Related papers: Recent developments of the Uniform Mordell-Lang Co…

We prove some cases of the Zilber-Pink conjecture on unlikely intersections in Shimura varieties. Firstly, we prove that the Zilber-Pink conjecture holds for intersections between a curve and the union of the Hecke translates of a fixed…

One of the aims of this article is to provide a class of polynomial mappings for which the Jacobian conjecture is true. Also, we state and prove several global univalence theorems and present a couple of applications of them.

The Beilinson-Bloch type conjectures predict that the low degree rational Chow groups of intersections of quadrics are one dimensional. This conjecture was proved by Otwinowska. Making use of homological projective duality and the recent…

Nous montrons qu'un raffinement du th\'eor\`eme de Siegel sur les points entiers de courbes alg\'ebriques impliquerait la conjecture abc de Masser-Oesterl\'e. Nous formulons une hypoth\`ese "Siegel uniforme" qui est une majoration de la…

In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of…

We give a proof of the Zilber--Pink conjecture for $n$-fold self-products of a curve $X$ inside the self-product of its Jacobian, when $X$ has appropriate bad reduction, its Jacobian has no extra endomorphisms, and $n$ is sufficiently…

The Razumov-Stroganov conjecture relates the ground-state coefficients in the even-length dense O(1) loop model to the enumeration of fully-packed loop configuration on the square, with alternating boundary conditions, refined according to…

Any counterexample to the two-dimensional Jacobian Conjecture gives a rational map from one projective plane to another. We use some ideas of the Minimal Model Program to study the combinatorial structure of a rational surface, that is…

Using Jack polynomials, Goulden and Jackson have introduced a one parameter deformation $\tau_b$ of the generating series of bipartite maps, which generalizes the partition function of $\beta$-ensembles of random matrices. The Matching-Jack…

We present here some conjectures on the diagonalizability of uniform principal bundles on rational homogeneous spaces, that are natural extensions of classical theorems on uniform vector bundles on the projective space, and study the…

We first give a new proof and also a new formulation for the Abhyankar-Gurjar inversion formula for formal maps of affine spaces. We then use the reformulated Abhyankar-Gurjar formula to give a more straightforward proof for the equivalence…

In 2007, Dmytrenko, Lazebnik and Williford posed two related conjectures about polynomials over finite fields. Conjecture~1 is a claim about the uniqueness of certain monomial graphs. Conjecture~2, which implies Conjecture~1, deals with…

We will give a pure combinatorial proof of the Eisenbud-Goto conjecture for arbitrary monomial curves. Moreover, we will show that the conjecture holds for certain simplicial affine semigroup rings.

In the 1970's, George Mackey pointed out an analogy that exists between tempered representations of semisimple Lie groups and unitary representations of associated semidirect product Lie groups. More recently, Nigel Higson refined Mackey's…

We develop a black-box theory, which can be used to show that a sequence of Gibbsian line ensembles is tight, provided that the one-point marginals of the lowest labeled curves of the ensembles are tight and globally approximate an inverted…

We prove the (LC) conjecture of Hochster and Huneke in some non-trivial cases. This has several applications. Recently, Brenner and Caminata answered a numerical evidence due to Dao and Smirnov on the shape of generalized Hilbert-Kunz…

This article generalises to K\"ahler orbifolds general results on uniformisation of compact K\"ahler manifolds such as the Shafarevich conjecture for linear fundamental groups.

In this paper, we prove the geometric Bombieri-Lang conjecture for projective varieties which have finite morphisms to abelian varieties of trivial traces over function fields of characteristic 0. The proof is based on the idea of…

This paper is part of the general project of proof mining, developed by Kohlenbach. By "proof mining" we mean the logical analysis of mathematical proofs with the aim of extracting new numerically relevant information hidden in the proofs.…

In the early 1970s, Andrew Ogg made several conjectures about the rational torsion points of elliptic curves over $\mathbb{Q}$ and the Jacobians of modular curves. These conjectures were proved shortly after by Barry Mazur as a consequence…