Mathematical Colloquium

Tame (real) geometry is situated between general differential topology and real algebraic geometry. The objects of study (which I will call "tame structures") are categories of real sets and functions which have a "tame" topological behaviour: finite stratifications and a good dimension theory for sets in the category; almost everywhere regularity, factorization theorems, uniform asymptotics for parametric families of functions in the category.
An example of tame structure is given by the category $S$ of globally subanalytic sets and functions, generated by real analytic functions restricted to compact boxes. Another example is given by the category $S(\mathrm{exp})$, generated by $S$ and the unrestricted real exponential function.
Tame structures are stable under many natural geometric and analytic operations (composition, taking derivatives, extracting implicit functions, blow-ups...) but not, in general, under parametric integration, nor under integral transforms. Such objects arise naturally in many different domains (for example, the error function is a parametric integral and the Euler gamma function is a Mellin transform of functions in $S(\mathrm{exp})$). It is hence natural to investigate how much of the tameness of the function we integrate/transform is preserved after the process. I will address some specific questions in this direction.

A key goal in model theory is distinguishing between tame structures (e.g. the complex field) and wild structures (e.g. the ring of integers). This distinction was introduced by Shelah in the 1970s (Shelah’s Classification Theory) and is based on restricting combinatorial pattern given by “definable” binary relations (relations defined by first order formulas). At the apex are the stable theories, such as the complex field, in which one cannot define a linear order. Stable theories have two complementary extensions: NIP theories (including $p$-adic fields) and simple theories (such as pseudofinite fields). In this talk, we briefly discuss fields whose theory is tame in the above sense. Afterwards, we leave the binary world, and introduce “$n$-ary” Classification theory. Instead of “controlling” $2$-ary relation, we only assume that $n$-ary relations are tame. We give examples and discuss groups and fields within these classes of theories.

The tail behaviour of distributions has been studied much. However, the first logarithmic moment $E_I(X) = \int (−\mathrm{ln} |x|)f(x) \mathrm{dx}$, where $f(x)$ is the pdf of some r.v. $X$, is almost unknown – although it gives valuable information about all values that $X$ assumes. Since $−\mathrm{ln} |x|$ is positive close to the origin and negative `farther out’, $E_I(X)$ can be interpreted as a parameter that measures the distribution’s amount of centrality, or quite simply the `expected information’ in $X$ on the origin.
The expected information in most important statistical distributions can be expressed in rather elementary terms, such as $E_I(X) = (\gamma + \mathrm{ln} 2)/2$ if $X$ is standard Normal, where $\gamma \approx 0.577$ is Euler’s constant. Higher logarithmic moments, in particular the information variance $\sigma^2_I(X) = \int (−\mathrm{ln} |x|)^2 f(x) \mathrm{dx} − E^2_I(X)$, also reveal interesting properties of (seemingly) well-known distributions. For instance, the information variance of a (standard) Cauchy is $\sigma^2_I = \pi^2/4$ and twice as large as that of a Normal.
The latter moments have beautiful properties, most importantly $E_I(X^k) = k E_I(X)$, and $E_I(YZ) = E_I(Y) + E_I(Z)$. Thus they may be used to study products and ratios of r.v.’s, scale-shape families of distributions, and their discrete analogues. As a result, the complicated web of distributions becomes a well-organized pattern that builds on the Cauchy, and whose centrepiece is a stochastic version of the Askey scheme.

In this talk I will survey old and recent results on profinite rigidity, the study of a group via its finite quotients. Time permitting I will discuss recent advances related to 3-manifolds, projective varieties, and more.