REU Apprentice
First Friday
Instructor: Laszlo Babai
Scribe: Benjamin McKenna
2
Second class: Friday, June 27
2.1
Abelian groups, vector spaces: axioms
Recall: If v1 , . . . , vk V for V a vector space, you can make a linear combination of them using vect
REU 2007 Apprentice Program Lecture 3
Instructor: Laszlo Babai
Scribe: Gabriel Kerr
June 27, 2007. Revised by instructor.
Last updated June 27, 10 p.m.
For basic definitions and facts about graphs and digraphs (directed graphs), see Chapter
6 of the instr
BABAI DISCRETE MATHEMATICS REU 2008
EXERCISES FROM LECTURE 3, THURSDAY, JUNE 26
SCRIBE: MATTHEW J. THIBAULT
Exercise 1 (Clearing the Corner.). A game is played on a grid in the first
quadrant of the plane. Each square in the grid can contain at most one c
REU 2007 Apprentice Class Lecture 2
Instructor: Laszlo Babai
Scribe: Michael Geline
June 26, 2007.
COMPLETE notes. Revised by instructor. Last updated June 28, 1:30 a.m.
A2.1
A Determinant Calculation
We examine
.
det
,
.
the determinant of the n n matri
REU 2007 Discrete Math Lecture 6
Instructor: Laszlo Babai
Scribe: Shawn Drenning
June 26, 2007.
Last updated June 26, 11:59 p.m.
Revised by the instructor.
6.1
Finite Probability Spaces
For an introduction to finite probability spaces, see Chapter 7 of DL
REU 2007 Transfinite Combinatorics Lecture 7
Instructor: Laszlo Babai
Scribe: Travis Schedler
August 6, 2007. NOT revised by instructor.
Last updated August 6, 5:30 PM
7.1
Matchings
(with all edges having
Suppose we have a bipartite graph G = (V, E) with
REU 2007 Transfinite Combinatorics Lecture 4
Instructor: Laszlo Babai
Scribe: Travis Schedler
July 30, 2007. Revised by instructor.
Last updated July 31, 10:00 AM
4.1
K
onig Path Lemma revisited
Recall the statement of the Konig Path Lemma: Let V = V0 V1
REU 2007 Apprentice Class Lecture 8
Instructor: Laszlo Babai
Scribe: Ian Shipman
July 5, 2007. Revised by instructor.
Last updated July 5, 5:15 p.m.
A8.1
The Cayley-Hamilton Theorem
Recall that for a square matrix A, the characteristic polynomial of A is
REU 2007 Apprentice Program Lecture 5
Instructor: Laszlo Babai
Scribe: Ian Shipman
June 29, 2007. NOT PROOFREAD by instructor.
Last updated June 29, 3 p.m.
A5.1
Vector space review
Let V be a vector space over a field F . A basis is a linearly
independent
REU 2007 Transfinite Combinatorics Lecture 3
Instructor: Laszlo Babai
Scribe: Damir Dzhafarov
July 23, 2007. Revised by instructor.
Last updated July 29, 10:40 PM
3.1
Directed Graphs
In a previous lecture, we assigned the non-trivial following exercise:
E
REU 2007 Discrete Mathematics Lecture 9
Instructor: Laszlo Babai
Scribe: Damir Dzhafarov
July 6, 2007. Revised by instructor.
Last updated July 8, 3am
9.1
General and Special Linear Groups
Recall that if F is a finite field and n N, the set Mn (F ) of all
REU 2007 Transfinite Combinatorics Lecture 1
Instructor: Laszlo Babai
Scribe: Damir Dzhafarov
July 23, 2007. Partially revised by instructor.
Last updated July 26, 12:00 AM
1.1
Transfinite Combinatorics and Toy Problems
Lamp and switches. Imagine a lamp w
REU 2007 Discrete Math Lecture 2
Instructor: Laszlo Babai
Scribe: Shawn Drenning
June 19, 2007. Proofread by instructor.
Last updated June 20, 1 a.m.
