Elementary number theory, group theory, and ramanujan graphs. As mentioned above, the ramanujan peterson conjecture was generalized to other groups, and some of its generalizations to gld instead of only gl 2 led to higher dimensional versions of ramanujan graphs, the so called ramanujan complexes. Download for offline reading, highlight, bookmark or take notes while you read ramanujans notebooks. Elementary number theory, group theory, and ramanujan. Cambridge core discrete mathematics information theory and coding elementary number theory, group theory and. Expander graphs are both highly connected but sparse, and besides their interest within combinatorics and graph theory, they also find various applications in computer science and engineering. Download for offline reading, highlight, bookmark or take notes while you read number theory in the spirit of ramanujan. Lubotzky, phillips and sarnak lps86 defined a ramanujan graph to be a. Examples of ramanujan graphs include the clique, the biclique, and the petersen graph. In spectral graph theory, a ramanujan graph, is a regular graph whose spectral gap is almost as large as possible see extremal graph theory. Elementary number theory,group theory,and ramanujan graphs guiliana davidoff, peter sarnak,alain valette. As observed by sunada, a regular graph is a ramanujan graph if and only if its ihara zeta function satisfies an analogue of the riemann hypothesis.
Besides their interest in combinatorics and graph theory. The ramanujan graphs of lubotzky, phillips and sarnak lps88 were shown. Buy elementary number theory, group theory and ramanujan graphs london mathematical society student texts on free shipping on qualified orders. The spectra of matrices such as the adjacency and edge adjacency matrices of a graph are essential to the plot of this book, which makes connections with quantum chaos and random matrix theory and also with expander and ramanujan graphs, of interest in computer science. Ramanujan graphs department of mathematics and statistics. This subfield of number theory is about the different ways in which whole numbers can be written as sums of 1s, 2s, 3s, etc. One of the main reasons for this phenomenon is the applicability of graph theory in other.
Srinivasa ramanujan was raised in a poor family in a poor nation, and showed early aptitude by mastering a trigonometry text when he was about ten years old. Rather, my goal is to introduce the main ideas and to provide intuition. The fact that all these topics are used to prove graph theory results is what makes this book so interesting. Graph theory i graph theory glossary of graph theory list of graph theory topics 1factorization 2factor theorem aanderaakarprosenberg conjecture acyclic coloring adjacency algebra adjacency matrix adjacentvertexdistinguishingtotal coloring albertson conjecture algebraic connectivity algebraic graph theory alpha centrality apollonian. On the whole, considered as an inspiration for a boy of such abnormal gifts, carr was not too bad, and ramanujan responded amazingly. Borwein 19871994 a book and a series of papers proved all 17 of ramanujans series for 1. This text is a self contained treatment of expander graphs and in particular their explicit construction. Ramanujan graphs and shimura curves what follows are. Ramanujans theories of elliptic functions to alternative. One of the main reasons for this phenomenon is the applicability of graph theory in other disciplines such as physics, chemistry, psychology, sociology, and theoretical computer science. Hardy perhaps the most famous collaboration in the history of mathematics is the subject of the 1991 book the man who knew infinity by robert kanigel. This page is currently inactive and is retained for historical reference. To begin, designate the number of primes less than or equal to n by. International conference on the theory and application of cryptology and information.
Thus one way to show that a graph is ramanujan is to identify an algebraic variety v such that p gu p v u. Buy the elementary number theory, group theory and ramanujan graphs ebook. This is largely a survey paper on the theory of ramanujan graphs. Through this book, ramanujan went ranging with delight.
Ramanujans alternative theories of elliptic functions k. An introduction to algebraic graph theory and ramanujan graphs ashwin k 12026 background algebraic graph theory is a branch of mathematics in which algebraic methods, particularly those employed in group theory and linear algebra, are use to solve graphtheoretic problems. Apr 27, 2016 i actually went to college because of ramanujan. Most of ramanujan s work in number theory arose out of \q\series and theta functions. Elementary number theory, group theory and ramanujan graphs london mathematical society student texts set up a giveaway. Terras, audrey university of california, san diego.
Charlesgorenlauter in 2006 proposed two hash functions based on the hardness of finding paths in ramanujan graphs. Ramanujan graphs in cryptography cryptology eprint archive iacr. Spectral graph theory, expanders, and ramanujan graphs. However, formatting rules can vary widely between applications and fields of interest or study. The math book ramanujan got at age 16by the time ramanujan got to college, all he wanted to do was mathematics and he failed his other classes, and at one point ran away, causing his mother to. The book is broken up into four chapters covering graph theory, number theory, group theory, and the ramanujan graphs. Then, we will learn about applications to the study of expanders and ramanujan graphs, and more generally, to computer science as a whole. Library of congress cataloging in publication data davidoff, giuliana p.
