By Frances Kirwan, Jonathan Woolf

A grad/research-level creation to the ability and sweetness of intersection homology concept. available to any mathematician with an curiosity within the topology of singular areas. The emphasis is on introducing and explaining the most rules. tough proofs of significant theorems are passed over or simply sketched. Covers algebraic topology, algebraic geometry, illustration idea and differential equations.

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I n ) the set of all (i 1 , . . , i n ) badly approximable points; that is the set of (x1 , . . , xn ) ∈ Rn such that there exists a positive constant c(x1 , . . , xn ) > 0 so that max{ q x1 1/i 1 , . . , q xn 1/i n } > c(x1 , . . , xn ) q −1 ∀q ∈ N. 9 Let n = 2 and note that if (x, y) ∈ Bad(i, j ) for some pair (i, j), then it would imply that lim inf q q x q→∞ qy = 0. Hence ∩i+ j =1 Bad(i, j) = ∅ would imply that Littlewood’s conjecture is true. 4. 10 Geometrically speaking, Bad(i 1 , .

Velani The following theorem is a consequence of a more general result established in [29]. 4 Let ( , d) be a compact metric space equipped with a Borel probability measure μ. Let E q (q ∈ N) be a sequence of balls in such that μ(E ∞ ) > 0. Then, there exists a strictly increasing sequence of integers (qk )k∈N such that ∞ k=1 μ(E qk ) = ∞ and the balls E qk (k ∈ N) are quasiindependent on average. 1 is the ‘easy case’) provides an elegant criterion for the ‘size’ of the set W (ψ) expressed in terms of Lebesgue measure.

Then dim I = 1 and ⎧ ⎪ 0 if s > 1 ⎪ ⎪ ⎨ 2Hs (I) = 1 if s = 1 ⎪ ⎪ ⎪ ⎩ ∞ if s < 1. Thus, 2H1 (I) = m(I) and I is an example of an s-set with s = 1. Note that the presence of the factor ‘2’ here is because, in the definition of the Hausdorff measure, we have used the radii of balls rather than their diameters. g. W (τ ) (this is the Jarník–Besicovitch theorem discussed below), and is easier than determining the Hausdorff measure. Further details regarding Hausdorff measure and dimension can be found in [50, 72].