# Research

## Classical Analysis

### Friday, October 20, 2006

Title: Algebra of Interpolation
Speaker: Boris Shekhtman
Time: 3:00pm–4:00pm
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

Title: Sharkovsky's Theorem
Speaker: Eric Lundberg
Time: 3:00pm–4:00pm
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 $$n$$th 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

Title: The Schwarz reflection principle revisited, Part III
Speaker: Dmitry Khavinson
Time: 3:00pm–4:00pm
Place: ENG 004

### Friday, September 22, 2006

Title: The Schwarz reflection principle revisited, Part II
Speaker: Dmitry Khavinson
Time: 3:00pm–4:00pm
Place: ENG 004

### Friday, September 15, 2006

Title: The Schwarz reflection principle revisited, Part I
Speaker: Dmitry Khavinson
Time: 3:00pm–4:00pm
Place: ENG 004

#### 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!