Difference between revisions of "User:Eotvos"

Revision as of 05:13, 24 June 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

Historical note about development of homological algebra.

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.

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>.

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.

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 future, in 1941. That year, Saunders Mac Lane gave a series of lectures on group extensions at the University of Michigan. Most of the lectures concerned applications to Galois groups and class field theory, but 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.

Pages I wrote

1. Gioco_di_Ehrenfeucht-Fraïssé (italian)