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We show the map $\sigma : T_g \to C_g$ sending a compact hyperbolic surface $X$ to a random simple closed geodesic on $X$ determines a proper embedding of Teichm\"uller space into the space of geodesic currents. The proof depends on a…

Contour integrals of rational functions over ${\cal M}_{0,n}$, the moduli space of $n$-punctured spheres, have recently appeared at the core of the tree-level S-matrix of massless particles in arbitrary dimensions. The contour is determined…

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$…

Topological quantum field theories with gauge group $\textrm{SU}_2$ associate to each surface with marked points $\Sigma$ and each integer $r>0$ a vector space $V_r (\Sigma)$ and to each simple closed curve $\gamma$ in $\Sigma$ an Hermitian…

We describe algebraic curves $ X : F(x, y) = 0 $ defined over $\overline{\mathbb{Q}}$ that satisfy the following property: there exist a number field $k$ and an infinite set $S \subset k$ such that, for every $y \in S$, the roots of the…

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using…

Let $G$ be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of $G$, subject to a local assumption at one place, stronger than supercuspidality, and assuming the…

Let $k$ be a perfect field and let $C_0:f=0$ be a smooth curve in the torus $\mathbb{G}_{m,k}^2$. Let $\mathbb{T}_\Delta$ be the toric variety associated to the Newton polygon of $f$. Extending the toric resolution of $C_0$ on…

Let C be an algebraic curve in a power of an elliptic curve, both defined over the algebraic numbers. We show that the set of algebraic points of C which satisfy certain conditions is a finite set. This result has implications with the…

Let SU_C(2) be the moduli space of rank 2 semistable vector bundles with trivial de terminant on a smooth complex algebraic curve C of genus g > 1, we assume C non-hyperellptic if g > 2. In this paper we construct large families of pointed…

We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and…

In this note, we will show that Bogomolov conjecture holds for a non-isotrivial curve of genus 2 over a function field.

Godel's theory T can be understood as a theory of the simply-typed lambda calculus that is extended to include the constant 0, the successor function S, and the operator R_tau for primitive recursion on objects of type tau. It is known that…

We formulate a conjecture on the behavior of the minimal free resolutions of sets of general points on arbitrary varieties embedded by complete linear series, in analogy with the well-known Minimal Resolution Conjecture for points in…

Given a function f in the exterior of a convex curve in the real plane, we prove that if the restrictions of f to the tangent lines to the curve extend as entire functions, then the function f is an entire function of two variables.

Let $C_1,C_2\subseteq\mathbb{G}_m^N(\mathbb{C})$ be irreducible closed algebraic curves, with $N\geq 3$. Suppose $C_1$ is not contained in an algebraic subgroup of $\mathbb{G}_m^N(\mathbb{C})$ of dimension $1$ and $C_1\cup C_2$ is not…

Let $K$ be an algebraically closed field of characteristic $2$, $G$ be the algebraic group $\mathrm{SL}_2$ over $K$, and $V$ be the natural representation of $G$. Let $b_k^{G,V}$ denote the number of $G$-indecomposable factors of…

Let $\mathcal{C}$ be a representable 2-category, and $\mathfrak{T}_\bullet$ a 2-endofunctor of the arrow 2-category $\mathcal{C}^\downarrow$ such that (i) $\mathsf{cod} \mathfrak{T}_\bullet = \mathsf{cod}$ and (ii) $\mathfrak{T}_\bullet$…

Projective structures on topological surfaces support the structure of 2d CFTs with a degree of technical simplification. We propose a complex analytic space $\mathcal{P}_g$ biholomorphic to $T^*_{(1,0)} \mathcal{M}_g$ as a candidate moduli…

We state and check the analogue of Faber's conjectures for the tautological ring of the moduli spaces $M_{g,n}$ of $n$-pointed smooth curves of genus $g$.