# Program

- Complexity of numerican computations
- homotopy methods
- optimization and interior point methods
- differential equations.

_{f}with the following property. If z is a point close enough to a root of f then the sequence defined by z

_{0}= z , z

_{ i+1}= N

_{f}(z

_{i}) , converges to . Moreover, in general, this convergence is very fast: the distance between and z

_{I}is at most that between and z

_{0}divided by 2

^{2i-1}. In more recent years mathematicians have focused in questions such as what exactly "close enough'' or "in general'' mean in the description above. Can one decide whether a point z is close enough? Will the sequence starting with a specific z

_{0}converge to a root of f ?

Image courtesy of Scott Sutherland, SUNY Stony Brook

- Complexity: What is the 'cost' inherent in a computational problem that no algorithm can circumvent?
- Optimization: How to find the best value of a function (or a functional) subject to constraints?
- Homotopy: How does the knowledge of the solution of a `nearby' problem assist us to compute the problem in hand?
- Geometric integration: How to compute approximate solutions that share qualitative properties with the true solution of the problem?

November 02, 1998 - November 06, 1998 | Complexity of Continuous and Algebraic Mathematics |

August 17, 1998 - August 26, 1998 | Introductory Workshop on Foundations of Computational Mathematics and Symbolic Computation in Geometry and Analysis |