VIEWING THIS PAGE CORRECTLY: I HAVE A PROBLEM YET MAKING IT SO IT READS WELL IN ANY BROWSER. IN INTERNET
EXPLORER MAKE SURE YOUR BROWSER IS SET FOR “WESTERN EUROPEAN (WINDOWS)”. I HOPE
TO SOLVE THIS PROBLEM SHORTLY.
last updated: April 11, 2007
Page URL: http://www.math.mcgill.ca/goren/courses06-07.html
Algebra 1, Topology and Geometry 1, Honors Algebra 2
Instructor: Dr. Eyal Goren
Time of course: MWF 8:35-9:25 in Arts Building
W-120.
Office hours: Friday 10:00-12:00, BURN
1108. SPECIAL OFFICE HOURS: WEDNESDAY DECEMBER 20,
09:00 – 11:00.
TA: Gabriel Chenevert, Shahab Shahabi
Tutorial
Hours: Monday 12:35
- 1:25, BURN
1214 (GC), Tuesday
4:05 - 4:55, BURN 1205 (SS)
TA office hours: Thursday 12:20 - 13:50 and Friday 15:30 - 17:00 BURN 1034
(GC), Tuesday 2:25 -
3:55 pm, and Thursday 8:55 - 10:25 am BURN 1033 (SS)
Syllabus (Calendar description): Sets and
relations. Rings and fields. Integers, rationals,
real and complex numbers; modular arithmetic. Polynomials over a field.
Divisibility theory for integers and polynomials. Linear equations over a
field. Introduction to vector spaces.
Prerequisites: MATH 133 or
equivalent
Text book: Hungerford,
Thomas W.: Abstract algebra : an introduction (2nd edition). Thomson
Brooks/Cole. http://www.amazon.ca/exec/obidos/ASIN/0030105595/ref=pd_rhf_p_2/702-6311809-4508854
Additional
textbooks: Abstract
algebra / W.E. Deskins.
Evaluation method:
·
10% Assignments. Assignments are
posted here on Mondays and should be submitted the next Monday by 12:00 to the
secretaries (Burnside Hall 10th floor).
·
10% Quiz. (In class) Parts 1, 2, 3
in the notes. This is more or less the
appendices, Chapter 1, 2 and sections 3.1, 3.2 in Hungerford (though we did a
bit more! check the notes as well!). The quiz is on FRIDAY, OCTOBER 6. The Quiz
will consist of multiple choice questions based on exercises in Hungerford, or
the assignments.
·
20%
Midterm. The topics to be covered in the midterm are sections 1 to 17 (inclusive) in the course notes. The midterm exam has been booked for
Thursday November 2, 2006 in ADAMS AUD from 6:30 to 8:30 pm. This midterm exam
is priority #2. Priority#1 is CHEM 170. Students taking CHEM 170 please inform
me at once.
·
60% Final.
Calculation of final grade: The better of the method
described here and the grade of the final exam alone. Same rule applies to
deferred/supplemental. Students will NOT have the option of make-up/additional
work to improve their grades.
Academic integrity:
Course
Notes (pdf) (Final version: November
28, 2006) ERRATA
(version: December 15, 2006)
Detailed syllabus (To be updated during the semester)
Date |
Material |
Assignment |
Misc. |
September 6-8 |
Sets, Methods of Proof |
|
We covered appendices A, B, C in Hungerford |
September 11 - 15 |
Functions, Complex numbers, Polynomials and the
fundamental thm of Algebra |
|
|
September 18-22 |
Rings and Fields. Divisibility, gcd,
Euclidean algorithm for integers. Primes and the sieve of Eratosthenes. |
|
|
September 25-29 |
The Fundamental Thm of
Arithmetic, infinity of primes, 2^(1/2) is irrational, equivalence relations. |
|
|
October 2-6 |
Congruences,
Fermat’s little theorem, computing and solving equations in Zn.
Public Key crypto and RSA. |
No assignment this week J |
FRIDAY OCTOBER 6 IS THE QUIZ. QUIZ starts at
9:00 and ends 9:30. |
October 9-13 |
The ring of polynomials over a field. Degree. Division
with residue. GCD’s. The Euclidean algorithm for
polynomials. Irreducible polynomials. Unique factorization. |
|
No class Monday (thanksgiving). There’s a class on Tuesday
– same time and place. |
October 16-20 |
Roots of polynomials. Roots of rational and real
polynomials. Roots of polynomials over Z_p. Rings
(recall). Ideals. Z and F[x] are principal ideal rings. |
|
There’s a typo in the last question. It should read 2*x^4+4*x^3-5*x^2-5*x+2 |
October 23 - 27 |
Homomorphisms and
kernels. Quotient rings. |
|
|
October 30 – November 3 |
F[x]/(f(x)) and constructing finite fields. Roots in
extension fields. First isomorphism theorem. Chinese remainder theorem. |
J Two
weeks to submit assignment J |
Thursday
November 2 is the midterm. (See above for details) |
November 6 - 10 |
Applications of CRT. Groups: the basic definition and
examples. The symmetric group. |
|
|
November 13 - 17 |
The dihedral group. Cosets and
Lagrange’s theorem. Homomorphisms and isomorphisms; Cayley’s theorem.
