## EVENT DETAILS AND ABSTRACT

**Colloquium**
**Title:** Smale's 17th Problem: Recent progress

**Speaker:** Professor Mike Shub

**Speaker Info:** University of Toronto

**Brief Description:**

**Special Note**:

**Abstract:**

In a series of papers written in the first half of the 1990's Steve Smale and I
studied the complexity of solving systems of n polynomial equations in n complex
variables. We studied path following techniques. A system with known solution is
connected by a path to the system we want to solve and the solution is "continued" along
the path. The path we chose was the straight line connecting the systems. We proved that
"on average" systems can be solved with polynomial cost but we did not prove the
existence of a uniform algorithm. The question of the existence of a uniform algorithm is
Smale's 17th problem. Recently, Beltran and Pardo have made significant progress on this
problem. Moreover, Jointly with Beltran I have linked the complexity to the length of the
(problem,solution) path in the condition number Riemannian structure. Surprisingly short
paths exist! So the study of the geodesics of this Riemannian structure presents
interesting challenges.

**Date:** Wednesday, May 23, 2007

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Prof. Jeff Xia

**Contact email:** xia@math.northwestern.edu

**Contact Phone:** 847-491-5487

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