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2016 | OriginalPaper | Buchkapitel

1. Prologue

verfasst von : Letterio Gatto, Parham Salehyan

Erschienen in: Hasse-Schmidt Derivations on Grassmann Algebras

Verlag: Springer International Publishing

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Abstract

This chapter lays down a non-technical and expository pathway to two non-linear PDEs: the Korteweg–de Vries (KdV) equation, modelling solitary waves, and the Kadomtsev–Petviashvili (KP) equation, a generalization of the KdV originally motivated by applications to plasma physics. Although the chapter’s contents might not seem related to the rest of the book in a straightforward manner, the KdV and KP equations are, surprisingly, linked to a number of subjects that any mathematician has had early close encounters with. To name but a few, cubic plane curves, linear recurrence sequences, linear ODEs and (generalized) Wronskians associated with fundamental systems of solutions, exterior algebras of free modules, Plücker embeddings of finite-dimensional Grassmannians and Schubert calculus. In sum, these two equations provide us with an excuse to return to the early stages of our scholarly education and shed a new light on it. The chapter culminates with the appearance, as a kind of deus ex machina, of the bosonic expression of the vertex operators occurring in the so-called vertex algebra of free charged fermions. These operators act on a polynomial ring in infinitely many indeterminates and encode the full system of PDEs known under the name of KP hierarchy, which arises as compatibility conditions for another system of infinitely many PDEs (expressed in Lax form, see [101, 140] or [78, p. 73]).

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Nächstes Kapitel Generic Linear Recurrence Sequences
Fußnoten
1
Diederik J. Korteweg (Den Bosch, 1848–Amsterdam, 1941) and Gustav de Vries (Amsterdam, 1866–Harlem, 1934) were Dutch mathematicians, after whom the Institute of Mathematics of the University of Amsterdam is currently named.
 
2
Boris B. Kadomtsev (1928–1998) and Vladimir I. Petviashvili (1936–1993), Russian physicists
 
3
Józef Hoene-Wroński (Wolsztyn, 1776–Neuilly-sur-Seine, 1853). A Polish mathematician with multifaceted interests. He was also an inventor, a lawyer and an economist. See [122].
 
4
Julius Plücker (1801–1868), German mathematician and physicist. His formulas computing the geometric genus of a plane curves with singularities are particularly renowned.
 
5
After Hermann Günther Grassmann, great German mathematician born in Stettin in the year 1809, when the town was part of the Kingdom of Prussia. He died in 1877 in the same town, by which time it was part of the German Empire.
 
6
Hermann Cäsar Hannibal Schubert (Potsdam, 1848–Hamburg, 1911), the creator of the celebrated calculus which bears his name
 
7
Jules Henri Poincaré was one of the most famous French mathematicians, who left important contributions in nearly every field of mathematics, besides theoretical physics and philosophy of science. He was born in Nancy in 1854 and died in Paris in 1912.
 
8
Born in Verona in 1879, Giovanni Zeno Giambelli was an Italian mathematician, one of the most brilliant students of Corrado Segre. The formula bearing his name is extremely important in Schubert calculus and its variants (quantum, equivariant or for other homogeneous spaces). He died in Messina in 1953, probably without suspecting the future influence of his work. See [91] for more details on his biography.
 
9
Niels Henrik Abel (Finnøy, 1802–Froland, 1829) was a Norwegian mathematician. His name is linked to many different contributions, including the impossibility to solve the quintic equation by radicals. He died a couple of days before being appointed professor at the University of Berlin.
 
10
Joseph Liouville (Saint-Omer, 24 marzo 1809–Parigi, gave outstanding contributions in nearly all the branches of mathematics, from number theory (he provided a proof of the existence of transcendental numbers) to theoretical mechanics (Liouville tori)).
 
11
Karl Theodor Wilhelm Weierstrass (1815–1897), German mathematician often cited as the “father of modern analysis”. However, his contributions to mathematics are countless. For instance Weierstrass points are very important in the geometry of algebraic curves.
 
12
A discrete subgroup \(\varLambda\) of \(\mathbb{C}\) such that \(\dim _{\mathbb{R}}\varLambda \otimes _{\mathbb{Z}}\mathbb{R} = 2\).
 
13
A modular form is a complex-valued function f, defined on the complex upper half-plane \(\mathbb{H}\), such that f(gz) = (cz + d) k f(z), where for any \(g = \left (\begin{array}{*{10}c} a& b\\ c &d \end{array} \right ) \in Sl_{2}(\mathbb{Z})\) one defines \(gz:={ az+b \over cz+d}\). See on this the exciting survey [152].
 
14
Ferdinand Gotthold Max Eisenstein (Berlin, 1823–1852), German mathematician who gave outstanding contributions to number theory.
 
15
Carl Ludwig Siegel (Berlin, 1896 – Gottingen, 1981), German mathematician known for his deep contributions to analytic number theory.
 
16
After Charles Hermite (Dieuze, 1822–Paris, 1901), French mathematician well known for his studies on complex-valued quadratic forms bearing his name.
 
17
Leonhard Euler (Basel, 1707–Saint Petersburg, 1783) was a Swiss polymath mathematician; famous also for discovering the T-shirt slogan \(e^{\pi \sqrt{-1}} + 1 = 0\).
 
18
A function that is constant along the integral curves of a vector field.
 
19
More precisely it is the total energy divided by the mass of the point P and by the length ℓ of the supposedly massless string.
 
20
August Ferdinand Möbius (Bad Kösen, 1790–Leipzig, 1868), German mathematician and astronomer, well known to the general public for an important non-orientable surface (the Möbius strip)
 
21
Marius Sophus Lie (Nordfjordeid, 1842–Oslo, 1899), the Norwegian mathematician who introduced the notion of the homonymous algebra, which is still one of the building blocks of modern mathematics
 
22
Pierre Alphonse Laurent (Paris, 1813–1854), French mathematician. His paper on the generalization of Taylor series was published posthumously.
 
23
The definition we adopt of \(\Gamma ^{{\ast}}(z)\) is 1∕z times the one used, for instance, in [78].
 
24
Werner Karl Heisenberg (Würzburg, 1901–Munich, Bavaria, 1976), German theoretical physicist and one of the pioneers of quantum mechanics, known for his celebrated indeterminacy principle
 
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Metadaten
Titel
Prologue
verfasst von
Letterio Gatto
Parham Salehyan
Copyright-Jahr
2016
Buch
Hasse-Schmidt Derivations on Grassmann Algebras

Print ISBN: 978-3-319-31841-7

Electronic ISBN: 978-3-319-31842-4

Copyright-Jahr: 2016

DOI
https://doi.org/10.1007/978-3-319-31842-4_1