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 $S(\exp )$, 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 $S(\exp )$). 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.

TBA (Here is the abstract of the talk.....)

The tail behaviour of distributions has been studied much. However, the first logarithmic moment $E_I(X)=\int (-\ln |x|)f(x)\mathrm{dx}$, 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 $-\ln |x|$ 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 $E_I(X)=(\gamma +\ln 2)/2$ if $X$ is standard Normal, where $\gamma \approx 0.577$ is Euler’s constant. Higher logarithmic moments, in particular the information variance ${\sigma }_I^2(X)=\int (-\ln |x|{)}^{2}f(x)\mathrm{dx}-E_I^2(X)$, also reveal interesting properties of (seemingly) well-known distributions. For instance, the information variance of a (standard) Cauchy is ${\sigma }_I^2={\pi }^2/4$ 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.