Linear algebra methods in combinatorics, with applications to geometry and computer science. More precisely we will look at some theorems from the area of discrete geometry,extremal combinatorics and nite elds constructions. Algebraic and geometric methods in enumerative combinatorics federico ardila. A magic theorem in linear algebra theorem 1 below says that, if m vectors v1. Linear algebraic proofs of vcdimension based inequalities.
Designed for advanced undergraduate and beginning graduate students in linear or abstract algebra, advanced linear algebra covers theoretical aspects of the subject, along with examples, computations, and proofs. Topics in combinatorics algebraic methods in combinatorics michael krivelevich spring semester 20 course number. Mathematical and algorithmic applications of linear algebra. Three major resultant methods due to euler, bezout, sylvester, cayley, dixon and macaulay, and a sparse reformulation of macaulay will be discussed and compared. Frankl, department of computer science, university of chicago, preliminary version, 1992.
Our proof is a simple application of the linear algebra bound method. Linear algebraic methods are some of the most beautiful and powerful techniques in combinatorics. Additional algebraic background will be introduced in the course. University of chicago, department of computer science, 1988 algebras, linear 102 pages. In the following pages we will examine the use of linear algebra in combinatorics. Laszl o babai and p eter frankl, linear algebra methods in combinatorics, preliminary version 2 september 1992, dept. Infinitude of primes and the riemann zeta function pdf exercise 1 hints on page 2, but dont peek before giving a try pdf. The theorem states that the edge set of the complete graph k n cannot be written as the disjoint union of n. Frankl, linear algebra methods in combinatorics, department of computer. The polynomial method in combinatorics larry guth ams joint meetings 4 january 2012 larry guth the polynomial method in combinatorics. This question is really just a linear algebra problem. Linear algebra methods in combinatorics with applications to geometry and computer science preliminary version 2 september 1992 216 pages laszlo babai and peter frankl to obtain copies of this volume please write to. Incidentally, the use of linear algebra in combinatorics has become a welldeveloped art, and the examples i present are just the tip of a large iceberg.
This thesis will be concerned with such issues applied to a particular area of mathematics. Book, chapter of book, survey articles april 12, 2014 124 l aszl o babai and p eter frankl. Linear algebra methods in combinatorics, by laszlo babai and peter frankl, department of computer science, university of chicago, preliminary version, 1992. Frankl, department of computer science, university of chicago, preliminary version, 1992 combinatorial nullstellensatz, by n. Algebraic methods in combinatorics, fall 2014 boris bukh. In the second part, two other methods gr obner basis method based on polyno. Linear algebra and its applications, fifth edition, by david lay, et al. We shall see a bit more about linear algebra in the following lectures. The author presents a wide spectrum of the most powerful combinatorial tools together with impressive applications in computer science. All clubs must have even cardinality, their pairwise in tersection must be even as well, and no two clubs can have the same members.
Linear algebra methods in combinatorics with applications to geometry and computer science babai, le. Begin with a problem about some points in a vector space. Even though the ideas used in applications of linear algebra to combinatorics may be very simple, the results obtained can be very strong and surprising. Monte carlo algorithms in graph isomorphism testing. Sep 07, 2015 we discuss linear combinations in linear algebra. With applications to geometry and computer science, preliminary version 2, university of chicago, september 1992. Requires only knowledge that an average matrix algebra student would be expected to have i. List of my favorite publications university of chicago.
The polynomial method in combinatorics mit mathematics. Frankl, linear algebra methods in combinatorics preliminary version 2, department of computer science, the university of chicago, 1992. Linear algebra methods in combinatorics by babai and frankl i have also placed a walk through combinatorics by mikl os b ona on reserve in the library. Basis a minimal set of vectors b that can be used to represent any vector v in a vector space v as the sum of scalar multiples of the elements in b. We will consider two widely known groups of proof methods in combinatorics, namely, probabilistic methods and methods using linear algebra. The attractivity of maxalgebra is related to the fact that r. Having said all this, naturally, the prime application area of the methods presented remains combinatorics, especially the theory of extremal set systems. Byrne department of mathematical sciences university of massachusetts lowell applied and computational linear algebra. This example and its exposition is taken from babai and frankls book titled linear algebra methods in combinatorics. References and additional reading these books and papers cover most of the topics that will be discussed in the course. Ma5350 the methods of linear algebra, polynomials, nullstellensatz and the incidence geometry. Introduction to linear algebraic methods in combinatorics. In a nutshell, algebraic combinatorics is the study of combinatorial objects via algebraic structures associated with them. Free algebra 2 worksheets created with infinite algebra 2.
