# Talk Titles and Abstracts

## Invited Speakers

- Matthias Aschenbrenner (University of California, Los Angeles)
- Fabrizio Barroero (Scuola Normale Superiore di Pisa)
- Alessandro Berarducci (University of Pisa)
- Lou van den Dries (University of Illinois at Urbana-Champaign)
- Philipp Habegger (University of Basel)
- Gareth Jones (University of Manchester)
- Tobias Kaiser (University of Passau)
- Salma Kuhlmann (University of Konstanz)
- Jana Maříková (Western Illinois University)
- Jean-Philippe Rolin (Université de Bourgogne)
- Tamara Servi (University of Pisa)
- Patrick Speissegger (McMaster University)
- Alex Wilkie (University of Manchester)
- Yosef Yomdin (Weizmann Institute of Science)

## Contributing Speakers

- Laura Capuano (Scuola Normale Superiore di Pisa)
- Christopher Daw (Institut des Hautes Etudes Scientifiques (IHES))
- Sabrina Lehner (University of Passau)
- Vincenzo Mantova (Scuola Normale Superiore di Pisa)
- Julia Ruppert (University of Passau)
- Erik Walsberg (University of California, Los Angeles)

## Invited Talk Abstracts

Title: Model theory of transseries

Slides from M. Aschenbrenner's talk

Abstract: The concept of a "transseries" is a natural extension of that of a Laurent series, allowing for exponential and logarithmic terms. The germs of many naturally occurring real-valued functions of one variable - in particular, many that live in o-minimal expansions of the ordered field of reals - have asymptotic expansions which are transseries. In the last few years, our knowledge about the algebraic and model-theoretic aspects of these intricate but fascinating mathematical objects has increased substantially. My goal for this talk is to introduce transseries without prior knowledge of the subject, and to explain these recent results. (Joint work with L. van den Dries and J. van der Hoeven.)

Title: Unlikely intersections in families of powers of elliptic curves

Abstract: Let $E_{t}$ be the Legendre elliptic curve of equation $Y^{2} = X(X-1)(X-t)$. In 2010 Masser and Zannier proved that, given two points on $E_{t}$ with coordinates algebraic over $Q(t)$, there are at most finitely many specializations of $t$ such that the two points become simultaneously torsion on the specialized elliptic curve, unless they were already generically linearly dependent. One of the main ingredients of the proof is a result of Pila about counting rational points of bounded height on subanalytic surfaces, which is a special case and pre-dates the celebrated Pila-Wilkie theorem.

As a natural higher-dimensional analogue, we considered the case of $n$ generically independent points on $E_{t}$ with coordinates algebraic over $Q(t)$. Then there are at most finitely many specializations of $t$ such that two independent relations hold between the specialized points. Here one needs a more sophisticated counting theorem: relying on results of Pila, we estimate the number of points on some subanalytic surfaces lying on certain linear affine varieties defined by equations with rational coefficients of bounded height. This is joint work with L. Capuano.

Title: Transserial derivations on the surreal numbers

Abstract: I shall present recent work with Vincenzo Mantova on ordered differential fields and Conway's surreal numbers, leading to a solution of the conjecture that Conway's surreal numbers admit the structure of a field of transseries and a compatible derivation. The logarithmic-exponential series admit a natural embedding of differential fields into the surreal numbers endowed with this new differential structure.

Title: Transseries, Hardy fields, and surreal numbers

(joint work with M. Aschenbrenner and J. van der Hoeven)

Slides from L. van den Dries' talk

Abstract: This is a follow-up on Aschenbrenner's talk, and concerns applications and connections of our work on the model theory of the differential field $\mathbb{T}$ of transseries to related areas: Hardy fields and surreal numbers. As to the surreals, Berarducci and Mantova have recently managed to equip Conway's field $\textbf{No}$ of surreals with a distinguished derivation that makes it a Louville closed $H$-field with the field of reals $\mathbb{R}$ as constant field.

It is easy to show that there is a unique strongly $\mathbb{R}$-linear embedding of the ordered exponential field $\mathbb{T}$ into the ordered exponential field $\textbf{No}$. Old result: this is an elementary embedding of ordered exponential fields. New result: it is also an elementary embedding of differential fields.

Title: O-Minimality and Diophantine Approximation

Abstract: I will talk about work in progress towards applications of o-minimality to approximations of definable sets by rational points. The method is based on work of Pila and Wilkie and also a variation of the Lojasiewicz inequality.

Title: Some effective instances of relative Manin-Mumford

Abstract: In a series of recent papers David Masser and Umberto Zannier proved the relative Manin-Mumford conjecture for abelian surfaces, at least when everything is defined over the algebraic numbers. In a further paper with Daniel Bertrand and Anand Pillay they have explained what happens in the semiabelian situation, under the same restriction as above.

