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Alexis Marin

Publications and source records attributed to Alexis Marin.

7 recordsLinked to original sources
arXiv2025

Uniformisation des surfaces de Riemann

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 characterization, due to Brown, of the sphere as variety covered by two discs, a generalization of the construction of double of a Riemann surface with boundary and the arithmetic, due to Jordan, of separation in surfaces

math.CV↗
arXiv2025

C'est un petit val qui mousse de fonctions

Easy lowering local minima, after introducing valley functions on smooth manifolds gives, without gluing technicities, "M. Morse's lemma's" canonical form, moving critical values and eliminating pair of critical points theorems, reducing the last to its one dimensional case.

math.DG↗
arXiv2025

Alg\`ebre lin\'eaire

After the language of module and theirs morphisms, this short course presents matricial calculus and determinants in a commutative ring as appliction of ``remarquable identities'' in the ring of polynomials with integer coefficients with variable coefficients and second member of the ``general Gauss method'' requiring the factoriality of such polynomial rings, here obtained by a modification of Zermolo's proof of the factoriality of the integers. As third part, using the, today almost forgotten, presentation in the first edition of N. Bourbaki's Algebra of rational structures for a subfield, one gets, without central simple algebras, the baba of skewfields, and the theorems of Wederburn and Erd\"os-Kaplanski (commutativity of finite fields and dimension of the dual).

math.HO↗
arXiv2025

$\displaystyle SL(2, {\Bbb Z})$, les tresses \`a trois brins, le tore modulaire et $Aut^{+}(F_{2})$

The action of $SL(2, {\bf Z})$ on the integer torus and its quotient by central symmetry and Artin's presentation of three strings braid group $B_{3}$, produces a presentation with parabolic generators $\pmatrix{1& -1\cr 0& 1\cr}$ and $\pmatrix{1& 0\cr 1& 1\cr}$. This braided presentation describes the action of the derived group on Poincar\'e's half plane and its quotient the modular torus, just as Nielsen's theorem giving the group of direct automorphisms of the free group on two generators as semi-direct product, amalgamated on the index $2$ subgroup of the center of $B_{3}$, of inner automorphisms with $B_{3}$.

math.GR↗
arXiv2022

$SL(2,\mathbb{Z})$, les tresses \`a trois brins et le tore modulaire

The action of $SL(2,\mathbb{Z})$ on the integer torus and its quotient by central symmetry and Artin's presentation of three strings braids, produces a presentation with parabolic generators $\pmatrix{1& -1\cr 0& 1\cr}$ and $\pmatrix{1& 0\cr 1& 1\cr}$ and describes the action of the derived group on Poincar\'e's half plane.

math.GR↗
arXiv2013

Le capo du troupeau des séries de Fourier

A glimpse at $\sum{{\sin(kx)}\over{k}}$ gives a few lines exposition of Fourier series's quadratic mean convergence for square integrable functions and Dirichlet's convergence theorem of the Fourier serie of a piecewise differentiable function with integrable derivative.

math.CA↗