Expander graphs are both highly connected but sparse, and besides their interest within. Pdf contributions of srinivasa ramanujan to number theory. In graph theory, the girth of a graph is the length of a shortest cycle contained in the graph. This book is mostly based on lecture notes from the spectral graph theory course that i have. This textbook provides a solid background in the basic topics of graph theory, and is intended for an advanced undergraduate or. One of the supreme achievements of 19thcentury mathematics was the prime number theorem, and it is worth a brief digression.
Venkatachaliengar 1988, republished 2012 initial investigations into the alternative theories j. This three part lecture series is based primarily on the book elementary number theory, group theory, and ramanujan graphs by davidoff, sarnak, and valette. A life of the genius ramanujan robert kanigel page 2 of 4 bookofthemonth club selection quality paperback book. I noticed an apparent conflict in the definition in literature about what is a ramanujan graph, which i was wondering if someone could kindly clarify. One is based on lubotzkyphillipssarnak lps graphs and the other one is based on supersingular isogeny graphs. Use features like bookmarks, note taking and highlighting while reading elementary number theory, group theory and ramanujan graphs.
Clark what follows are some long, rambling notes of mine on ramanujan graphs. If x is a kregular graph, then d k is an eigenvalue with multiplicity equal to the number of connected components. As murtys survey paper notes, ramanujan graphs fuse diverse branches of pure mathematics, namely, number theory, representation theory, and algebraic. It treats a melange of topics from combinatorial probability theory, number theory, random graph theory and combinatorics. Every dregular graph has a 2lift in which all the new eigenvalues have absolute value at most 2 p d 1 would give infinite families of ramanujan graphs. Elementary number theory, group theory and ramanujan. Here now is the first book to provide an introduction to his work in number theory. Only a knowledge of elementary algebra, analysis and combinatorics is required because the authors provide the necessary background from graph theory, number theory, group theory and representation theory. They are close to regular random graphs, and hence expand well. Apr 28, 2016 some things you probably did not know about 1729 and the man who knew infinity. It also includes a simplified proof of the celebrated alonbopanna theorem and discusses the future direction of the theory of zeta functions of regular graphs. Ramanujan graphs and shimura curves introduction q. In mathematics, ramanujan s master theorem named after srinivasa ramanujan is a technique that provides an analytic expression for the mellin transform of an analytic function. Buy elementary number theory, group theory and ramanujan graphs london.
Elementary number theory, group theory, and ramanujan graphs guiliana davidoff, peter sarnak, alain valette. A graph is called kregular if every vertex has degree k. Elementary number theory, group theory and ramanujan graphs london mathematical society student texts book 55 kindle edition by davidoff, giuliana, sarnak, peter, valette, alain. Research, the number theory foundation and the nsfhrd 1500481 awm advance grant. A mathematicians practical guide to mentoring undergraduate research.
However ramanujan may have acquired his very peculiar knowledge of this theory, it was not from carr. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. This three part lecture series is based primarily on the book elementary number theory, group theory, and ramanujan graphs by davido, sarnak, and valette. Thus the text can be used as a brief introduction to these subjects and their synthesis in modern mathematics. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions. Alain valette this text is a self contained treatment of expander graphs and in particular their explicit construction. Unfortunately, ramanujan soon fell ill and was forced to return to india, where he died at the age of 32. Download it once and read it on your kindle device, pc, phones or tablets. An introduction to algebraic graph theory and ramanujan graphs ashwin k 12026 background algebraic graph theory is a branch of mathematics in which algebraic methods, particularly those employed in group theory and linear algebra, are use to solve graphtheoretic.
What does this connection between chebyshev, ramanujan, ihara and riemann mean. Hardy immediately recognised ramanujans genius, and arranged for him to travel to cambridge in england. An introduction to algebraic graph theory and ramanujan graphs ashwin k 12026 background algebraic graph theory is a branch of mathematics in which algebraic methods, particu. Jun 16, 2016 the collaboration that took place in the years 19141919 between the indian mathematician srinivasa ramanujan and the english mathematician g. An introduction to ramanujans magic squares georgep. There are three tasks that one must accomplish in the beginning of a course on spectral graph theory. Together, they made numerous discoveries in number theory, analysis, and infinite series. Ramanujans interests include in nite series, integrals, asymptotic expansions and approximations, gamma function, hypergeometric. Spectral and algebraic graph theory computer science yale. Elementary number theory, group theory and ramanujan graphs giuliana davidoff, peter sarnak, alain valette this text is a selfcontained study of expander graphs, specifically, their explicit construction. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Ramanujan was one of the greatest mathematical geniuses ever to emerge from india. Ramanujanpetersson conjecture project gutenberg self. Graph theory experienced a tremendous growth in the 20th century.
