Related papers: Another Simple Proof for the Existence of the Smal…

We give three constructions of a vertex-minimal triangulation of $4$-dimensional real projective space $\mathbb{R}P^4$. The first construction describes a $4$-dimensional sphere on $32$ vertices, which is a double cover of a triangulated…

We codify a short self-contained proof of the existence of $K_q^r$-absorbers implicit in Keevash's original proof of the Existence Conjecture. Combining this with the work of the first and third authors in yields a proof of the Existence…

We give a $K$-theoretic account of the basic properties of Witt vectors. Along the way we re-prove basic properties of the little-known Witt vector norm, give a characterization of Witt vectors in terms of algebraic $K$-theory, and a…

We establish new and different kinds of proofs of properties that arise due to the orthogonal decomposition of the Hilbert space, including projections, over the unit interval of one dimension. We also see angles between functions,…

We prove the High Girth Existence Conjecture - the common generalization of the Existence Conjecture for Combinatorial Designs originating from the 1800s and Erd\H{o}s' Conjecture from 1973 on the Existence of High Girth Steiner Triple…

Using techniques from the fields of symbolic computation and satisfiability checking we verify one of the cases used in the landmark result that projective planes of order ten do not exist. In particular, we show that there exist no…

First we refine the duality theory for Witt divisorial sheaves on smooth projective varieties over a perfect field of positive characteristic. Building on previous work [Lem22], we remove the residual derived limit to obtain a cleaner…

We give a simple, short and self-contained presentation of Bourgain's discretised projection theorem from 2010, which is a fundamental tool in many recent breakthroughs in geometric measure theory, harmonic analysis, and homogeneous…

We prove a second main theorem for elliptic projective planes.

Existence theorem is proven for the generating equations of the split involution constraint algebra. The structure of the general solution is established, and the characteristic arbitrariness in generating functions is described.

Wedderburn's theorem on the structure of finite dimensional semisimple algebras is proved by using minimal prerequisites.

The Stable Reduction Theorem guarantees that any smooth, projective, geometrically irreducible curve of genus $g \geq 2$ over a discretely valued field admits a unique stable model after a finite field extension. Computing this model is a…

This is an elementary geometrical proof of Birkhoff theorem. It is hardly important, but the pictures behind are quite nice.

We prove the rationality of a $\k$-form $X$ of the product $S$ of projective spaces provided the existence of a $\k$-point on $X$. The method of the proof is to find a Galois-invariant birational projection of $S$ to the projective space.…

We study minimally Terracini finite sets of points in the projective plane and we prove that the sequence of the cardinalities of minimally Terracini sets can have any number of gaps for degree great enough.

Results on $8$-dimensional topological planes are scattered in the literature. It is the aim of the present paper to give a survey of these geometries, in particular of information obtained after the appearance of the treatise Compact…

We give a short proof of a theorem of J.-E. Pin (theorem 1.1 below), which can be found in his thesis. The part of the proof which is my own (not Pin's) is a complete replacement of the same part in an earlier version of this paper.

In this article, we establish the existence of a good minimal model for a compact K\"ahler klt pair $(X, B)$ when the Albanese map of $X$ is a projective morphism and the general fiber of $(X, B)$ has a good minimal model.

A point set $\mathrm X_N$ on the unit sphere is a spherical $t$-design is equivalent to the nonnegative quantity $A_{N,t+1}$ vanished. We show that if $\mathrm X_N$ is a stationary point set of $A_{N,t+1}$ and the minimal singular value of…

We give a new simple geometric proof that any seven points in the plane have four Tverberg partitions into three sets. This is the only confirmed non-trivial case of Sierksma's conjecture. Earlier proofs, by Stephan Hell, relied on…