Knowledge of basic linear and abstract algebra is desirable, as is some background in graph theory andor combinatorics. Hence many of the basic tools we are used to from linear algebra are available in maxalgebra as well. Linear algebra methods in combinatorics, by laszlo babai and peter frankl, department. Has a solution that requires cleverness, but not so much cleverness that only students with contest math backgrounds will be able to solve the problem. Maxalgebra enables us to describe and study a class of nonlinear problems ap. First of all i express my thanks to the active students of this course through the years from 1988 to the present time.
Linear algebra methods in combinatorics babai and frankl. Lecture 10 linear algebra methods in combinatorics cadmo. Extremal combinatorics with applications in computer science, by s. We will consider certain applications of such methods, both of which are significant to ramsey theory. Alon, combinatorics, probability and computing 8 1999, 729. Quantitative methods for treating problems arising in management, economic sciences, related areas. Linear algebra methods in combinatorics with applications to geometry and computer science. Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have gained increased representation in college mathematics curricula in recent years. A few linear algebraic methods in combinatorics syed. Math 535 combinatorics i, fall 2016 emory university. References to publications of university of south carolina. A spectacular application of linear algebra to prove a combinatorial statement is the grahampollak theorem 3. Simple linear algebra methods are a surprisingly powerful tool to prove combinatorial results. Find materials for this course in the pages linked along the left.
Linear algebraic techniques in combinatoricsgraph theory linear algebra and matrix theory provide one of the most important tools sometimes the only toolin combinatorics and graph theory. Let us nish by taking an example in extremal combinatorics which shows the power of linear algebra. Algebraic and geometric methods in enumerative combinatorics. Graph rigidity, including lower bounds on face numbers of triangulated manifolds. Each term is treated as an unknown and the theory of linear algebra can be applied. The dates of the results span from the beginning of the 20th century the dehn theorem to recent years the dvir theorem. Linear algebra in combinatorics espen slettnes walkthrough.
Computer science, university of chicago, preliminary version, 1992. These two conditions 56 imply that the vectors are linearly independent. Over the last fty years, combinatorics has undergone a radical transformation. The geometry of graphs and some of its algorithmic applications by nathan linial, eran london, yuri rabinovich. Frankl, linear algebra methods in combinatorics, 1988. Some examples of combinatorial theorems with linear algebra proofs are the weak perfect graph theorem, the franklwilson theorem, and fishers inequality.
By maxalgebra we understand the analogue of linear algebra developed for the pair of operations. We have made an effort to motivate each combinatorial application area and to give some idea about the alternative non linear algebra approaches to the same area. By applying this to the transpose at we get the opposite inequality. Preliminary version 2 1992 by l babai, p frankl add to metacart.
Algebraic methods in combinatorics l12 nonexaminable part iii level natasha morrison linear algebraic methods are some of the most beautiful and powerful techniques in combinatorics. I have a preliminary version of a wonderful book, linear algebra methods in combinatorics by laszlo babai, which indeed makes nice use of the above inner product over finite fields, even in characteristic 2. Math 535 combinatorics i, fall 2016 textbook course description. Algebraic methods in combinatorics school of mathematical. Find a polynomial of lowest possible degree that vanishes on. Combinatorial nullstellensatz and the polynomial method. I will assume some very basic knowledge of linear algebra. Dimension maximum number of linearly independent vectors in a vector space. Numerical linear algebra has also been called applied linear algebra, and because many properties of matrices and vectors also apply to functions and operators, it can also be viewed as a type of functional analysis that has a particular emphasis on practical algorithms ix. Pdf intersection patterns of linear subspaces with the. Algebraic methods in combinatorics natasha morrison impa and cambridge linear algebraic methods are some of the most beautiful and powerful techniques in combinatorics. If you are curiousinterested, you can have a look at cur90, chapter 1. Linear algebra methods in combinatorics nonuniform modular raychaudhuri wilson theorem a corollary corollary let l be a set of s integers and f a family of kelement subsets of. It is enough to show that the incidence vectors v i are linearly independent over f 2.