Group actions on sets: first definitions and properties. |
|
|
November 20 - 24 |
Group actions on sets: Examples and the Cauchy-Frobenius formula. Applications to Combinatorics. |
|
|
November 27 – December 1 |
Homomorphisms, normal
subgroups, quotient groups and the first isomorphism theorem. Examples.
Groups of low order. |
|
|
December 4 |
On the notion of cardinality. |
|
I will discuss the structure of the final exam. |
Instructor: Dr. Eyal Goren
Time of course: MW 10:35-11:55 BURN 1205
Office hours: Friday 10:00-12:00, BURN 1108,
or by appointment.
Syllabus (Calendar description): Basic
point-set topology, including connectedness, compactness, product spaces,
separation axioms, metric spaces. The fundamental group and covering spaces.
Simplicial complexes. Singular and simplicial homology. Part of the material of
MATH 577 may be covered as well.
Remark on Syllabus. At this point I am tending to
spend more time on the fundamental group and covering spaces and some
applications to group theory. I don’t believe will get to the material of 577
and our discussion of homology will be sketchy (a thorough discussion is a
course by itself).
Prerequisites: MATH354 or instructor’s approval
(IN fact for 576 just the basic theory of metric spaces is needed)
Text book: James R. Munkres: Topology (2nd edition). Prentice Hall. ISBN
0131816292
Additional
textbooks: Hocking and Young/Topology,
Massey/Algebraic Topology: An Introduction. Lipschutz/General
Topology.
Evaulation method:
·
25 % Assignments.
25% Midterm. MIDTERM
EXAM WILL TAKE PLACE WEDNESDAY OCTOBER 25, 10:30-12:30. IN CLASS. IT WILL BE A
2 HOURS EXAM CONSISTING MOSTLY OF PROBLEMS
MIDTERM
GRADES (the absolute grade is how much you got out of the maximum
possible of 125 points. I will however give you at most 100 points towards the
calculation of the final grade).
·
50% Final. (Take home)
Calculation of final grade: As described above.
Students achieving a grade (strictly) lower than B will have the option of
make-up/additional work to improve their grade to B.
Academic integrity:
Detailed syllabus (To be updated during the semester)
Date |
Material |
Assignment |
Misc. |
September 6-8 |
Def. Top. spaces, examples, bases |
|
|
September 11-15 |
closed sets, closure, boundary, subspaces, cont’s maps |
||
September 18-22 |
Products, compactness, sequential compactness and
compactness in metric spaces. |
At this point we would have finished Chapter 2 and started
Chapter 3. I advise you read Chapter 2 in its entirety and solve all the
exercises. |
|
September 25 - 29 |
Tychonoff’s
theorem, topological groups, quotient spaces. |
|
|
October 2-6 |
Topological groups (cont’d). Connected, pathwise-connected, local properties. |
|
October 2 there’s no class. A make-up class is scheduled
for October 3 10:00 – 11:30 BURN 920. |
October 9-13 |
One point compactification. Separation axioms. Urysohn’s Lemma, The embedding theorem, Uryshohn’s metrization theorem. |
October 9 there’s no class (Thanksgiving). There’s a class
October 10, same place, same time. |
|
October 16-20 |
Stone-Cech compactification. The
fundamental group - overview. Homotopy of maps; the homotopy groups \pi_n(X,x). |
|
|
October 23 – 27. |
Retract and deformation retract. Homotopic
spaces and the fundamental group. A topological group has an abelian fund.
group. Survery of results on homotopy groups of
spheres. |
|
|
October 30 – November 3 |
The fund. gp. of S^n (n>1) is trivial. Covering spaces and the
fundamental group. The fund. gp. of S^1 is Z.
Lifting lemmas. |
|
|
November 6 - 10 |
Classification of covering spaces, and deck transformations.
(Higher homotopy groups of covering spaces) |
|
|
November 13 – 17 |
Existence of covering spaces. Applications of the
fundamental group, including Brouwer’s fixed point
theorem, Borsuk-Ulam, the ham sandwich theorem, fund’t theorem of arithmetic. Free groups, free products,
amalgamated products. |
|
|
November 20-24 |
The Seifert-Van Kampen theorem
and the fundl gp of a
bouquet of circles. Applications to group theory. Simplicial complexes and
the chain complex. Simplicial homology: def’n and
examples. The Euler Poincare characteristic. |
|
|
November 27 – December 1 |
Singular homology, excision, relative homology, Mayer-Vietoris, examples. No retract B^n
-> S^{n-1}. CW complexes. Cellular homology. Examples. |
|
|
December 4 |
Functoriality. |
|
Take home exam will be handed out. |
Instructor: Dr. Eyal Goren
Time of course: MWF 8:35-9:25 in Burnside Hall
1B36
Office hours: Monday 10:00 – 12:00.