At present it is not clear that these results are effective. I'll discuss work in progress with various people in which we show that certain very special cases can be made effective. For instance, we can effectively compute a bound on the order of a root of unity $t$ such that the point with abscissa $2$ is torsion on the Legendre curve with parameter $t$.

Title: Integration on Nash manifolds over real closed fields and Stoke's theorem

Abstract: We extend the recently developed Lebesgue measure and integration theory on real closed fields to integrate semialgebraic differential forms on Nash manifolds. We formulate and prove Stoke's theorem in this setting which is of independent interest in the case of reals.

Title: Power series expansions for germs in Hardy fields of o-minimal expansions of real closed fields

Abstract: Let $T$ be the theory of a polynomially bounded o-minimal expansion $P$ of the reals by a set of real valued functions, and $T(exp)$ the theory of $(P,exp)$. Under some mild assumptions, $T(exp)$ remains o-minimal. We consider an elementary extension $R$ of $(P,exp)$, and its associated Hardy field $H(R)$ of definable germs. We analyze the structure of $H(R)$ as a field equipped with convex valuations. We introduce an intrinsic form of power series expansions for the elements of $H(R)$ using monomials of an arbitrary value group cross section together with coefficients from significant residue fields. From such an expansion we define the principal part of a germ $h$ which determines the asymptotic class of the germ $\exp h$. We derive some applications.

Joint work with Franz-Viktor Kuhlmann [arXiv:1206.0711 in June 2012].

Title: Measures and metrics in o-minimal fields

Abstract: Let $R$ be an o-minimal field. In [MS], the authors defined a measure on $R$-definable sets, which has the property that a set $X\subseteq R^n$ is assigned positive measure iff the interior of $X$ is non-empty. We shall discuss some consequences of [MS] that will be essential in establishing basic properties of Hausdorff measure of metric spaces definable in o-minimal expansions of the real field, and which will be presented in a subsequent talk by E. Walsberg.

[MS] J. Maříková, M. Shiota Measuring definable sets in o-minimal fields, to appear in Israel J. Math.

Title: Normal forms for log-power transseries.

Abstract: Given a formal power series in one variable, classical methods allow to find a normal form and to embed it in the flow of a formal vector field. We extend these results to formal transseries involving real powers and logarithms, and give some applications in fractal analysis.

Title: Lebesgue integration of oscillating and subanalytic functions

Abstract: We consider the algebra generated by all global subanalytic functions, their logarithms and the functions $\exp (if)$, where $f$ is subanalytic. We aim to understand the nature of the parametric integrals of the functions in the algebra. Is this family stable under integration? What is the nature of the locus of integrability? We show that the answer to these questions can be obtained by adding certain "transcendental" functions to the original algebra. Joint work with R. Cluckers, G. Comte, D. Miller and J.-P. Rolin.

Title: Holomorphic extensions of functions definable in $\mathbb{R}_{an, exp}$

P. Speissegger's blog posts on this topic

Abstract: Allowing for compositional iterates of $\log$ in asymptotic expansions, Ilyashenko's construction of the quasianalytic class of almost regular maps can be extended to obtain a quasianalytic algebra. If we want to extend this construction to several variables, we need to adapt it first to series using any function definable in $\mathbb{R}_{an, exp}$ as monomial. After a quick introduction to the problem, I will discuss the holomorphic extension properties of these definable functions required for this construction. Joint work with Tobias Kaiser.

Title: The fine structure of $\mathbb{R}_{an, exp}$-definable unary functions of exponential growth

Abstract: I consider three convex subrings of the Hardy field of germs of $\mathbb{R}_{an, exp}$-definable unary functions, namely the (germs determined by) functions of polynomial growth, subexponential growth and exponential growth. In each case I give an explicit description of the residue fields. I shall also discuss to what extent these functions have complex analytic extensions to a right half plane in $\mathbb{C}$.

Title: Smooth parametrizations of analytic curves, and Remez-type inequalities

Abstract: Smooth parametrization consists in a subdivision of mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the talk is to provide some new results on analytic parametrization of the graphs $G_{f} = \{y = f(z)\}$ of meromorphic functions $f(z)$, stressing the uniformity of the bounds on the number of analytic charts required in the parametrization.

These results turn out to be very closely connected with "Remez-type" (or "Norming") inequalities for polynomials $P(x,y)$ of degree $d$, restricted to analytic curves $G_{f}$. Such inequalities provide a bound on the maximum of $|P|$ on, say, the part of $G_{f}$ inside the unit ball, through the maximum of $|P|$ on a certain subset $Z$ of $G_{f}$. These inequalities are, in their turn, closely related to counting zeroes of $P$ restricted to $G_{f}$ (Bezout-like theorems).

