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1993 | Buch

The 0th Week: No Prerequisites Kapitel lesen Erstes Kapitel lesen

Galois’ Dream: Group Theory and Differential Equations

verfasst von: Michio Kuga

Verlag: Birkhäuser Boston

Enthalten in: Professional Book Archive

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Über dieses Buch

First year, undergraduate, mathematics students in Japan have for many years had the opportunity of a unique experience---an introduction, at an elementary level, to some very advanced ideas in mathematics from one of the leading mathematicians of the world. Michio Kuga’s lectures on Group Theory and Differential Equations are a realization of two dreams---one to see Galois groups used to attack the problems of differential equations---the other to do so in such a manner as to take students from a very basic level to an understanding of the heart of this fascinating mathematical problem. English reading students now have the opportunity to enjoy this lively presentation, from elementary ideas to cartoons to funny examples, and to follow the mind of an imaginative and creative mathematician into a world of enduring mathematical creations.

Inhaltsverzeichnis

Frontmatter

Pre-Mathematics

Frontmatter
The 0th Week: No Prerequisites
Abstract
The title of this series of lectures is “Group Theory and Differential Equations”. The contents are briefly explained in the preface. The 0th week is a summary of the preliminary meeting of the course. (In the lecture, I outlined everything in the course, and left the students in a fog.)
Michio Kuga
The First Week: Sets and Maps
Abstract
This week I will explain the concepts of “set” and “map”, which are fundamental in mathematics. I am sure that many of you know these terms already, so my explanation will be just an appetizer before the main course.
Michio Kuga
The Second Week: Equivalence Classes
Abstract
(Instructor’s Soliloquy: I do not like teaching abstract concepts at the beginning of a course, but since I have to make a choice, this is better than not being able to explain important concepts later.)
Michio Kuga
The Third Week: The Story of Free Groups
Abstract
We shall start with a set of 2n + 1 “letters”,
$$E; {A_1},{A_2},{A_3},{A_4},\ldots,{A_n},$$
$$A_1^{ - 1},A_2^{ - 1},A_3^{ - 1},A_4^{ - 1}, \ldots,A_n^{ - 1}$$
Michio Kuga

Heave Ho! (Pull it Tight)

Frontmatter
The Fourth Week: The fundamental group of a surface
Abstract
The scene of today’s lecture is set in a region in a plane. We define a region to be a part of a plane surrounded by some closed curves. For example, the portion D in Figure 4.1 surrounded by the closed curves C 1, C 2, and C 3 is a region (i.e., the unshaded part of the figure).
Michio Kuga
The Fifth Week: Fundamental Groups
Abstract
Last week we were considering curves in a region. A region was a part of the plane surrounded by closed curves (for example, the unshaded part of figure 5.1). We decided to call such a region land and the outside region a sea or a lake (such as the shaded part of Figure 5.1)
Michio Kuga
The Sixth Week: Examples of fundamental groups
Abstract
The region D is the plane minus one point. There is no sea, but there is an infinitely small lake P 0. (See Figure 6.1.) According to our rule, we are not allowed to move the curve across P 0. No matter how small, a lake is a lake, so we may not move our rubber railroad across P 0.
Michio Kuga
The Seventh Week: Examples of fundamental groups, continued
Abstract
So far we have discussed fundamental groups of plane regions. Needless to say, we can define fundamental groups for surfaces in space.
Michio Kuga

Men Who Don’t Realize That Their Wives Have Been Interchanged

Frontmatter
The Eighth Week: Coverings
Abstract
Let us start with an example. Take two distinct points O’ and O in the plane. Let a and b be real numbers such that 0 < a < b. Let D’ be the set of points P whose distance from O’ is between a and b. Similarly, let D be the region bounded by two circles of radii a and b and center O. See Figure 8.1.
Michio Kuga
The Ninth Week: Covering surfaces and fundamental groups
Abstract
Let f: D’ → D be a covering. Since f is a continuous map, when a point P in D’ moves continuously over a figure F of D’, f (P) = Q moves continuously in D. Let f (F) be the figure in D which Q traces out. In particular, if P traces a curve C in D’, then the trace f (C) of the point f (P) = Q is again a curve in D. If C is a closed curve, then f (C) is also a closed curve.
Michio Kuga
The Tenth Week: Covering surfaces and fundamental groups, continued
Abstract
Let’s recall what a covering map is:
Cl: A continuous surjective map f from D’ to D.
C2: For every point Q of D,f -1 (Q) consists of finitely or countably many points ;{P 1, P 2, P 3,…} of D’. Also, there exists a small neighborhood U around each point Q such that there is a small neighborhood V i around each P i satisfying
Michio Kuga
The Eleventh Week: The Group of Covering Transformations
Abstract
Let f : D’ → D be a covering. We say that points P 1 and P 2 of D’ are conjugate if f(P 1) = f(P 2). The number of points which are conjugate to P 1 (including P 1 itself) is equal to n = deg(f). Similarly, we say that curves C 1’and C2’ are conjugate if they are lifts of the same curve. In order to construct a curve starting at P 2 which is conjugate to a curve C 1’ starting at P 1, first we project C 1’ to make a curve C. Then take the lift of C which starts at P 2. The number of curves conjugate to C l’ (including C l’ itself) in D’ is also equal to n = deg(f).
Michio Kuga