Exercise 2.0.1. Let G be an abelian group and A G be a subset with |A| = n. Show
there exists a product-f
REU: Apprentice
First Monday
Instructor: Laszlo Babai
Scribes: Benjamin McKenna and Julie J. Huh
Email: Laszlo Babai laci at cs.etc, Sean Howe (TA) seanpkh at math.etc
1
First Class: Mon June 23
1.1
First session: Card shuffling, Eventown/Oddtown, linear
REU Apprentice
Third Friday
Section 8.3: Benjamin McKenna
8
Eighth Class: Fri. 7/11/14
8.1
Fun problems
Your online resource is the following two sets of problems:
2007 REU Linear Algebra puzzles (Lin07)
2012 REU Linear Algebra problem set (Lin12)
(Click
REU Apprentice
Third Monday
Benjamin McKenna
6
Sixth Class: Mon. 7/7/14
6.1
Fundamentals of arithmetic: divisibility, greatest common divisors, unique
prime factorization, Euclids algorithm.
A little number theory: The basic concept of number theory is di
REU Apprentice
Fourth Wednesday
Benjamin McKenna
10
10.1
Tenth Class: Wed. 7/16/14
First session: Graph isomorphisms, graph colorings, girth, regularity
Recall: A graph G = (V, E) is composed of a set V of vertices (singular: vertex) and a set E of edges
REU Apprentice
Third Wednesday
Benjamin McKenna
7
Seventh Class: Wed. 7/9/14
7.1
Roots vs. coefficients of a polynomial, elementary symmetric polynomials,
roots vs. minors of the characteristic polynomial
Rotation matrix: With respect to any orthonormal1
REU 2007 Transfinite Combinatorics Lecture 6
Instructor: Laszlo Babai
Scribe: Sundeep Balaji
August 3, 2007. Revised by instructor.
Last updated August 5, 2pm
Exercise 6.0.1. Prove that 20 6= . (Hint: use the following result.)
Exercise 6.0.2 (Gyula [Juli
REU 2007 Apprentice Class Lecture 9
Instructor: Laszlo Babai
Scribe: Courtney Morris
July 6, 2007. Revised by instructor.
Last updated July 8, 12:50 a.m.
A9.1
Circulant Matrices
Recall that a circulant matrix is one where the
the following row. For exampl
REU 2007 Transfinite Combinatorics Lecture 2
Instructor: Laszlo Babai
Scribe: Damir Dzhafarov
July 23, 2007. NOT revised by instructor.
Last updated July 27, 6:00 AM
2.1
More Problems and Examples in Transfinite Combinatorics
Disjoint and Almost Disjoint
REU 2007 Transfinite Combinatorics Lecture 9
Instructor: Laszlo Babai
Scribe: Travis Schedler
August 10, 2007. Revised by instructor.
Last updated August 11, 3:40pm
Note: All (0, 1)-measures will be assumed to be nontrivial, even when not explicitly so st
REU 2007 Apprentice Class Lecture 12
Instructor: Laszlo Babai
Scribes: Courtney Morris and Gabriel Kerr
July 11, 2007. Revised by instructor.
Last updated July 11, 9:30 p.m.
A12.1
Rational roots of polynomials
Consider the polynomial f (x) = x4 + ax3 + bx
REU 2007 Apprentice Program Lecture 1
Instructor: Laszlo Babai
Scribe: Michael Broshi
June 25, 2007.
Last updated June 26, 3:30 p.m.
Proofread by the instructor.
A1.1
The Determinant of an n n matrix
Let F be a field and n N, and identify the set of n n m
REU 2007 Discrete Math Lecture 5
Instructor: Laszlo Babai
Scribe: Sundeep Balaji
June 22, 2007.
Last updated June 24, 2 p.m.
Proofread by the instructor.
5.1
Puzzle problems
Exercise 5.1.1 (Dimension Invariance). If the groups Zk and Z` are isomorphic the
REU 2007 Apprentice Program Lecture 4
Instructor: Laszlo Babai
June 28, 2007. Instructors notes.
INCOMPLETE NOTES. Greatly expanded on June 29, 7 p.m.
Look for updates on the weekend.
For basic definitions and facts about finite probability spaces (sample