The classical Helly Theorem (1913) states that if a finite family of convex subsets of $\mathbb{R}^d$ satisfies that any $d+1$ sets have nonempty intersection then the whole family has nonempty intersection. By a simple compactness argument the same is true for infinite families of convex compact sets. Since this first result other Helly-type theorems have been proved for different families of sets (notably families with finite VC-dimension). The underlying thesis being that, in a family of "low complexity", a weak intersection property implies a strong one. We present a Helly-type result for definable families of sets in o-minimal structures (these include in particular the family of fibers of any semialgebraic or subanalytic set). Let $\mathcal{S}$ be a definable family of subsets of $R^d$ in some o-minimal structure $\mathcal{R}=(R,\ldots)$ and $d\geq d+1$. There is $n$ such that, if a finite subfamily of $\mathcal{S}$ has the $(p,d+1)$-property, meaning that, for every $p$ sets in the subfamily, some $d+1$ intersect, then it admits a transversal of size $n$ (i.e. there is a finite set of size $n$ that intersects every set in the subfamily). We use a suitable notion of definable compactness to generalize this result to infinite families.

Using a widely known example of a family of semi-algebraic curves, I explain how to construct $C^r$-parametrizations, also known as Yomdin-Gromov parametrizations. I will briefly highlight the applications of these type of parametrizations and give some recent results. The goal is to understand how these techniques can be refined, additionally using Weierstrass preparation, to obtain the $C^r$-parametrization theorem of Cluckers, Pila and Wilkie, which shows that the amount of charts in their construction is polynomial in $r$. Finally, if time permits, I will also discuss mild parametrizations.

.pdf abstract with references

In 1971, the related notions VC dimension and "not the indepence property" (NIP) were established independently of each other. The former was introduced by Vapnik and Chervonenkis in the context of linear learning theory, the latter by Shelah in the study of stable theories - a highly abstract concept from model theory. It took 21 years until a connection between VC dimensions and NIP was noticed, describing how the two notions give different descriptions of the same idea. This connection resulted in an interesting application of a purely model theoretical and combinatorial concept to neural network learning.

In my talk, I will firstly outline the mathematical idea behind artificial neural networks and in this regard describe the formal learning process for such a network. I will then highlight the theorem which links neural network learning to the model theoretic concept of NIP theories. Finally, I will present our recent progress in the study of NIP ordered fields and definable valuations motivated by the Shelah-Hasson Conjecture.

The talk should be accessible to anyone with a background in mathematics, and all relevant notions will be introduced.

Random groups are proposed by Gromov as a model to study the typical behavior of finitely presented groups. They share many properties of the free group, and Knight conjectured that random groups satisfy a strong zero-one law and have the same first-order theory as the free group. In joint work with Franklin and Knight, we study this zero-one law in other classes of structures. In particular, we consider random presentations in algebraic varieties in the sense of universal algebra. We will discuss some examples where the zero-one law holds and some other examples where the zero-one law fails. We will also discuss some general results.

I'll discuss some joint work, still in progress, with Gal Binyamini, Harry Schmidt, and Margaret Thomas, in which we prove effective forms of the Pila-Wilkie Theorem for various structures involving pfaffian functions. I'll also show how, in combination with some earlier work with Schmidt, this can be used to prove some effective results in diophantine geometry.

We study outer Lipschitz geometry of surface germs definable in a polynomially bounded o-minimal structure (e.g., semialgebraic or subanalytic).
By the finiteness theorem of Mostowski, Parusinski and Valette, any definable family has finitely many outer Lipschitz equivalence classes. Our goal is classification of definable surface germs with respect to the outer Lipschitz equivalence.
The inner Lipschitz classification of definable surface germs was described by Birbrair.
The outer Lipschitz geometry is much more complicated.
Using the $K$-equivalence classification of Lipschitz functions ("pizza decomposition") of Birbrair et al. and the theory of abnormal surface germs ("snakes") by Gabrielov and Souza, we obtain a decomposition of a surface germ into normally embedded Holder triangles, unique up to outer Lipschitz equivalence. This triangulation, with some additional data ("pizza toppings") is a complete discrete invariant of an outer Lipschitz equivalence class of surface germs.
Joint work with L. Birbrair, A. Fernandes, R. Mendes and E. Souza (UFC Fortaleza, Brazil).