Related papers: Degrees bounding principles and universal instance…

As is well known, the Lefschetz theorems for the \'etale fundamental group of SGA1 do not hold. We fill a small gap in the literature showing they do for tame coverings. Let $X$ be a regular projective variety over a field $k$, and let…

Let (G, +) be an abelian group. A subset of G is sumfree if it contains no elements x, y, z such that x +y = z. We extend this concept by introducing the Schur degree of a subset of G, where Schur degree 1 corresponds to sumfree. The…

We are concerned with the problem of the stability of the syzygy bundles associated to base point free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from Mark Green's vanishing…

A complete analysis is given of the computable reductions that hold between $\mathsf{SRT}^2_2$, $\mathsf{SPT}^2_2$, and $\mathsf{SIPT}^2_2$. In particular, while $\mathsf{D}^2_2\le_{\rm sW}\mathsf{SIPT}^2_2\le_{\rm…

The prevalent interpretation of G\"odel's Second Theorem states that a sufficiently adequate and consistent theory does not prove its consistency. It is however not entirely clear how to justify this informal reading, as the formulation of…

We study a pinned variant of Bourgain's theorem, concerning the occurrence of affine copies of $k$-point patterns in $\mathbb{R}^d$. Focusing on the case $k=2$, which corresponds to pinned distances, we show that the classical conclusion…

This paper introduces new notions of asymptotic proofs, PT(polynomial-time)-extensions, PTM(polynomial-time Turing machine)-omega-consistency, etc. on formal theories of arithmetic including PA (Peano Arithmetic). This paper shows that P…

For a graph $G$, we write $G\rightarrow \big(K_{r+1},\mathcal{T}(n,D)\big)$ if every blue-red colouring of the edges of $G$ contains either a blue copy of $K_{r+1}$, or a red copy of each tree with $n$ edges and maximum degree at most $D$.…

In this note, we show various minimality results for a geometric theory of fields $T$: $T$ is stable if and only if it is strongly minimal, $T$ is simple if and only if it has SU-rank 1, and $T$ is rosy if and only if $T$ is surgical.…

For a cuspidal automorphic representation of GL2/Q associated to a modular form, the local and global Langlands correspondences are compatible at all finite places of Q. On the p-adic Coleman-Mazur eigencurve this principle can fail (away…

The principle $ADS$ asserts that every linear order on $\omega$ has an infinite ascending or descending sequence. This has been studied extensively in the reverse mathematics literature, beginning with the work of Hirschfeldt and Shore. We…

A coarse description of a subset A of omega is a subset D of omega such that the symmetric difference of A and D has asymptotic density 0. We study the extent to which noncomputable information can be effectively recovered from all coarse…

The aim of this paper is to give a proof of the restriction theorems for principal bundles with a reductive algebraic group as structure group in arbitrary characteristic. Let $G$ be a reductive algebraic group over any field $k=\bar{k}$,…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

The inverse conjecture for the Gowers norms $U^d(V)$ for finite-dimensional vector spaces $V$ over a finite field $\F$ asserts, roughly speaking, that a bounded function $f$ has large Gowers norm $\|f\|_{U^d(V)}$ if and only if it…

We investigate big Ramsey degrees of finite substructures of the universal countable homogeneous meet-tree and its binary variant. We prove that structures containing antichains have infinite big Ramsey degrees, and the big Ramsey degree of…

Let $K$ be a number field, $S$ a finite set of places. For $\mathbb{G}_m$ or an elliptic curve $E$ defined over $K$, we establish uniformity results on the number of $S$-integral torsion points relative to a non-torsion point $\beta$, as…

While every matrix admits a singular value decomposition, in which the terms are pairwise orthogonal in a strong sense, higher-order tensors typically do not admit such an orthogonal decomposition. Those that do have attracted attention…

We construct an $\ll^2$-solution (also known as a weakly low solution) to ${\mathrm{D}^2}$ within ${\mathrm{B}\Sigma^0_{3}}$ and prove the $\ll^2$-basis theorem for $\mathrm{RT}^2$ over ${\mathrm{B}\Sigma^0_{3}}$. The $\ll^2$-basis theorem…

We introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a…