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The purpose of this article is to give an interpretation of real projective structures and associated cohomology classes in terms of connections, sections, etc. satisfying elliptic partial differential equations in the spirit of Hodge…
In two recent papers, we described some Siegel modular threefolds which admit a weak Calabi--Yau model. Not all of them admit a {\it projective} model. The purpose of this paper is to exhibit criterions for the projectivity, to treat…
We discuss the recently developed method of refined absorption and how it is used to provide a new proof of the Existence Conjecture for combinatorial designs. This method can also be applied to resolve open problems in extremal and…
In this paper we study plus-one generated arrangements of conics and lines in the complex projective plane with simple singularities. We provide several degree-wise classification results that allow us to construct explicit examples of such…
We show that every tiling of a convex set in the Euclidean plane $\mathbb{R}^2$ by equilateral triangles of mutually different sizes contains arbitrarily small tiles. The proof is purely elementary up to the discussion of one family of…
In this work we show that the only White surface in the projective 5-space having an excess of trisecant lines is the polygonal surface constructed by C. Segre. The proof follows the line of B.Gambier's beautiful approach to this question…
Throughout this paper we study the existence of irreducible curves C on smooth projective surfaces S with singular points of prescribed topological types S_1,...,S_r. There are necessary conditions for the existence of the type \sum_{i=1}^r…
Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow…
We consider the problem of constructing triangulations of projective planes over Hurwitz algebras with minimal numbers of vertices. We observe that the numbers of faces of each dimension must be equal to the dimensions of certain…
We give a brief proof of a recent result of Avron, Seiler and Simon.
The group of piecewise projective homeomorphisms of the line provides straightforward counter-examples to the so-called von Neumann conjecture. The examples are so simple that many additional properties can be established.
In this article we prove the topological minimality of unions of several almost orthogonal planes of arbitrary dimensions. A particular case was proved in arXiv:1103.1468, where we proved the Almgren minimality (which is a weaker property…
We present a Geometric Invariant Theory (GIT) construction which allows us to construct good projective degenerations of Hilbert schemes of points for simple degenerations. A comparison with the construction of Li and Wu shows that our GIT…
This paper studies how spatial thinking interacts with simplicity in [informal] proof, by analysing a set of example proofs mainly concerned with Ferrers diagrams (visual representations of partitions of integers, and comparing them to…
We provide a simple and efficient algorithm for computing the Euclidean projection of a point onto the capped simplex---a simplex with an additional uniform bound on each coordinate---together with an elementary proof. Both the MATLAB and…
A fundamental question is whether Turing machines can model all reasoning processes. We introduce an existence principle stating that the perception of the physical existence of any Turing program can serve as a physical causation for the…
This paper is devoted to construct a minimal toric embedded resolution of a rational singularity via jet schemes. The minimality is reached by extending the concept of the profile of a simplicial cone given in 6.
From a commutative associative algebra $A$, the infinite dimensional unital 3-Lie Poisson algebra~$\mathfrak{L}$~is constructed, which is also a canonical Nambu 3-Lie algebra, and the structure of $\mathfrak{L}$ is discussed. It is proved…
We prove that many seemingly simple theories have Borel complete reducts. Specifically, if a countable theory has uncountably many complete 1-types, then it has a Borel complete reduct. Similarly, if $Th(M)$ is not small, then $M^{eq}$ has…
A new large-cardinal property is introduced which enables one to give a relative consistency proof of restricted versions of the reflection principles discussed by Tait in his essay "Constructing Cardinals from Below".