Related papers: The first thirty years of Andersen-Lempert theory
Condensed mathematics, developed by Clausen and Scholze over the last few years, is a new way of studying the interplay between algebra and geometry. It replaces the concept of a topological space by a more sophisticated but better-behaved…
Extreme debris discs (EDDs) are bright and warm circumstellar dusty structures around main sequence stars. They may represent the outcome of giant collisions occuring in the terrestrial region between large planetesimals or planetary…
Despite the success of diffusion models (DMs), we still lack a thorough understanding of their latent space. To understand the latent space $\mathbf{x}_t \in \mathcal{X}$, we analyze them from a geometrical perspective. Our approach…
For $k,s\geq2$, the $s$-stable Kneser graphs are the graphs with vertex set the $k$-subsets $S$ of $\{1,\ldots,n\}$ such that the circular distance between any two elements in $S$ is at least $s$ and two vertices are adjacent if and only if…
The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real…
In 1977 L.T. Ramsey showed that any sequence in $\mathbb{Z}^2$ with bounded gaps contains arbitrarily many collinear points. Thereafter, in 1980, C. Pomerance provided a density version of this result, relaxing the condition on the sequence…
We derive precise formulas for the archimedean Euler factors occurring in certain standard Langlands $L$-functions for unitary groups. In the 1980s, Paul Garrett, as well as Ilya Piatetski-Shapiro and Stephen Rallis (independently of…
Using the works of Gervais, Harer, Hatcher and Thurston and others, we show that the mapping class group of a compact orientable surface has a presentation so that the generators are the set of all Dehn twists and the relations are…
We give several results showing that different discrete structures typically gain certain spanning substructures (in particular, Hamilton cycles) after a modest random perturbation. First, we prove that adding linearly many random edges to…
Every complex surface of Markov type, i.e.\ the variety given by $x^2 + y^2 + z^2 + Exyz - Ax - By - Cz - D = 0$, has the symplectic density property and the Hamiltonian density property. We prove a singular symplectic version of the…
Some time ago, the chiral algebra theory of Beilinson-Drinfeld was expected to play a central role in the convergence of divergence in mathematical physics of superstring theory for quantization of gauge theory and gravity. Naively, this…
Condensed mathematics, developed by Clausen and Scholze over the last few years, proposes a generalization of topology with better categorical properties. It replaces the concept of a topological space by that of a condensed set, which can…
Shear bands are narrow zones of intense shear observed during plastic deformations of metals at high strain rates. Because they often precede rupture, their study attracted attention as a mechanism of material failure. Here, we aim to…
Uchillo-Ibanez et al. introduced a topology on the sets of shape morphisms between arbitrary topological spaces in 1999. In this paper, applying a similar idea, we introduce a topology on the set of coarse shape morphisms $Sh^*(X,Y)$, for…
Archdeacon, in his seminal paper $[1]$, defined the concept of Heffter array to provide explicit constructions of biembeddings of the complete graph $K_v$ into orientable surfaces, the so-called Archdeacon embeddings, and proved that these…
We introduce the symplectic holomorphic density property and the Hamiltonian holomorphic density property together with the corresponding version of Anders\'en-Lempert theory. We establish these properties for the Calogero-Moser space…
The aim of these notes is to acquaint the reader with important objects in complex algebraic geometry: K3 surfaces and their higher-dimensional analogs, hyperk\"ahler manifolds. These manifolds are interesting from several points of view:…
AdS black holes with planar event horizon topology play a central role in AdS/CFT holography, and particularly in its applications. Generalisations of the known planar black holes can be found by considering the Plebanski--Demianski…
We systematically study the so-called auto-arc spaces. Auto-arc spaces were originally introduced by Schoutens in 2012 and later generalized and studied by the author in his PhD Thesis and subsequent work. In that aforementioned work, only…
Shape theory was founded by K.~Borsuk 50 years ago. In essence, this is spectral homotopy theory; it occupies an important place in geometric topology. The article presents the basic concepts and the most important, in our opinion, results…