The unreasonable effectiveness of the convexity assumption in high dimensions
We survey progress from the past five years on the distribution of mass in high-dimensional convex bodies and in probability distributions with convexity properties. The concentration of measure phenomenon has traditionally been studied in highly regular or structured settings, such as spheres, Hamming cubes, Gaussian measures, Markov chains, and martingales. It turns out that convexity assumptions provide an alternative source of regularity in high dimensions with remarkably similar features: Lipschitz functions are highly concentrated, the isoperimetric problem is nearly saturated by half-spaces (up to logarithmic factors), and the central limit theorem is nearly as strong as in the setting of independent random variables. The main developments discussed include the resolution of Bourgain’s slicing problem and the Variance Conjecture, as well as recent progress on the isoperimetric problem for high-dimensional convex bodies. Based on joint work with P. Bizeul and J. Lehec.
We survey progress from the past five years on the distribution of mass in high-dimensional convex bodies and in probability distributions with convexity properties. The concentration of measure phenomenon has traditionally been studied in highly regular or structured settings, such as spheres, Hamming cubes, Gaussian measures, Markov chains, and martingales. It turns out that convexity assumptions provide an alternative source of regularity in high dimensions with remarkably similar features: Lipschitz functions are highly concentrated, the isoperimetric problem is nearly saturated by half-spaces (up to logarithmic factors), and the central limit theorem is nearly as strong as in the setting of independent random variables. The main developments discussed include the resolution of Bourgain’s slicing problem and the Variance Conjecture, as well as recent progress on the isoperimetric problem for high-dimensional convex bodies. Based on joint work with P. Bizeul and J. Lehec.
Model theory of Abelian groups is extensively studied in the literature also in recent
years. An identical inclusion is a formula that can be expressed as a (possibly infinitary)
disjunctive identity
u = v1 ∨ u = v2 ∨ u = v3 ∨ . . . ,
or, equivalently, as a universally closed identical equality of subsets of words (terms). For
groups and rings, the classes defined by identical inclusions and by infinitary disjunctive
identities are coincide, for semigroups they do not coincide. A class of algebras defined
by a set of identical inclusions is called an inclusive variety. An inclusive variety that
can not be defined by first order formulas is called a nonelementary inclusive variety. An
inclusive variety defined by a system of identical inclusions - each depending on a finite set
of variables - is called a quasielementary inclusive variety.
We describe elementary, nonelementary and quasielementary inclusive varieties of Abelian
groups. There exist continuum many inclusive varieties of each of these kinds. We also
determine Abelian groups defined by identical inclusions up to isomorphism and classify
Abelian groups up to inclusive equivalence.