch the standard notions of Lagrange Interpolation in one variable
in purely algebraic terms,(very elementary). Next, I will attempt to recast
these results in the multivariate setting.
I will analyze the difference and similarity of multivariate and univariate
interpolation through the interplay between Approximation Theory, Commutative
Algebra and Algebraic Geometry. Interestingly, many simple univariate results
become difficult (but still valid) bivariate results and fail in three or more
variables.
This is the first part of a four part project:
- Algebra of Interpolation.
- Geometry of Interpolation.
- Combinatorics of Interpolation.
- Analysis of Interpolation.
Friday, October 13, 2006
| Speaker |
Eric Lundberg |
| Topic |
Sharkovsky's Theorem |
| Time |
3:00-4:00 p.m. |
| Place |
ENG 004 |
Abstract
Choose a point on the real line and iterate a continuous function. If the point
is brought back to itself on the nth iteration it is called periodic of
order n. Sharkovsky’s Theorem gives a complete account of the
relationship of different orders. A special case of the Theorem says if you have
a periodic point of order 3, then you have points of every order. Although
considered a theorem of dynamical systems, it has gained general popularity and a
reputation for its impressive conclusion-to-hypothesis ratio. In this talk, we
will see a background and history. We will see a full statement of the theorem but
a proof of just the special case which will rely on two innocuous facts from
elementary analysis. Finally, we will see a statement without proof of a recent
addendum to Sharkovsky’s Theorem. The talk will be especially accessible to
undergraduates.
Friday, September 29, 2006
Friday, September 22, 2006
Friday, September 15, 2006
Abstract
The celebrated H. A. Schwarz reflection principle says that a harmonic function
(a potential) vanishing on a line takes opposite values at points that are
symmetric about that line. A similar point-to-point reflection holds when the
line is replaced by a circle and also, in higher dimensions, for planes and
spheres. Yet, virtually very little has been known in higher dimensions for
other surfaces. In the plane, Schwarz himself has already extended this reflection
principle to arbitrary analytic curves.
In this talk we shall discuss the complete solution of this problem in all
dimensions obtained in the last decade, connections with reflection principles for
other partial differential equations, Huygens’ principle, the celebrated
“antenna problem” in mathematical physics and more.
The talk WILL be made accessible to all graduate students and to the
undergraduate math, physics and engineering majors who have had multivariate
calculus. All are welcome!