Fearless Symmetry: Exposing the Hidden Patterns of Numbers - New Edition - Softcover

Ash, Avner; Gross, Robert

Softcover

ISBN 10: 0691138710 ISBN 13: 9780691138718

Verlag: Princeton Univers. Press, 2008

Inhaltsangabe

Mathematicians solve equations, or try to. But sometimes the solutions are not as interesting as the beautiful symmetric patterns that lead to them. Written in a friendly style for a general audience, Fearless Symmetry is the first popular math book to discuss these elegant and mysterious patterns and the ingenious techniques mathematicians use to uncover them.


Hidden symmetries were first discovered nearly two hundred years ago by French mathematician évariste Galois. They have been used extensively in the oldest and largest branch of mathematics--number theory--for such diverse applications as acoustics, radar, and codes and ciphers. They have also been employed in the study of Fibonacci numbers and to attack well-known problems such as Fermat's Last Theorem, Pythagorean Triples, and the ever-elusive Riemann Hypothesis. Mathematicians are still devising techniques for teasing out these mysterious patterns, and their uses are limited only by the imagination.


The first popular book to address representation theory and reciprocity laws, Fearless Symmetry focuses on how mathematicians solve equations and prove theorems. It discusses rules of math and why they are just as important as those in any games one might play. The book starts with basic properties of integers and permutations and reaches current research in number theory. Along the way, it takes delightful historical and philosophical digressions. Required reading for all math buffs, the book will appeal to anyone curious about popular mathematics and its myriad contributions to everyday life.

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Über die Autorin bzw. den Autor

Avner Ash is professor of mathematics at Boston College and the coauthor of Smooth Compactification of Locally Symmetric Varieties. Robert Gross is associate professor of mathematics at Boston College.

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"All too often, abstract mathematics, one of the most beautiful of human intellectual creations, is ground into the dry dust of drills and proofs. Useful, yes; exciting, no. Avner Ash and Robert Gross have done something different--by focusing on the ideas that modern mathematicians actually care about. Fearless Symmetry is a book about detecting hidden patterns, about finding definitions that clarify, about the study of numbers that has entranced some of our great thinkers for thousands of years. It is a book that takes on number theory in a way that a nonmathematician can follow-systematically but without a barrage of technicalities. Ash and Gross are two terrific guides who take the reader, scientist or layman, on a wonderful hike through concepts that matter, culminating in the extraordinary peaks that surround the irresistible, beckoning claim of Fermat's Last Theorem."--Peter Galison, Harvard University

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Fearful Symmetry

By A. Ash R. Gross

Princeton University Press

Copyright © 2008 Princeton University Press
All right reserved.
ISBN: 978-0-691-13871-8

Chapter One

REPRESENTATIONS

Road Map

To start our journey, we discuss the basic concept of representation from a formal point of view. This is the key concept underlying the number-theoretic methods of Galois representations that are our goal. To flesh out the abstract formalism, we go through an example: The ordinary act of counting can be viewed as a representation of sets. So we give (or review) mathematical definitions of sets, functions, morphisms, and representations, which will be with us for the whole book.

The Bare Notion of Representation

Before we narrow our focus to mathematical concepts, we start by discussing the general concept of a representation. In philosophy, the concept of one thing representing or misrepresenting another thing is a central concern. The distinction between truth and appearance, the thing-in-itself and its representation, is a keynote of philosophy. It plays a critical role in the works of such figures as Plato, Kant, Schopenhauer, and Nietzsche. Generally speaking, for these philosophers the "appearance" of something is thought to be an impediment or veil, which we wish to penetrate through to the reality acting behind it. But in mathematics, matters stand somewhat differently.

Consider, in an abstract way, the relationship that occurs when one thing represents another. Say B represents A. We have three terms that stand together in some kind of relationship: A, B, and the fact that B represents A. We can call this fact X. It is important to remember that, in a representation, the three terms A, [B, and X are usually distinct.

