**Panel Discussion on the**
**FUTURE OF THE FOUNDATIONS OF
COMPUTATIONAL MATHEMATICS**
**August 13, 2002**

Moderator: **Endre
Suli**

Panelists: **Steve
Smale, Lenore Blum, Peter
Olver, Ron DeVore**

**Steve Smale**

Views on the mathematics of the future.

Smale talked about a debate that he had with Michael Atiyah concerning the influence upon mathematics of fields outside of mathematics. Atiyah argued that the main influence has been and will continue tio be from physics. Smale argued for the increasing influence of the engineering, biological and social sciences on mathematics. Atiyah responded that these were ``soft" sciences, especially the social sciences. Smale rejoined that the soft sciences can be made ``tight."

In this context Steve discussed several problems from his famous list of 18 ( to which 3 minor problems were added later). For each problem he emphasized two fundamental issues:

Smale problems (Postscript file) The problems Steve emphasised during the panel discussion were (in the order that he discussed them):Approximation

Probability

1) Problem 18: The Limits of Intelligence

2) Problem 17: Solving polynomial systems in polynomial time (progress reported at this meeting)

3) Problem 3: P not equal to NP?

4) Problem 19: Mean Value Conjecture (probably solved)

5) Problem 14: Chaos theory and the Lorentz attractor (solved by Warwick Tucker in a paper in the FoCM Journal)

6) Problem 9: Linear Programming problem. Determine if Ax>= b is feasible over R in polynomial time.

7) Problem 7: Equidistant points on the sphere

**Lenore Blume**

Lenore Blum talk (PDF file)

**Ron DeVore**

Devore talk (Power Point)

**Peter Olver**

Peter Olver contrasted two modes of research that contribute to the mission of the society:

1) Problem driven research (as exemplified by Smale)He concentrated on the second mode where he pointed out three emerging themes:2) Research driven by applications, but not a specific problem (as exemplified by Lie and Noether)

A) The increasing interplay between the continuous and the discrete.B) The incorporation of structure into algorithms (geometry, symmetry, conservation laws, etc.)

C) The reunification between algebra (computational algebra) and analysis (differential equations). [Example: Grobner bases <--> overdetermined systems of PDEs.]