Organizers
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 biochemistry.
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 polynomialtime algorithms for detecting real solutions in certain instances where only exponentialtime 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.
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 exponentialtime algorithms. We reveal a completely different approach, arising from seminal work of Pascal Koiran, that uses number theory to get subexponential 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. 