For example, A may be a citizen of Massachusetts, B her state representative, and X the legal fact that B represents A by voting in the legislature on her behalf. Or, to jump ahead, A may be an abstract group, B a group of matrices, and X a morphism from A to B. (We will define these terms later.)

It can happen, though, that A = B. For instance, B may be said to (also) represent herself in the state legislature. Or A may be a group of matrices and B the same group of matrices. But whether A = B or A [not equal to] B, we call these relationships "representations." Note that the fact of representation, X, is always going to be different from A and B, because A and B are objects and X is a fact of representation.

Now, what would be a good picture of A, B, and X? We can view X as an arrow going from A to B. This captures the one- way quality of the relationship, showing that B is representing A, not vice versa:

A [right arrow] B.

We can abstract even further, if we do not want to name A and B and we just want to visualize their relationship. We can picture them with dots. Then the picture of a representation becomes

[right arrow]

which is the ultimate in abstraction. The dots are just placeholders for the names of the objects. The two dots can stand for two different objects or the same object. The dot or object from which the arrow emanates is called the source of that arrow, and the dot or object to which the arrow goes is called the target of that arrow.

In normal life, if A represents B, B and A can be very different kinds of things. For instance, a flag can represent a country, a slogan on a T-shirt can represent an idea, and a mental image can represent a beloved person. In mathematics, the situation is different. All the mathematical entities we encounter or invent are considered to be on the same plane and have the same degree and type of reality or ideality: They are all mathematical entities.

What are representations used for? They explain one thing by means of another. The object we want to understand is the "thing": the thing-in-itself, the source. The object that we know quite a bit about already, to which we compare the source via a representation, we call the standard object. It is the site of appearance, the target.

Our conventions might not correspond to your expectations. The target, the object at the head of the arrow, is the piece of the picture that we understand better. We will derive information about the source by using properties of both the arrow and the target.

An Example: Counting

We look at the simplest possible example, one that goes back to prehistory: counting. Suppose we have a sack of potatoes or a flock of sheep. We want to know how many potatoes or sheep we have.

This is a much more sophisticated question than knowing whether they are the same in number as another sack of potatoes or another flock of sheep. We start with the less sophisticated question. Suppose we want to know whether the flock of sheep being herded home this evening is the same size as the herd we let out to the pasture in the morning. In the morning, we put a small pebble in our pouch for each sheep as it went out of the fold. Now we take a pebble out of the pouch as each sheep returns to the fold.

We were careful to make sure the pouch was empty in the morning before we began, and careful not to put anything in or take anything out during the day. So if the pouch becomes empty exactly as the last sheep comes in, we are happy. A mathematician says that we have demonstrated the existence of a one-to-one correspondence from the sheep in the morning to the sheep in the evening.

To make this mathematically precise, we make two definitions:

DEFINITION: A set is a collection of things, which are called the elements of the set.

For example, the collection of all odd numbers is a set, and the odd number 3 is an element of that set.

DEFINITION: A one-to-one correspondence from a set A to a set B is a rule that associates to each element in A exactly one element in B, in such a way that each element in B gets used exactly once, and for exactly one element in A.

Digression: Definitions

A mathematician uses the term "definition" in a way that might be surprising to nonmathematicians. The Oxford English Dictionary defines "definition" as "a precise statement of the essential nature of a thing." Mathematicians agree that a definition should be "precise," but we are not so sure about capturing the "essential nature." Our definition of one-to-one correspondence above will let you recognize a one-to-one correspondence if one is shown to you. Suppose that A is the set {red, blue, green} and B is the set {1, 2, 3}. Then a one-to-one correspondence between the two sets is given by

red [right arrow] 1

blue [right arrow] 2

green [right arrow] 3.

You can check that this associates to each element of the set A a different element of the set B, and that...

Details zur Ausgabe

• Verlag: Princeton Univers. Press
• Erscheinungsdatum: 2008
• Sprache: Englisch
• ISBN 10: 0691138710
• ISBN 13: 9780691138718
• Einband: Taschenbuch
• Anzahl der Seiten: 272
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: Nicht verfügbar