Related papers: Combinatorial geometry takes the lead
This is a popular article about the work of Hugo Duminil-Copin, 2022 Fields medalist.
This is a popular article about the work of Maryna Viazovska, 2022 Fields medalist.
This is a survey of recent developments in combinatorics. The goal is to give a big picture of its many interactions with other areas of mathematics, such as: group theory, representation theory, commutative algebra, geometry (including…
In this short review we introduce group field theory, a particular class of random tensor models, which represents nowadays one of the candidates for a fundamental theory of quantum gravity. We insist on the combinatorial richness of…
This is a survey on coarse geometry with an emphasis on coarse homology theories.
About twenty years ago, Green wrote a survey article on the utility of looking at toy versions over finite fields of problems in additive combinatorics. This article was extremely influential, and the rapid development of additive…
This is a survey paper about a selection of results in complex algebraic geometry that appeared in the recent and less recent litterature, and in which rational homogeneous spaces play a prominent r{\^o}le. This selection is largely…
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite…
We prove a precise version of a general conjecture on the polar degree stated by June Huh. We confirm Huh's conjectural list of all projective hypersurfaces with isolated singularities and polar degree equal to 2.
This article is an account of the scientific work of Hugo Duminil-Copin at the time of his award in 2022 of the Fields Medal "for solving longstanding problems in the probabilistic theory of phase transitions in statistical physics,…
The past decade has seen tremendous progress in our understanding of the behaviour of many probabilistic models at or near their "critical point". On the 5th of July 2022, Hugo Duminil-Copin was awarded the Fields medal for the crucial role…
A survey of new geometric flows motivated by string theories is provided. Their settings can range from complex geometry to almost-complex geometry to symplectic geometry. From the PDE viewpoint, many of them can be viewed as intermediate…
Heffter arrays were introduced by Archdeacon in 2015 as an interesting link between combinatorial designs and topological graph theory. Since the initial paper on this topic, there has been a good deal of interest in Heffter arrays as well…
In this article, we treat G_2-geometry as a special case of multisymplectic geometry and make a number of remarks regarding Hamiltonian multivector fields and Hamiltonian differential forms on manifolds with an integrable G_2-structure; in…
In recent years, the intersection of algebra, geometry, and combinatorics with particle physics and cosmology has led to significant advances. Central to this progress is the twofold formulation of the study of particle interactions and…
These notes follow my articles [1, 6], and give some new important details. We propose here a new combinatorial method of encoding of measure spaces with measure preserving transformations, (or groups of transformations) in order to give…
There is no field with only one element, yet there is a well-defined notion of what projective geometry over such a field means. This notion is familiar to experts and plays an interesting role behind the scenes in combinatorics and…
This is my laudation for Scholze's Fields medal 2018.
Recent uses of differential geometry in materials science are reviewed here, in particular the September issue of the Phil. Trans. Royal Soc., entitled ``Curvature and chemical Structure.''
In this article we are introducing combinatorial spectra of graphs, this is a generalization of $H$-Hamiltonian spectra. The main motivation was to made from $H$-Hamiltonian spectra an operation and develop some algebra in this field. An…