This award provides funding to help defray the expenses of participants, especially women, graduate students, postdocs, and junior faculty, in the "From Dynamics to Chaos" conference that will be held from May 7--11, 2012, at the Fields Institute for Research in Mathematical Sciences in Toronto, Canada.

This conference will cover a variety of topics in analysis (e.g., dynamical systems, complexity theory, mathematical physics), with partially hyperbolic dynamics serving as a key unifying theme. All of the conference topics are central to analysis and extremely active subjects of current research. The format of the meeting is such that young people will have ample opportunities to speak and be otherwise engaged in the various conference activities.

Project Report

This grant supported participants from the United States to attend a conference entitled "From Dynamics to Complexity," which was held at the Fields Institute in Toronto from May 7-11, 2012. The conference was designed with the innovative goal of combining researchers from two distinct areas: the Computational Complexity field from Computer Science, and Dynamical Systems from Mathematics. The expertise of the organizers spanned both areas, with Jean-Pierre Dedieu (University Paul Sabatier) and Teresa Krick (University of Buenos Aires) representing the Complexity side, and Charles Pugh (University of California, Berkeley) and Amie Wilkinson (University of Chicago) the Dynamical Systems side. The grant supported a total of 25 US researchers, all of them without alternate means of direct support. The lectures at the conference were pitched at a general audience and represented a diverse set of topics and research backgrounds. Intellectual Merit Complexity theory is the formal study of methods used in the solution to computational problems that arise in everyday life, for example the solutions to systems of polynomial equations in Econonmics, Engineering and Medicine. These complexity and algorithmic questions are –needless to say– central. The development of computers in the last 50 years has made their consideration crucial. The practical resolution of the polynomial questions requires the constant development of more high-performance algorithms, which in their turn require the development of a suitable original mathematical theory and the use of techniques coming from different fields. Even a proper foundational setting is still missing for some aspects of the analysis of these problems. The field of Dynamical Systems has its origins in early efforts toestablish the stability of the solar system. Early pioneers such as Henri Poincare and George Birkhoff discovered mechnanisms for stability and instability, thus initiating the study of hyperbolic dynamical systems. Around the same time, the geometer Eberhard Hopf used hyperbolic methods to study the long-term behavior of geodesics (straight lines) in negatively-curved spaces (such as those modelled on Minkowski spacetime). A century later, the study of hyperbolic dynamical systems has blossomed into a mature field, one of the cornerstones of dynamics. Parallel issues arise in the rigorous study of both Complexity in Dynamics: robustness (stability) of results, estimation of rate of convergence of algorithms, and prediction of "typical" (i.e. generic) behavior. The intersections of these methods emerged as a central theme and was addressed in several talks, among them: Shmuel Weinberger From Complexity to Geometry: Disordered Solids. Ke Zhang Arnold Diffusion via Invariant Cylinders and Mather Variational Method Jana Rodriguez Hertz Partial hyperbolicity and the topology of 3-manifolds Saugata Basu A B-S-S analogue of Valiant: complexity theory of constructible functions and sheaves Jim Renegar Optimization, Then and Now Pascal Koiran Progress on the real tau conjecture Maria Jose Pacifico A toy model for flows with equilibria attached to regular orbits A total of 31 talks were held, with plenary talks in the morning followed by parallel sessions in the afternoon. Broader Impact The conference was attended by approximately 200 participants and was widely and effectively advertised on the conference website: The grant subsized participation by 7 women who had no direct grant support, and many additional early-career researchers without outside support. Two of the four main organizers were women, and this was reflected in the high turnout of women and a general atmosphere of incluseiveness at the conference.

National Science Foundation (NSF)
Division of Mathematical Sciences (DMS)
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Edward Taylor
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Northwestern University at Chicago
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