Related papers: A really simple elementary proof of the uniform bo…
We present a simple extension of Lindeberg's argument for the Central Limit Theorem to get a general invariance result. We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields.
We give a pen and paper and (comparatively) much simpler proof to verify of the Four Colour Theorem.
We explore a definition of uniformity on noncompact manifolds that does not require a Riemannian metric, but is equivalent to bounded gemetry. These are unfinished research notes (and will likely never be published), but since they were…
Herbrand's Theorem is a fundamental result in mathematical logic which provides a reduction of first-order formulas satisfied by a universal class to formulas free of existential quantifiers. In this work, a simpler and self-contained…
We provide proofs for the fact that certain orders have no descending chains and no antichains.
This paper presents a short and simple proof of the Four-Color Theorem that can be utterly checkable by human mathematicians, without computer assistance. The new key idea that has allowed it and the global structure of the proof are…
Let A be a basic connected finite dimensional algebra over an algebraically closed field k. Assuming that A is monomial and that the ordinary quiver Q of A has no oriented cycle and no multiple arrows, we prove that A admits a universal…
A short and elementary proof is given of a celebrated eigenvalue-perturbation result due to Alfred Brauer.
We prove the boundedness of $n$-complements for surface pairs in a generalized case without restrictions on multiplicities or the Fano type assumption.
Boolos's proof of incompleteness is extended straightforwardly to yield simple ``diagonalization-free'' proofs of some classical limitative theorems of logic.
We give a simple proof of MacLane's algebraic planarity criterion for graphs. This proof does not use any other known planarity criteria. Keywords: graph, planarity, cycle space, a simple basis of a graph.
We give an elementary, self-contained and quick proof of Belyi's theorem. As a by-product of our proof we obtain an explicit bound for the degree of the defining number field of a Belyi surface.
We give effective bounds for the uniformity of the Iitaka fibration. These bounds follow from an effective theorem on the birationality of some adjoint linear series. In particular we derive an effective version of the main theorem in [17].
A proof of the uniformization theorem of Riemann surface is given with only elementary properties of holomorphic functions and not using the paracompacity of the surface. This proof leans on an holomorphic version of the topological…
For families of continuous plurisubharmonic functions we show that, in a local sense, separately bounded above implies bounded above.
The "unit theorem" to which the present mini-course is devoted is a theorem from algebra that has a combinatorial flavour, and that originated in fact from algebraic combinatorics. Beyond a proof, the course also addresses applications, one…
A very short proof of the Fej\'er-Riesz lemma is presented in the matrix case
I give a simpler proof of the generalisation of Engel's Theorem to Leibniz algebras.
We prove that for a non-isotrivial abelian scheme over a smooth curve, the genus of a generic sequence of multi-sections with small heights tends to infinity. As an application, we give a new proof of the uniform boundedness of…
Pisier's inequality is central in the study of normed spaces and has important applications in geometry. We provide an elementary proof of this inequality, which avoids some non-constructive steps from previous proofs. Our goal is to make…