Diophantine properties of subsets of $\mathbb{R}^n$ definable in an o-minimal expansion of the ordered field of real numbers have been much studied over the last few years and several applications to purely number theoretic problems have been made. One line of inquiry attempts to characterise the set of definable functions $f : \mathbb{R} \to \mathbb{R}$ having the property that $f(\mathbb{N}) \subset \mathbb{N}$. For example, a result of Thomas, Jones and myself shows that if the structure under consideration is $\mathbb{R}_{exp}$ (the real field expanded by the exponential function) and if, for all positive $r, f(x)$ eventually grows more slowly than $exp(x^r)$, then $f$ is necessarily a polynomial with rational coefficients. In this talk I shall improve this result in two directions. Firstly, I take the structure to be $\mathbb{R}_{an,exp}$ (the expansion of $\mathbb{R}_{exp}$ by all real analytic functions defined on compact balls in $\mathbb{R}^n$) and secondly, I allow the growth rate to be $x^N \cdot 2^x$ for arbitrary (fixed) $N$. The conclusion is that $f(x) = p(x) \cdot 2^x + q(x)$ for sufficiently large $x$, where $p$ and $q$ are polynomials with rational coefficients.

I should mention that over ninety years ago Pólya established the same result for entire functions $f : \mathbb{C} \to \mathbb{C}$ and that in 2007 Langley weakened this assumption to $f$ being regular in a right half-plane of $\mathbb{C}$. I follow Langley's method, but first we must consider which $\mathbb{R}_{an,exp}$-definable functions actually have complex continuations to a right half-plane and, as it turns out, which of them have a definable such continuation.[-]

Let $(V,p)$ be a complex isolated complete intersection singularity germ (an ICIS). It is well-known that its Milnor number $\mu$ can be expressed as the difference:
$$\mu = (-1)^n ({\rm Ind}_{GSV}(v;V) - {\rm Ind}_{rad}(v;V)) \;,$$
where $v$ is a continuous vector field on $V$ with an isolated singularity at $p$, the first of these indices is the GSV index and the latter is the Schwartz (or radial) index. This is independent of the choice of $v$.
In this talk we will review how this formula extends to compact varieties with non-isolated singularities. This depends on two different ways of extending the notion of Chern classes to singular varieties. On elf these are the Fulton-Johnson classes, whose 0-degree term coincides with the total GSV-Index, while the others are the Schwartz-McPherson classes, whose 0-degree term is the total radial index, and it coincides with the Euler characteristic. This yields to the well known notion of Milnor classes, which extend the Milnor number. We will discuss some geometric facts about the Milnor classes.[-]

The Jacobian algebra, obtained from the ring of germs of functions modulo the partial derivatives of a function $f$ with an isolated singularity, has a non-degenerate bilinear form, Grothendieck Residue, for which multiplication by $f$ is a symmetric nilpotent operator. The vanishing cohomology of the Milnor Fibre has a bilinear form induced by cup product for which the nilpotent operator $N$, the logarithm of the unipotent part of the monodromy, is antisymmetric. Using the nilpotent operators we obtain primitive parts of the bilinear form and we compare both bilinear forms. In particular, over $\mathbb{R}$, we obtain signatures of these primitive forms, that we compare.[-]

A real algebraic domain is a closed topological subsurface of a real affine plane such that its boundary consists of disjoint smooth connected components of real algebraic plane curves. Our goal is to study the nonconvexity of real algebraic domains relative to the vertical direction. To this end, we collapse all vertical segments contained in the algebraic domain, yielding a Poincar´e–Reeb graph which is naturally transversal to the foliation by vertical lines. Our main result is the following: any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines arises up to isomorphism as a Poincar´e–Reeb graph of a real algebraic domain. We also give a purely topological description of the setting in which our construction of Poincar´e–Reeb graphs may be applied, with no differentiability assumptions. This is a joint work with Arnaud Bodin and Patrick Popescu-Pampu (Université de Lille, France).[-]

In the last years there has been an increasing interest into the statistical behaviour of algebraic sets over non-algebraically closed fields: when the notion of 'generic' is no longer available, one seeks for a 'random' study of the objects of interest. In this course, divided into four lectures, I will present the major ideas in the subject (lecture notes will be made available):

1. Generic and random. In the first lecture I will discuss how to switch from the notion of generic, from classical algebraic geometry, to the notion of random. Of course, this depends on the choice of the probability distribution on the 'moduli space' of the objects of interest. I will discuss what are the reasonable choices in the case $\mathbb{K}=\mathbb{C}$ (where still these questions make sense, and 'random' and 'generic' are synonymous) and in the case $\mathbb{K}=\mathbb{R}$ (where spherical harmonics play a crucial role).[-]

3. The square-root law and the topology of random hypersurfaces. In the third lecture I will focus on the case $\mathbb{K}=\mathbb{R}$ and explain in which sense random real algebraic geometry behaves as the 'square root' of complex algebraic geometry. I will discuss a probabilistic version of Hilbert's Sixteenth Problem, following the work of Gayet & Welschinger (introducing a local random version of Nash and Tognoli's Theorem and of Morse theory for the study of Betti numbers of random hypersurfaces) and of Diatta $\&$ Lerario (showing that 'most' hypersurfaces of degree $d$ are isotopic to hypersurfaces of degree $\sqrt{d \log d}$ ).[-]

4. The zonoid ring and the nonarchimedean world. In the last lecture I will explain a ring-theoretical interpretation of the computations in random algebraic geometry, using a recently discovered ring structure on special convex bodies. This leads to the construction of a probabilistic version of Schubert calculus. In the final part of the lecture I will export some of the ideas from the previous lectures to the case $\mathbb{K}=\mathbb{Q}_{p}$, leaving with some open questions.[-]