Difference between revisions of "User:Eotvos"

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As a pretext to type some mathematical text, an  
 
As a pretext to type some mathematical text, an  
  
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==Historical note about development of homological algebra.==
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==Stuff I wrote (mainly in Italian)==
  
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Despite their geometric origin, homological methods slowly but firmly arose as the perfect method in order to study a great number of algebraic problems. In fact, studying "all the possible extensions turning <m>A\to B</m> into <m>0\to A\to E\to B\to 0</m>" is quite an ancient idea, but the homological point of view promoted by Cartan, Eilenberg and Mac Lane fastly brought out a lot of natural and elegant results.
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* [http://killingbuddha.altervista.org/DIDAFILES/GHEMATRIA/gedif.pdf Notes on elementary Differential Geometry] (curves, surfaces); the study of curves and surfaces culminates with Gauss' masterpiece ''Disquisitiones generales circa superficies curvas'', where he defines the concept of ''intrinsical'' geometric property. Can a small ant lying on a cylinder notice it is walking on a globally non-flat surface? And what if it was on a sphere? And what if it was on a torus?
  
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The Baer point of view, as exposed in his ''Erwiterung von Gruppen und ihren Isomorphismen'', is the following. Suppose that we fix a ''presentation'' of an abelian group <m>A</m> by generators and relations: write <m>A = F/R</m>, where <m>F</m> is a free abelian group, say generated by <m>\{e_i\}</m>, and <m>R</m> is the subgroup of relations. If <m>E</m> is any extension of <m>B</m> by <m>A</m>, then by lifting the generators of <m>A</m> to elements <m>a(e_i)</m> of <m>E</m> we get an element <m>a(r)</m> of <m>B</m> for every relation <m>r</m> in <m>R</m>.  
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* [http://killingbuddha.altervista.org/DIDAFILES/SDR/sdr.pdf Lecture notes about Riemann surfaces]; A Riemann surface can be characterized as a complex one-dimensional manifold: asking the transition functions between charts to be ''(bi)holomorphisms'' between domains of the complex line obstructs the general (even smooth) two-dimensional manifold to be a RS. Algebraic, analytical and geometrical methods work in sinergy to give a beautiful and (at least in the case of compact spaces) complete theory.
  
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Brauer thought of this as a function from the defining relations of <M>A</M> into <M>B</M>, so he called the induced homomorphism <m>a\colon R\to B</m> a relations function. Conversely, he observed that every relations function a gives rise to a factor set, and hence to an extension <m>E(a)</m>, showing that two relations functions a and a gave the same extensions if and only if there are elements <m>b_i</m> (corresponding to a function <m>b \colon F \to B</m>) so that <m>a (r) = b(r) + a(r)</m> (''ibi'', p. 394--395). Finally, collected all the extensions in a set <m>\text{Ext}(A,B)</m>, Baer observed that the formal sum <m>a + a'</m> of two relations functions defined an addition law on <m>\text{Ext}(A, B)</m>, making it into an abelian group. In his honor, we now call the extension <m>E(a + a )</m> the ''Baer sum'' of the extensions.
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* [http://killingbuddha.altervista.org/DIDAFILES/THE/tesi.pdf My (undergraduate) thesis] in '''Category Theory'''; the main result is the so-called ''Beck's monadicity theorem'': Huber proved in 1961 that any pair of adjoint functors <m>\mathcal F\dashv \mathcal G</m> gives rise to a suitable monad <m>(\mathcal G\circ\mathcal F, \mathcal G(\boldsymbol{\varepsilon}_\mathcal F),\boldsymbol\eta)</m>. Hilton rapidly conjectured that ''every monad arises from a suitable pair of adjoint functors''. This turns out to be "essentially" true, thanks to Kleisli (1965) and Eilenberg-Moore (1965), who independently proved this theorem exhibiting two ''extremal'' ways to factor the endofunctor <m>\mathcal T</m> of a given monad <m>\mathfrak M</m> in the product of two mutually adjoint functors <m>\mathcal F_\mathfrak M\dashv \mathcal G_\mathfrak M</m>: the collection of all the ways to factor <m>\mathcal T</m> can be organized to be a (rarely small) category, whose initial object is Klesli's adjunction, and whose terminal one is Eilenberg-Moore's. [...]
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Baer's presentation <M>A = F/R</M> amounted to a free resolution of <M>A</M>, and his formulas were equivalent to the modern calculation of <m>\text{Ext}(A, B)</m> as the cokernel of <m>\hom(F, B) \to \hom(R, B)</m>. But working with free resolutions was still a decade away, and using them to calculate <m>\text{Ext}(A, B)</m> was even further in the
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future, in 1941. That year, Saunders Mac Lane gave a series of lectures
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on group extensions at the University of Michigan. Most of
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the lectures concerned applications to Galois groups and class field theory, but
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Mac Lane ended with a calculation of the abelian extensions of <m>\mathbb{Z}</m> by <m>A = \mathbb{Z}[1/p]</m>. Samuel Eilenberg, who had recently emmigrated from Poland and was an Instructor at Michigan, could not attend the last lecture and asked for a private one. Eilenberg immediately noticed that the group <m>\mathbb{Z}[1/p]</m> was dual to the topological <m>p-</m>adic solenoid group <m>\Sigma</m>, which Eilenberg had been studying, and that Mac Lane's algebraic answer <m>\text{Ext}(\mathbb{Z}[1/p],\mathbb{Z})\simeq \hat{\mathbb{Z}}_p/\mathbb{Z}</m> coincided with certain homology groups calculated by Steenrod. After an all-night session, followed by several months of puzzling over this observation, they precisely figured out how <m>\text{Ext}</m> plays a role in cohomology.
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* [http://killingbuddha.altervista.org/DIDAFILES/GEOMALG1/project/funderiv.pdf A note on derived functors]; Homological Algebra firmly arose as the ''Golden Bridge'' between Algebra and Geometry. This brief note is intended to give a basic account of the formal definitions necessary to give the notion of ''derivative'' of a functor between abelian categories with "lots" of injective/projective objects. Exposition then turns to a sheaf-theoretic application of the theory, lying on the following fundamental result: ''The category <m>\text{Sh}(\mathbf C)</m> is abelian/has lots of injectives provided <m>\mathbf C</m> is
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abelian/has lots of injectives.
  