This book has many good exercises to try as practice. Stanleyreisner rings of simplicial complexes, and applications to face enumeration. Ma5510 algebra, analysis and probability in combinatorics new course course details prereq. Also, there are many combinatorialists who are much better practitioners of this art than i will ever be.
Linear algebra methods in combinatorics, by l aszl o babai and peter frankl, department of computer science, university of chicago, preliminary version, 1992. The applications of algebra to hungarianstyle combinatorics are relatively rare, but powerful. Prerequisites i will assume some very basic knowledge of linear algebra. Dec 05, 2017 laszi o babai and peter frankl, linear algebra methods in combinatorics. List of my favorite publications in order of my personal preference with clickable links l aszl o babai april 11, 2014. While asymptotic group theory took a boost from the classification of finite simple groups, elementary methods of combinatorics, graph theory, and linear algebra have added to the arsenal, expanding the scope of the study from highly symmetrical to highly regular.
Advanced combinatorics handouts andras gyarfas 2011 fall. Linear algebraic techniques in combinatoricsgraph theory. A friend of mine once made the outrageous claim but hear me out that most linear algebra proofs in combinatorics are not truly using linear algebra. It is helpful, but not essential, to have taken combinatorics or part ii graph theory. Maxalgebra, which has been studied for more than 40 years, is an attractive way of describing a class of nonlinear problems appearing for instance in machinescheduling, information technology and. Combinatorics with applications to geometry and cs. Some examples of combinatorial theorems with linear algebra proofs are the weak perfect graph theor. It is enough to show that the incidence vectors vi are linearly independent over f2. Donna brooms department of computer science the university of chicago ryerson hall, room 152 1100 e 58th street. Linear algebra methods in combinatorics laszlo babai. Laci babai s 2012 reu web page is now up, go take a look go number theory. Finding polynomials with prescribed zeroes is just linear algebra f a eld. It explores a variety of advanced topics in linear algebra that highlight the rich interconnections of the subject to geometry, algebra, analysis, combinatorics, numerical computation.
We will begin our journey through linear algebra by defining and conceptualizing what a vector is rather than starting with matrices and matrix operations like in a more basic algebra course and defining some basic operations like addition subtraction and scalar multiplication. Linear algebra methods in combinatorics university of chicago. We have made an effort to motivate each combinatorial application area and to give some idea about the alternative nonlinearalgebra approaches to the same area. In this course i will present some the most appealing applications of these. Algebraic methods in combinatorics poshen loh june 2009 1 linear independence these problems both appeared in a course of benny sudakov at princeton, but the links to olympiad problems. List of my favorite publications in order of my personal preference with clickable links l aszl o babai april 11, 2014 35 l aszl o babai. We will assume a solid knowledge of linear algebra, and a basic familiarity with groups and rings. Frankl, linear algebra methods in combinatorics, a soft copy of which is easy to find by an online search.
Home assignments will be given roughly every other week, and their solutions will be graded. Due perhaps to a recognition of the wide applicability of their elementary concepts and techniques, both combinatorics and linear algebra have. In this course i will present some the most appealing applications of these techniques. Linear algebra methods in combinatorics, with applications. C2 fn m and linear independence of their columns or rows over f. In a town of ncitizens, no more than nclubs can be formed under the rules each club have an odd number of members each pair of clubs share an even number of members. Applications of linear algebra in combinatorics, including use of. Frankl, linear algebra methods in combinatorics, a soft copy of which is easy to. Linear algebra methods in combinatorics with applications to geometry and computer science, preliminary version, by laszlo babai and peter frankl, 1992 website.
1601 791 654 301 793 464 998 191 880 95 107 322 614 1440 904 371 319 175 1334 946 950 263 893 608 451 1210 235 1617 1179 969 1317 1410 993 975 1110 1350 204 575 869 1291 1181 1026 260 364