In spectral graph theory, a ramanujan graph, is a regular graph whose spectral gap is almost. I read the book the man who knew infinity and i couldnt understand some of the advanced math in the book so i went to college to learn and i became sidetracked with computers and physics and i. Markus rost, and jeanpierre tignol, the book of involutions, american. Thus the text can be used as a brief introduction to. Some things you probably did not know about 1729 and the man who knew infinity. A catalog record for this book is available from the british library. Number theory in the spirit of ramanujan ebook written by bruce c. This and other features make them useful in communication. In fact, the first three chapters can be read independently and each one is interesting. The physical and mathematical foundations of the theory of relativity. Ramanujan graphs 3 since a kregular graph is one whose adjacency matrix has every row sum and hence every column sum equal to k, we clearly have that 0 d k is an eigenvalue of a with eigenvector equal to u d.
One must convey how the coordinates of eigenvectors correspond to vertices in a graph. Srinivasa ramanujan life and mathematics christian krattenthaler universit at wien christian krattenthaler srinivasa ramanujan. The more general ramanujanpetersson conjecture for holomorphic cusp forms in the theory of elliptic modular forms for congruence subgroups has a similar formulation, with exponent k. In graph theory, the helly property has been applied to families of sets, such as. An introduction to algebraic graph theory and ramanujan graphs. In the last two decades, the theory of ramanujan graphs has gained prominence primarily for two reasons.
Galois died at twentyone, abel at twentyseven, ramanujan at thirtythree, riemann at forty. Special topics in graph theory michelle delcourt abstract. At age 15, he was given a copy of carrs synopsis of pure mathematics. Extremal graph theory is a very deep and wide area of modern. What does this connection between chebyshev, ramanujan. Jan 01, 2006 number theory in the spirit of ramanujan ebook written by bruce c. The best indicator for this growth is the explosion in msc2010, field 05. We do this by proving a variant of a conjecture of bilu and linial about the existence of good 2lifts of every graph. Born into a poor brahmin family, he had limited formal education. To revive discussion, seek broader input via a forum such as the village pump. Srinivasan ramanujan a bursting supernova, illuminating the darkest and most profound corner of mathematics, has discovered the mathematics of the centuries in a few years. Elementary number theory, group theory and ramanujan graphs.
Number theory in the spirit of ramanujan by bruce c. Elementary number theory, group theory and ramanujan graphs hardcover at. Either the page is no longer relevant or consensus on its purpose has become unclear. Problems from the discrete to the continuous probability. This book provides an introduction to these two important subjects and to some of the topics in number theory that are inextricably intertwined with them, including the theory of partitions, sums of squares and triangular numbers, and the ramanujan tau function. The book takes a number of specific problems and solves them, the needed tools developed along the way in the context of the particular problems. The diameter of a connected graph is then the maximum value of the distance.
Prime members enjoy free twoday delivery and exclusive access to music, movies, tv shows, original audio series, and kindle books. Nonetheless, the lack of education had obvious adverse effects on ramanujan s mathematical accomplishments. In spectral graph theory, a ramanujan graph, named after srinivasa ramanujan, is a regular graph whose spectral gap is almost as large as possible see extremal graph theory. Our purpose in this book is to describe the ramanujan graphs of lubotzky et al. Magic squares, theory of partitions, ramanujans contribution to the concept of highly composite numbers, expressions for. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory see for example 2. If youve studied graph theory, it might be too basic, but the exercises are really wellchosen and so it might be worth it its a cheap book for exercises alone.
I read the book the man who knew infinity and i couldnt understand some of the advanced math in the book so i went to college to learn and i became sidetracked with computers and physics and i came up with the complete theory of everything. During his high school years he studied a book about advanced mathematics, synopsis of elementary results in pure mathematics, by george shoobridge carr 18371914. No mathematician should ever allow him to forget that mathematics, more than any other art or science, is a young mans game. Numerous and frequentlyupdated resource results are available from this search. A fimily of cubic ramanujan graph is explicitly constructed. They are realized as cayley graphs of a certain free group acting on the 3regular tree. Adam marcus crisply, yale daniel spielman yale nikhil. A grid has girth 4 as well, and a triangular mesh has girth 3. An introduction to algebraic graph theory and ramanujan. Many wellchosen illustrations and exercises, both theoretical and computer. Srinivasa ramanujan was an indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions. Mathematics institute and the number theory foundation for supporting the workshop. This acclaimed book by giuliana davidoff is available at in several formats for your ereader.
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