will speak on
Some easily stated open problems will be discussed.
(i) It is known (Zalcman, 1982) that the Radon transform is not injective: there exist
non-trivial continuous functions $f:\mathbb{R}^2\to \mathbb{R}$ with zero (proper) integral on every (doubly
infinite, straight) line. All known examples of such functions have extremely rapid overall
growth. Can such a function have slow growth, or even be bounded? Is it true that a
continuous function on $\mathbb R^3$ with zero integral on every line must be identically zero?
(ii) Every polygonal domain D in $\mathbb{4R}^2$ has the Pompeiu property (PP): if $f: \mathbb R^2 \to \mathbb R$ is continuous
and $\int_{\sigma(D)} f(x)dx = 0 $ for every rigid motion σ, then f ≡ 0. The corresponding assertion for functions on the sphere S2 is false: there are infinitely many
(non-congruent) regular spherical polygons that lack PP, and they can be characterised.
But it still seems unclear whether, for example, all non-trivial regular spherical triangles
have PP, and whether (up to congruence) the known example of a spherical square lacking
PP is unique.
(iii) One formulation of the maximum principle asserts that if h is a non-constant
harmonic function on a ball centred at the origin O in $\mathbb{R}^n$ and $h(O) = 0$, then $h$ takes
positive values and negative values on every neighbourhood of O. In the case $n = 2$, this
can be quantified: it is easy to show that, with $h$ as above, the subset of $ \{x:\| x\| < r \} $
where $h>0$ and the subset where $h < 0$ have roughly the same area. (The ratio of the
areas tends to 1 as $r \to 0^+$.) What can be said in the case $n \ge 3$?
(This talk is part of the IMS September Meeting 2007 series.)
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