This talk will be based on the well-studied problem of finding asymptotic bounds for the density of rational (or algebraic) points of bounded height on transcendental sets. The guiding philosophy here is that such sets should contain (in a quantifiable way) "few" such points. After pointing out the connections with model theory, we will focus on the particular instances where the sets in question are graphs of transcendental meromorphic functions. We adopted a Nevanlinna theoretic approach and obtained the polylogarithmic bound predicted by (the analogue of) Wilkie's conjecture in this setting.

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The interaction between model theory and diophantine geometry began with a new proof, by Pila and Zannier, of the Manin-Mumford conjecture. And this led to a breakthrough with Pila's proof of the André-Oort conjecture for products of modular curves. In a related direction, Jones, Thomas and Wilkie[2012] applied improvements of the Pila-Wilkie theorem for certain curves to prove results on integer-valued functions, that is, functions $f$ such that $f(n)$ is an integer for integer points in the domain of $f$ definable in the real exponential field. This gives a version of a 100 year old theorem due to Pólya, but with complex functions replaced by real functions.

Theorem [Pólya]
If $f:\mathbb{C} \to \mathbb{C}$ is entire with $f(\mathbb{N})\in\mathbb{Z}$ and $|f(z)|\leqslant dC^{|z|}$ with real $d$ and $C<2$, then $f$ is a polynomial.

More recently, Wilkie[2016] proved an almost exact analogue of Pólya's theorem in $\mathbb{R}_{an,\exp}$. This talk will show how to combine Wilkie's ideas with techniques from transcendental number theory and o-minimality in order to establish Pólya-type theorems in which the function is definable in o-minimal expansion of real ordered field and only assumed to be integer valued on a certain sequence of natural numbers. If time permits, I will also introduce some results about several variables and then mention further research that can be done in the future.

A surface germ at the origin of $R^n$ is normally embedded if its inner and outer metrics induced from $R^n$ are equivalent. A normally embedded $\beta$-Hölder triangle is the simplest building block in the Lipschitz geometry of real semialgebraic (or, more general, definable in a polynomially bounded o-minimal structure over $R$) surface germs. We study the next simplest case: a surface germ that is the union of two normally embedded $\beta$-Hölder triangles. We show that the outer Lipschitz geometry of such surface germs may be surprisingly nontrivial, and define discrete (piecewise constant in definable families) invariants of the outer Lipschitz equivalence classes of these surfaces.

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Let $f\in \mathbb{Z}[x_1,...,x_n]$ be a non-constant polynomial. Let $p$ be a prime number and $m$ be a positive integer. We associate to $f,p,m$ the exponential sum $$E_{f}(p,m):=\frac{1}{p^{mn}}\sum_{x\in(\mathbb{Z}/p^m\mathbb{Z})^n}\exp(2\pi if(x)/p^m).$$ Let $\sigma$ be a positive real number. Suppose that for each prime number $p$, there is a positive constant $c_p$ such that $$|E_{f}(p,m)|\leq c_pp^{-m\sigma}$$ for all $m\geq 2$. Igusa's conjecture for exponential sums predicts that one can take $c_p$ independent of $p$ in the above inequality. This conjecture relates to the existence of a certain adèlic Poisson summation formula and the estimation of the major arcs in the Hardy-Littlewood circle method towards the Hasse principle of $f$.

In this talk, I will recall Igusa's conjecture for exponential sums and discuss some new progress and open questions relating this conjecture to the singularities of the hypersurface defined by $f$.

This talk is based on recent joint work with Wim Veys and with Raf Cluckers