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2. Dimension theory (algebra) - Wikipedia

   en.wikipedia.org/wiki/Dimension_theory_(algebra)

   In mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme).The need of a theory for such an apparently simple notion results from the existence of many definitions of dimension that are equivalent only in the most regular cases (see Dimension of an algebraic variety).

3. Falconer's conjecture - Wikipedia

   en.wikipedia.org/wiki/Falconer's_conjecture

   Falconer (1985) proved that Borel sets with Hausdorff dimension greater than (+) / have distance sets with nonzero measure. [2] He motivated this result as a multidimensional generalization of the Steinhaus theorem, a previous result of Hugo Steinhaus proving that every set of real numbers with nonzero measure must have a difference set that contains an interval of the form (,) for some >. [3]

4. Kakeya set - Wikipedia

   en.wikipedia.org/wiki/Kakeya_set

   In 2000, Katz, Łaba, and Tao [12] proved that the Minkowski dimension of Kakeya sets in 3 dimensions is strictly greater than 5/2. In 2000, Jean Bourgain connected the Kakeya problem to arithmetic combinatorics [ 13 ] [ 14 ] which involves harmonic analysis and additive number theory .

5. Dimension (vector space) - Wikipedia

   en.wikipedia.org/wiki/Dimension_(vector_space)

   A diagram of dimensions 1, 2, 3, and 4. In mathematics, the dimension of a vector space V is the cardinality (i.e., the number of vectors) of a basis of V over its base field. [1] [2] It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension.

6. Higher-dimensional algebra - Wikipedia

   en.wikipedia.org/wiki/Higher-dimensional_algebra

   In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, [11] and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms.

7. Kissing number - Wikipedia

   en.wikipedia.org/wiki/Kissing_number

   Other names for kissing number that have been used are Newton number (after the originator of the problem), and contact number. In general, the kissing number problem seeks the maximum possible kissing number for n-dimensional spheres in (n + 1)-dimensional Euclidean space. Ordinary spheres correspond to two-dimensional closed surfaces in three ...