Mid Atlantic Algebra Conference
James Madison University

April 29-30, 2006


James Madison University
Department of Mathematics and Statistics


 Beth Arnold , Carter Lyons , Gary Peterson

 Principal Speaker
J. Maurice Rojas
 Texas A & M University

Understanding Polynomial Equations: Complexity and Reality

J. Maurice Rojas was born in Los Angeles, of Colombian immigrant parents. 
His degrees are from UCLA (BS in math/applied science) and UC Berkeley
(MS in computer science and a PhD in applied mathematics), and his
Ph.D. advisor was Fields Medalist Steve Smale.  Dr. Rojas was a winner
of the 1992 SIAM Best Paper Award, a National Science Foundation
Postdoctoral Fellowship (used to visit MIT, MSRI, and City University of
Hong Kong), and the 2000 Journal of Complexity Best Paper Award.

Dr. Rojas is now a tenured associate professor in the mathematics
department at Texas A&M University and the recipient of several NSF grants,
including an NSF CAREER award.  He works at the intersection of algebraic
geometry, complexity theory, and number theory.  He has also been involved  
in applying algebraic geometry methods to geometric modelling and

Talk 1: "Reality and Fewnomial Equations"  

Real Algebraic Geometry (RAG) is the theory behind the real
solutions of systems of polynomial equations.  It is also a beautiful
research area because it occurs in so many practical applications,
and so many basic questions in RAG remain open.

In this talk, we'll see how counting the real solutions of a
system of equations (an algebraic problem) is related to simple diagrams
one can draw by hand (polyhedral combinatorics). In particular, we will
derive polynomial-time algorithms for detecting real solutions in certain
instances where only exponential-time algorithms were known before.

We assume no background in algebraic geometry or algorithmic
complexity.  Some of the results we'll see are joint work with Frederic
Bihan, Alicia Dickenstein, Korben Rusek, Justin Shih, Frank Sottile, and
Casey Stella.

Talk 2: "Complexity: How Riemann Meets P=NP"

Deciding whether a polynomial in one variable has a complex 
root is easy, but for systems of multivariate polynomials, one 
quickly runs into complexity barriers: the best recent techniques 
(e.g., Grobner bases and resultants) lead only to exponential-time 

We reveal a completely different approach, arising from seminal 
work of Pascal Koiran, that uses number theory to get sub-exponential 
algorithms.  In particular, we'll see how the famous Riemann Hypothesis 
relates to the complexity of this new class of algorithms. 

While we thus reveal a link between P=NP, the Riemann hypothesis, 
and equation solving, we assume no background in complexity theory, 
number theory, or algebraic geometry. 




Local information, directions, parking information

Support for this conference is provided by the College of Science and Mathematics at James Madison University and the Department of Mathematics and Statistics at JMU.