Everyone has a tail

Frontmatter
The Twelfth Week: The Universal Covering Space
Abstract
When I was a schoolboy, I was seized by the idea that everyone has an invisible tail at his back. This tail is very light, a million miles long, very thin, and trails behind as we walk, from birth to death. The tail extends far away, beyond our world, and originates in the Land of the Dead. When we walk, the tail is reeled out from its source. When we have pulled out the entire tail, we die.
Michio Kuga
The Thirteenth Week: The correspondence between coverings of (D;O) and subgroups of π1(D;O)
Abstract
We first introduce a notation: \(\left( {D';O'} \right)\mathop \to \limits^f \left( {D,O} \right)\) means that \(D'\mathop \to \limits^f D\) is a covering and \(f\left( {O'} \right) = O\)
Michio Kuga

Seeing Galois Theory

Frontmatter
The Fourteenth Week: Continuous functions on covering surfaces
Abstract
Starting this week, we will look at functions defined on manifolds. Let D be a two-dimensional manifold. (If you still feel uncomfortable hearing the word “manifold”, think of D as a region in a plane.) We will denote by C the set of all complex numbers as usual, and consider a continuous function F: D → C that assigns a complex number F(P) to each point P ∈ D. The symbol C 0 (D) stands for the set of all continuous functions on D.
Michio Kuga
The Fifteenth Week: Function Theory on Covering Spaces
Abstract
This week we begin function theory on D and D’. As usual, C denotes the complex plane. For simplicity, we will assume that D is a region (i.e., open, connected subset) in C, and D’ is a covering surface of D. This restriction is necessary to rapidly develop function theory on these surfaces.
Michio Kuga

Solvable or not?

Frontmatter
The Sixteenth Week: Differential Equations
Abstract
If D is a simply connected region in C we have the following existence theorem for solutions of linear ordinary differential equations:
Michio Kuga
The Seventeenth Week: Elementary methods of solving Differential Equations
Abstract
Let us consider a set Σ of “known functions”. We regard every constant as a known function. The following procedures are used often to produce new functions out of old (known) functions F1, F2,...:
(i) The four operations of arithmetic:
$$\begin{array}{*{20}{c}} {{{F}_{1}},{{F}_{2}} \to {{F}_{1}} + {{F}_{2}}} \\ {{{F}_{1}},{{F}_{2}} \to {{F}_{1}} - {{F}_{2}}} \\ {{{F}_{1}},{{F}_{2}} \to {{F}_{1}} \cdot {{F}_{2}}} \\ {{{F}_{1}},{{F}_{2}} \to {{F}_{1}}/{{F}_{2}}} \\ \end{array}$$
Linear combinations:
$${F_1},{F_2} \to {\lambda _1}{F_1} + {\lambda _2}{F_2}$$
(ii) Differentiation:
$$F \to \frac{{dF}}{{dz}}$$
(iii) Integration:
$$F\left( z \right) \to \int {F\left( z \right)dz} $$
(iv) Exponentiation:
$$F\left( z \right) \to {e^{F\left( z \right)}}$$
Michio Kuga
The Eighteenth Week: Regular Singularities
Abstract
First, let us review some basic facts about differential equations with regular singularities. (For details, see Birkhoff-Rota [4], for example.) Consider an open disc U = U(a; ε)of radius ε and center a in the complex plane C. Let Ua denote U - {a}. We call such a region a “5-yen coin.” Choose a point b in U a and let z : \(\left( {{{\tilde U}_a},\tilde b} \right) \to \left( {{U_a},b} \right)\) be the universal covering. We call Ũ a a spiral staircase. For simplicity, we assume that b - a is a positive real number.
Michio Kuga
The Nineteenth Week: Fuchsian Differential Equations
Abstract
[A] Let D be the region obtained by removing the points a l, a 2,…, a n from the complex plane: D = C - {a l, a 2,…, a n }. It can also be obtained by removing the n + 1 points {a 1, a 2,…, a n, a n+1 = ∞} from the Riemann sphere C U {∞ }. Let z: \(\tilde D \to D\) be the universal covering surface of D. We take a “5-yen coin” Ua: around each a i (defined at the beginning of Week 18).Let \({\tilde U_{ai,1}},{\tilde U_{ai,2}},{\tilde U_{ai,3}}, \cdots \) be the connected components of the open subset \({z^{ - 1}}\left( {{U_{ai}}} \right)\) of \({\tilde D}\).Then each z : \({\tilde U_{ai,j}} \to {U_{ai}}\) is a covering of U ai ,.We call it a spiral staircase which covers U ai .
Michio Kuga
Backmatter
Metadaten
Titel
Galois’ Dream: Group Theory and Differential Equations
verfasst von
Michio Kuga
Copyright-Jahr
1993
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-0329-2
Print ISBN
978-1-4612-6710-2
DOI
https://doi.org/10.1007/978-1-4612-0329-2