## Contributed Talk Abstracts

Title: Unlikely Intersections in certain families of abelian varieties and the polynomial Pell equation

Abstract: The classical Pell's equation $X^{2}-DY^{2}=1$ to be solved in integers has a natural analogue for function fields, where $D=D(t)$ is a complex polynomial of positive even degree and we seek solutions in nonzero complex polynomials $X(t)$ and $Y(t)$. Here, solvability is no longer guaranteed by the non-squareness of $D$ and may be instead considered "exceptional". We will let $D=D_{\lambda}(t)$ vary in a pencil. In this context, Masser and Zannier proved that, if $D=t^{6}+t+\lambda$, there are at most finitely many lambda such that there is a nontrivial solution.

In a joint work with F. Barroero, we considered the so called "almost" Pell's equation, namely $X^2-DY^2=f$, where $f$ is a fixed complex polynomial, and proved the analogue finiteness result if $\deg f$ is at most two. This is related to questions about linear relations between points on Jacobians of genus two curves and to the more general setting of unlikely intersections. One of the main ingredients is a refinement of Pila-Wilkie counting theorem to estimate the number of points on some subanalytic surfaces lying on certain linear affine varieties defined by equations with rational coefficients of bounded height.

Title: Heights of pre-special points of Shimura varieties

Abstract: In this talk we discuss a bound for the height of the pre-image of a special point on a Shimura variety in a fundamental set of the associated Hermitian symmetric domain. This bound is polynomial in terms of standard invariants associated with the corresponding Mumford-Tate torus and generalises an earlier result of Pila and Tsimerman for the moduli space $\mathcal{A}_g$ of principally polarised abelian varieties of dimension $g$. The result constitutes the penultimate ingredient needed to complete a new proof of the André-Oort conjecture, using a strategy of Pila and Zannier (the final ingredient being lower bounds for the sizes of Galois orbits of special points, recently obtained for $\mathcal{A}_g$ by Tsimerman using new developments on the Colmez conjecture). This is joint work with Martin Orr (University College London).

Title: The Asymptotic Behaviour of the Riemann Mapping Function at Analytic Cusps

Abstract: The Riemann Mapping Theorem gives the existence of a conformal mapping of a simply connected proper domain of the complex plane onto the upper half plane. One of the main topics in geometric function theory is to investigate the behaviour of such mapping functions at the boundary. The first part of the talk is about the asymptotic behaviour at analytic corners with opening angle greater than $0$. In this case, the mapping function can be developed in a generalized power series. The second part of the talk is about the asymptotic behaviour of the mapping function at an analytic cusp. A simply connected domain has an analytic cusp if the boundary is locally given by two analytic curves such that the interior angle vanishes. Besides the asymptotic behaviour of the mapping function, the behaviour of its derivatives, its inverse, the derivatives of the inverse, and connections to o-minimality are discussed.

Title: Transserial derivations on the surreal numbers, part 2

Abstract: This is a sequel to the talk by Alessandro Berarducci. I will discuss some details of the techniques used in our work and some applications regarding composition of transseries.

Title: The Brownian Motion in o-minimal structures

Abstract: The Brownian Motion is a stochastic process $(B_t)_{t\geq 0}$ which describes random movements at time $t$. The probability of $B_t \in A$ for a Borel set $A\subset \mathbb{R}^n$ and a start value $z$ is given by \begin{eqnarray*} P_z(B_t\in A)= \begin{cases} \delta_z(A),& t=0 \\ \frac{1}{(2\pi t)^{\frac{n}{2}}}\int_{A} e^{\frac{-|y-z|^2}{2t}} dy, & t>0. \end{cases} \end{eqnarray*} We are interested whether the function above is definable in o-minimal structures for a semialgebraic family of sets $A\subset \mathbb{R}^m \times \mathbb{R}^n$. In case $n=1$ we obtain o-minimality. In higher dimensions we get with a result of Lion and Rolin that it is definable in an o-minimal structure for $t\rightarrow \infty$. For $t\rightarrow 0$ we can establish asymptotics.

Title: Measures and metrics in o-minimal fields II

Slides from E. Walsberg's talk

Abstract: I will discuss metric spaces which are definable in o-minimal expansions of the real field. The Hausdorff dimension of a definable metric space is a definable function of its defining parameters, and the Hausdorff dimension of a definable metric space is an element of the field of powers of the o-minimal structure. This is proven using basic results from my thesis and work of Maříková and Shiota on measure theory over nonarchimedean o-minimal structures. This talk is a continuation of Jana Maříková's.