Related papers: Survey on the geometric Bogomolov conjecture
A survey on recent developments in (algebraic) integral geometry is given. The main focus lies on algebraic structures on the space of translation invariant valuations and applications in integral geometry.
These brief notes record our puzzles and findings surrounding Givental's recent conjecture which expresses higher genus Gromov-Witten invariants in terms of the genus-0 data. We limit our considerations to the case of a projective line,…
In the last two decades there was a lot of progress in understanding the geometry of smooth Gaussian fields. This survey aims to cover one particular line of research: the large scale behaviour of level and excursion sets and their…
This survey deals with the construction of a category of spectral triples that is compatible with the Kasparov product in $KK$-theory. These notes serve as an intuitive guide to these results, avoiding the necessary technical proofs. We…
The purpose of this book is to build up the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for the researches of arithmetic geometry in several directions.
This is a survey of some recent results concerning polynomial inequalities and polynomial approximation of functions in the complex plane. The results are achieved by the application of methods and techniques of modern geometric function…
This is a survey on the theory of height zeta functions, written on the occasion of a French-Japanese winter school, held in Miura (Kanagawa, Japan) in Jan. 2008. It does not presuppose much knowledge in algebraic geometry. The last chapter…
We present a history of the Baum-Connes conjecture, the methods involved, the current status, and the mathematics it generated.
This is a survey paper on Geometric Manin's conjecture which was proposed by Brian Lehmann and the author. We introduce Geometric Manin's conjecture (GMC) and review some recent progress on this conjecture.
We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms…
We construct fields of algebraic numbers that have the Lehmer property but not the Bogomolov property. This answers a recent implicit question of Pengo and the first author.
We prove the Bogomolov conjecture for an abelian variety A over a function field which is totally degenerate at a place v. We adapt Zhang's proof of the number field case replacing the complex analytic tools by tropical analytic geometry. A…
The umbral restyling of hypergeometric functions is shown to be a useful and efficient approach in simplifying the associated computational technicalities. In this article, the authors provide a general introduction to the umbral version of…
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
In this expository article, we survey the rapidly emerging area of random geometric simplicial complexes.
Submanifold theory is a very active vast research field which plays an important role in the development of modern differential geometry. This branch of differential geometry is still so far from being exhausted; only a small portion of an…
The purpose of this note is to prove the existence of a remarkable structure in an iterated sumset derived from a set $P$ in a Cartesian square $\mathbb{F}_p^n\times\mathbb{F}_p^n$. More precisely, we perform horizontal and vertical sums…
We describe a new approach to the notion of general hypergeometric functions
This is a survey on the ongoing development of a descriptive theory of represented spaces, which is intended as an extension of both classical and effective descriptive set theory to deal with both sets and functions between represented…
This article discusses methods of geometric analysis in general relativity, with special focus on the role of "critical surfaces" such as minimal surfaces, marginal surface, maximal surfaces and null surfaces.