Author Archive: raphy

The topological KKMS Theorem is a powerful extension of the Brouwer’s Fixed-Point Theorem, which was proved by Shapley in 1973 in the context of game theory. We prove a colorful and polytopal generalization of the KKMS Theorem, and show that…
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The aim of this talk is to present an elementary approach to Fermat’s last theorem. We translate the condition of (x,y,z) being solution of Fermat’s equation x^n+y^n=z^n to a robust system of p-adic recursion relations (to which we refer as…
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Over the last 15 years, it has been noted that many combinatorial structures, such as real and complex hyperplane arrangements, interval greedoids, matroids, oriented matroids, and others have the structure of a left regular band, a certain kind of finite monoid….
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We give two simple generalizations of commutative rings. They form (co)-complete categories, that contain commutative (semi-) rings (e.g. f0; 1g ⊆ [0; 1] ⊆ [0; 1) with the usual multiplication x + y := maxfx; yg). But they also contains…
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The first part of the talk will be an introduction to geometric structures in the sense of Thurston. We will also review a bit of projective geometry, and take a virtual tour with computer visualizations through some interesting types of…
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The Kirchhoff and Boltzmann distributions date back to the nineteenth century, dealing respectively with the flow of electrical current in a circuit and with the distribution of particles in a gas. I will show how both can be derived as…
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Let N be a normal subgroup of a group G. When does a homomorphism from N to an abelian group A extend to a homomorphism from G to A?  When does an irreducible (projective) representation f of N extend to…
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The problem of bounding the number of rational or algebraic points of a given height in a transcendental set has a long history. In 2006 Pila and Wilkie made fundamental progress in this area by establishing a sub-polynomial asymptotic estimate…
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Let H be a self-adjoint operator defined on an infinite dimensional Hilbert space. Given some spectral information about H, such as the continuity of its spectral measure, what can be said about the asymptotic spectral properties of its finite dimensional…
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Non-Archimedean analytic spaces are analogues of complex manifolds when replacing the complex numbers by a non-Archimedean field, such as p-adic numbers or complex Laurent series. I will give several examples of situations involving degenerations in complex analysis and geometry that…
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