Related papers: The Manin conjecture in dimension 2

In this talk I review our present knowledge on neutrino masses and mixing as well as the expectations from near future experiments.

Lecture notes on selected topics in the theory of gravitation.

In this paper we give an elementary proof of the local sum conjecture in two dimensions. In a remarkable paper [CMN, arXiv:1810.11340], this conjecture has been established in all dimensions using sophisticated, powerful techniques from a…

Manin's conjecture predicts the distribution of rational points on Fano varieties. Using explicit parameterizations of rational points by integral points on universal torsors and lattice-point-counting techniques, it was proved for several…

In this paper we consider the remaining cases of Hebey-Vaugon conjecture.

We give a proof of some small weight and level cases of Serre's conjecture.

We prove Manin's conjecture for split smooth quintic del Pezzo surfaces over arbitrary number fields with respect to fairly general anticanonical height functions. After passing to universal torsors, we first show that we may restrict the…

In this paper I verify Manin's conjecture for a class of rational projective toric varieties with a large class of heights other than the usual one that comes from the standard metric on projective space.

For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q). We use this to establish the Manin conjecture for a singular del Pezzo surface of degree 2 defined…

We survey recent developments towards a proof of the Penrose conjecture and results on Penrose-type and other geometric inequalities for quasi-local masses in general relativity.

We discuss various recent advances on weak forms of the Twin Prime Conjecture.

For an optimal modular parametrization $J_0(n) \twoheadrightarrow E$ of an elliptic curve $E$ over $\mathbb{Q}$ of conductor $n$, Manin conjectured the agreement of two natural $\mathbb{Z}$-lattices in the $\mathbb{Q}$-vector space $H^0(E,…

Gamma conjecture I and the underlying Conjecture $\mathcal{O}$ for Fano manifolds were proposed by Galkin, Golyshev and Iritani recently. We show that both conjectures hold for all two-dimensional Fano manifolds. We prove Conjecture…

We study the space of rational curves on del Pezzo surfaces in positive characteristic. For most primes p we prove the irreducibility of the moduli space of rational curves of a given nef class, extending results of Testa in characteristic…

These are lecture notes on string theory at Fudan University.

In this paper, we give a survey of the recent develpoments of the DDVV conjecture.

We review the current status of dimensions, as the result of a long and controversial history that includes input from philosophy and physics. Our conclusion is that they are subjective but essential concepts which provide a kind of…

We classify generically transitive actions of semidirect products of an additive and a multiplicative group on the projective plane. Motivated by the program to study the distribution of rational points on del Pezzo surfaces (Manin's…

We give some results and conjectures about recurrence relations for certain sequences of binomial sums.

In this paper we give an asymptotic formula for the quantity of diagonal del Pezzo surfaces of degree 2 which have a Brauer-Manin obstruction to the Hasse principle when ordered by height.