Penn State - Maryland Semi-Annual Dynamics Workshops Charles Morgan, Lock Haven University of Pennsylvania

The Mathematics Department at Penn State University (State College, Pennsylvania, USA) held its semi-annual workshop on dynamical systems from October 18 through October 21. This was the eighteenth workshop held at Penn State since their inception in 1989. The semi-annual workshops are held jointly by Penn State and the University of Maryland. These workshops always bring the top names in the field, but this year organizers were compelled to hold the workshop on the same weekend as the Midwest Dynamical Systems Seminar at the University of Michigan. Since the Midwest seminars were founded and organized by the students of Stephen Smale, a lot of big names were missing from this year's workshop.

Although previous workshops have focused only on dynamical systems and directly related mathematics, recent workshops have become more general and now include geometry related to dynamical systems. In recent years, Smale's and Sinai's students have approached their 60th birthdays, and recent workshops have been held in honor of Sheldon Newhouse, Yakov Pesin, and others. In addition, the 2006 workshop announced the inauguration of the Journal of Modern Dynamics, published by the American Institute of Mathematical Sciences.

Generally, one sees the "cutting edge" of modern dynamical systems according to the Smale program. Among the speakers at the October workshop was Boris Hasselblatt (Tufts), who discussed recent joint work with Ya. Pesin and J. Schmeling in which they use invariant cone families which are not continuous to obtain uniform hyperbolicity everywhere for diffeomorphisms of a compact Riemannian manifold.

My good friend Huyi Hu (Michigan State) presented joint work with M. Jiang and Y. Jiang in which they show that any C^r uniformly hyperbolic attractor is topologically equivalent to a uniformly hyperbolic attractor in the C^r topology with arbitrarily small metric entropy with respect to the SRB measure supported on the attractor.

Arguably, one of the most interesting results presented was by Yulij Ilyashenko (jointly at Cornell and Independent University of Moscow). Generally, one expects that knowledge of a system's attractor allows one to predict long-term behavior of orbits. In this paper, which is joint work with his co-author, a senior at Princeton, Ilyashenko and Andrei Negut describe dynamical systems which possess the new notion of &epsilon attractor for 0 &le &epsilon &le 1, a set near which almost-every orbit spends on average at least a proportion of 1- &epsilon of its time in the future. The idea, according to the authors, is that dynamical systems often have attractors for which a large part is not observable by following orbits.

A recent announcement by Clifford Taubes of a result on the Weinstein Conjecture allowed Giovanni Forni (Maryland) to use recent work by J. Hertz to make more progress on the Greenfield-Wallach and Katok conjectures on certain vector fields.

Other notable speakers at this conference included Marcelo Viana (IMPA, Brazil), Benjamin Weiss (Hebrew University of Jerusalem), Larry Guth (Stanford), Vadim Kaloshin (Maryland and Penn State), John Mather (Princeton), Yaron Ostrover (MIT), Manfred Denker (Göttingen, visiting Case Western Reserve), and others. The upcoming workshop at the University of Maryland is scheduled for March 15-18, 2008. It is being held in honor of Michael Brin's 60th birthday. Information on this and past workshops can be found on the University of Maryland Dynamics or Penn State's Center for Dynamics and Geometry web sites.

Upcoming Maryland conference.