DIFFERENTIAL GEOMETRY/PDE SEMINAR

Tuesday, July 15, 2003 Padelford C-36, 1:30 pm

When are porous sets null on many curves or surfaces and why are we interested in this?

David Preiss (University College of London)

In spite of its title, the talk will try to explain recent progress in attempts to resolve the ’differentiability conjecture’. In ℝ^{n} this conjecture says that there is a null set which contains a point of differentiability of every Lipschitz map to ℝ^{n−1}, but there is no such null set for Lipschitz maps to ℝ^{n}. The most interesting case of the infinite dimensional version of the conjecture believes that Lipschitz functions on separable Hilbert spaces are (Frechet) differentiable “almost everywhere”; the natural setting is Banach spaces with separable dual.

Porous sets may be defined as sets of non-differentiability of their distance function (in the infinite dimensional case we consider Frechet differentiability). So they, and their countable unions (termed sigma-porous sets) are sets of non-differentiability of Lipschitz functions, but there are also simple examples of non-sigma-porous sets of non-differentiability of Lipschitz functions. The importance of studying porous sets in this connection stems from the intriguing basic fact (observed jointly with J. Lindenstrauss) that if porous sets are null on many (infinite dimensional) surfaces then all non-differentiability sets are null on many surfaces (and so Lipschitz functions are differentiable “almost everywhere”). This proved the infinite dimensional version of the conjecture in some spaces (such as C_{0}). In Hilbert spaces, however, porous sets need not be null even on three dimensional surfaces. Very recently, we investigated when are porous sets null on k-dimensional surfaces. This holds, for example, in Hilbert spaces for k = 2 and in finite dimensional spaces for all k. We have also shown that the analogue of the basic fact holds for finite dimensional surfaces and so obtained that two Lipschitz function in a Hilbert space necessarily have a common point of differentiability as well as the first part of the finite dimensional version of the differentiability conjecture.

Remarks.

1. ‘We’ is used to indicate various sets of people containing in particular J. Lindenstrauss and J. Tiser (especially in the infinite dimensional case) or G. Alberti and M. Csornyei (especially in the finite dimensional case).

2. The finite dimensional version of the differentiability conjecture is not universally accepted. In fact, the only program of proving the infinite dimensional version that I am aware of could be also understood as an attempt to show that its second part is false.

For more information about this seminar, visit the DG/PDE Seminar Web page (from the Math Department home page, www.math.washington.edu, follow the link Seminars, Colloquia, and Conferences).

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