Marianna Csörnyei

Structure of null sets in the plane and applications

We discuss a connection between two areas of mathematics which until recently seemed to be rather distant from each other: (1) noncommutative harmonic analysis on groups and (2) some topics in probability theory related to random point processes. In order to make the paper accessible to readers not familiar with either of these areas, we explain all needed basic concepts. This is an extended version of G.Olshanskis talk at the 4th European Congress of Mathematics.(1) ut + f(u)x = 0, (t, x) ∈ R × R, u ∈ R, Hereu = (u1, . . . , un) is the vector ofconserved quantities , while the components of = (f1, . . . , fn) are thefluxes. The system is said strictly hyperbolic if at each point u the Jacobian matrix Df(u) hasn real, distinct eigenvalues λ1(u) < · · · < λn(u). Several fundamental laws of physics take the form of a conservation equation. Weak solutions to the Cauchy problemIn this paper we survey some recent results in connection with the so called Painleves problem, the semiadditivity of analytic capacity and other related results.Catalan’s conjecture states that the equation x − y = 1 has no other integer solutions but 3 − 2 = 1. We prove a theorem which simplifies the proof of this conjecture.on the unit interval, where n is any natural number. In order to make this map continuous, we think of it as a map on the 1-torus T = R/Z. This system is very well understood, and it has many closed invariant sets and many invariant probability measures. Indeed, let τ : Σ = {0, . . . , n− 1} → R/Z be the map τ(a1, a2, . . . ) = ∑∞ i=1 n ai. Then any shift invariant probability measure ν on Σ, for example i.i.d. Bernoulli measure, gives rise to the ×n-invariant measure μ = τ∗ν (and similarly for sets). Every ×n invariant probability measure on R/Z is of this form, and moreover for measures μ for which μ({0}) = 0 the map τ∗ is also one-to-one. However, R/Z has additional structure: it is an abelian group, and for a fixed n, the map ×n is just one out of many endomorphisms of this group. In 1967, Hillel Furstenberg considered the joint action of two such endomorphisms ×n and ×m for n and m multiplicatively independent (i.e., not powers of the same integer). This Z+ action turns out to be much more subtle. In his landmark paper [8] Furstenberg introduced the notion of disjointness in dynamical systems and ergodic theory, a notion which has proven quite central in the modem theory of these subjects, and also proves as a byproduct that the closed subsets C ⊂ R/Z satisfying ×n(C) ⊂ C and ×m(C) ⊂ C are either R/Z or finite sets of rationals. The analogous question for measures has also been posed by Furstenberg (though apparently not in writing) in 1967, namely classifyingWe describe a decomposition result for Lebesgue negligible sets in the plane, and outline some applications to real analysis and geometric measure theory. These results are contained in (2).One important activity of theoretical meteorology involves the development of simplified model equations that describe selected scale-dependent phenomena observed in atmospheric flows. This paper summarizes a unified mathematical approach to the derivation of such models, based on multiple scales asymptotic techniques. First we motivate the approach by an example from fluid mechanics, the interaction of small-scale quasi incompressible flow with long-wave acoustics. In this case, the analysis proceeds via multiple scales asymptotics in terms of the Mach number, M, as the small expansion parameter. Then we discuss the particular setting of meteorology, where there is a host of singular small parameters to be taken into account. Examples are the Rossby, Froude, Mach, and Strouhal numbers. A particular distinguished limit among these parameters is introduced in combination with systematic multiple scales asymptotics. A wide range of simplified meteorological models can then be recovered by specializing this general ansatz to a single horizontal, a single vertical, and a single time coordinate. As a concrete example we report on the multiple scales derivation of boundary layer theories. In particular, we recover the classical Ekman boundary layer equations for flows on synoptic scales (∼ 500 km, 12 h), and find an extension of the nonlinear Prandtl boundary layer equations to atmospheric mesoscales (∼ 70 km,2h).For any tangle

Differentiability of Lipschitz functions, structure of null sets, and other problems

T

UPPER CONICAL DENSITY RESULTS FOR GENERAL MEASURES ON R-n

(up to isotopy) and integer

BV has the bounded approximation property

k\geq 1

A note on simultaneous polar and Cartesian decomposition

we construct a group

Scaled-oscillation and regularity

F(T)

Positive-measure self-similar sets without interior

(up to isomorphism). It is the fundamental group of the configuration space of

Small unions of affine subspaces and skeletons via Baire category

k

Scaling properties of Hausdorff and packing measures

points in a horizontal plane avoiding the tangle, provided the tangle is in what we call Heegaard position. This is analogous to the first half of Lawrences homology construction of braid group representations. We briefly discuss the second half: homology groups of

On planar sets with prescribed packing dimensions of line sections

F(T)