 
==Pages I wrote==
 
==Pages I wrote==
 
1. [[Gioco_di_Ehrenfeucht-Fra%C3%AFss%C3%A9]] (italian)
 
1. [[Gioco_di_Ehrenfeucht-Fra%C3%AFss%C3%A9]] (italian)

Revision as of 11:05, 2 October 2011

Before i was corrupted by the world

Someone on internet knows me as Eotvos, others know me as tetrapharmakon. However my best avatar is killing_buddha.

I am an italian mathematician living in Padua, where I was born in May 23, 1987. My scientific interests are pretty wide, I spend much time reading about category theory, universal algebra, differential geometry. I've ever been fascinated by the epistemological problem of figuration of space: I hope to find out a coherent geometrical model of Borges' Library of Babel. I chose to study Geometry, and Algebra therefore, for a simple reason. When I was still a child, I dreamed to become an artist, a painter or a sculptor. Later I discovered Mathematics, and I found out the same feelings through the infinitely malleable and ideal shape of a manifold, or the polished and perfect matter which Riemann surfaces are made of. When I have a pencil or a chalk in my hand, when I write on a blank sheet or a blackboard, when I plot a graph or I draw a commutative diagram, then I feel the same artistic sensations.

With the passing of time, Mathematics revealed me the deep and majestic identity between the shape of an object (its geometrical nature, its physical and plastic properties) and its gist (its purest essence, its algebraic and axiomatic construction). Following Klein's point of view, for example, I can perceive that what we call a geometry is nothing else than the result of applying the action of a suitable group over an appropriate set: on changing the shape of a space, we modify the relationships among the objects, not really caring about the objects themselves. Moving its early steps from a basic intuition (i.e., the identity principle), Mathematics elevates itself to a superior conception focusing its attention on the relations among a wide spectrum of entities, whereas it is important the way in which the objects at hand are related to each other and not their particular nature.



As a pretext to type some mathematical text, an

Stuff I wrote (mainly in Italian)

  • Notes on elementary Differential Geometry (curves, surfaces); the study of curves and surfaces culminates with Gauss' masterpiece Disquisitiones generales circa superficies curvas, where he defines the concept of intrinsical geometric property. Can a small ant lying on a cylinder notice it is walking on a globally non-flat surface? And what if it was on a sphere? And what if it was on a torus?
  • Lecture notes about Riemann surfaces; A Riemann surface can be characterized as a complex one-dimensional manifold: asking the transition functions between charts to be (bi)holomorphisms between domains of the complex line obstructs the general (even smooth) two-dimensional manifold to be a RS. Algebraic, analytical and geometrical methods work in sinergy to give a beautiful and (at least in the case of compact spaces) complete theory.
  • My (undergraduate) thesis in Category Theory; the main result is the so-called Beck's monadicity theorem: Huber proved in 1961 that any pair of adjoint functors <m>\mathcal F\dashv \mathcal G</m> gives rise to a suitable monad <m>(\mathcal G\circ\mathcal F, \mathcal G(\boldsymbol{\varepsilon}_\mathcal F),\boldsymbol\eta)</m>. Hilton rapidly conjectured that every monad arises from a suitable pair of adjoint functors. This turns out to be "essentially" true, thanks to Kleisli (1965) and Eilenberg-Moore (1965), who independently proved this theorem exhibiting two extremal ways to factor the endofunctor <m>\mathcal T</m> of a given monad <m>\mathfrak M</m> in the product of two mutually adjoint functors <m>\mathcal F_\mathfrak M\dashv \mathcal G_\mathfrak M</m>: the collection of all the ways to factor <m>\mathcal T</m> can be organized to be a (rarely small) category, whose initial object is Klesli's adjunction, and whose terminal one is Eilenberg-Moore's. [...]
  • A note on derived functors; Homological Algebra firmly arose as the Golden Bridge between Algebra and Geometry. This brief note is intended to give a basic account of the formal definitions necessary to give the notion of derivative of a functor between abelian categories with "lots" of injective/projective objects. Exposition then turns to a sheaf-theoretic application of the theory, lying on the following fundamental result: The category <m>\text{Sh}(\mathbf C)</m> is abelian/has lots of injectives provided <m>\mathbf C</m> is

abelian/has lots of injectives.

Pages I wrote

1. Gioco_di_Ehrenfeucht-Fraïssé (italian)

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