TA: Gabriel Chenevert. Office: BURN 1034. TA office hours: Friday 1:00 – 2:00.
Tutorial
hours and location: Monday 17:30 – 18:30 in BURN 920.
Syllabus (Calendar description): Linear maps and their matrix
representation. Determinants. Canonical forms. Duality. Bilinear and quadratic
forms. Real and complex inner product spaces. Diagonalization
of self-adjoint operators.
Prerequisites: MATH 235 or permission of the
department.
Text book: Schaum's Outline of
Linear Algebra by Seymour Lipschutz and Marc Lipson (3rd
edition)
Additional
textbooks: Not really necessary. I plan to
have on-line notes and I think that together with the text-book it will
suffice.
Evaulation method:
·
20% Assignments. Assignments are
handed out and submitted on Mondays. Deadline: 12:00.
·
80% Final.
Calculation of final grade: The better of the method
outlined here and the grade of the final exam alone. Same rule applies for
deferred/supplemental. Students will NOT have the option of make-up/additional
work to improve their grades.
Academic integrity:
Notes
(version: April 10, 2007). This is the final version. Please point out any
typos you find while reviewing the material. The corrections will be published here.
Final Exam and Deffered/Supplemental Exam: The structure and difficulty of the exams is the
same. They cover, in principle, all the material appearing in the course notes
and assignments (including the last lecture!). There are two parts; in the
first you need to answer 3 out of 4 questions (for a total of maximum 51
points). You would be asked to prove “Theorems” we’ve proven in class, where
“Theorem” here refers to any statement we’ve proven, even if it was called
“Lemma” or “Proposition”. In the second part you need to solve 5 out of 5
questions (for a total of maximum 50 points). It is more computational in nature,
but may include easy “theoretical” exercises as well. The questions here are
similar in style and difficulty to questions that were given in the
assignments.
For the purpose of
calculation of the final grade, for the contribution of the assignments I shall
use the best 8 out of 10 grades you have.
Office hours
prior to the exam: Monday, April 16, 10:00 – 12:00. Thursday,
April 19, 08:30 – 10:00.
Detailed syllabus (To be updated during the semester)
Date |
Material |
Assignment |
Misc. |
January 3 - 5 |
Introduction and
motivation. Defn of v. space and subspace.
Examples. |
|
|
January 8 - 12 |
Sum and intersection,
direct sum. Span. Linear dependence.
Basis and dimension. |
|
|
January 15 - 19 |
Coordinates and change of basis.
Linear maps: definitions and first examples.
Dim(V) = dim(Ker) + dim(Im).
Applications. |
||
January 22 - 26 |
Proof of Dim(V) = dim(Ker) + dim(Im). Quotient
spaces. Direct sums. Nipotent operators and
projections. |
Monday January 22, make-up class from 5:30 –
6:30 in BURN 920. |
|
January 29 – February 2 |
Linear maps and matrices. Signs
of permuations. The existence of determinant.
Uniqueness and multiplicativity of determinants. |
Monday January 29, make-up class from 5:30 –
6:30 in BURN 920. |
|
February 5 - 9 |
Geomteric interpretation of determinants. Laplace’sformulas and the adjoint
matrix. Systems of linear equations. Row and column space and rank. |
|
|
February 12 – 16 |
Two matrices in REF with same
row-space are equal. Rank_r(A) = Rank_c(A). Cramer’s rule, calculating the inverse matrix.
The dual vector space. |
|
|
February 19 -23 |
J |
J |
Study break |
February 26 – March 2 |
The dual vector space. Inner
product spaces. Cauchy-Schwartz inequality. Gram-Schmidt. |
Note: In the meanwhile I put a
new version of the notes. Don’t use Prop. 7.2.9 (in the current version) when
solving question 1. |
|
March 5- 9 |
Orthogonal projection. Eigenvalues and eigenvectors. The characteristic
polynomial. Diagonalization. |
|
|
March 12 – 16 |
The characteristic polynomial.
Arithmetic and geometric multiplicities. The minimal polynomial. Cayley-Hamilton. |
||
March 19 – 23 |
The primary decomposition
theorem. Diagonalization again. Examples. The |
||
March 26 – 30 |
The |
March 26 is elections day. No
class or tutorial. |
|
April 2 - 4 |
Symmetric, self adjoint and normal operators. (Cont’d). Applications: the
Principal Axis Theorem, Inner products and more. |
(Do not submit) |
April 6 and 9: no classes (Easter) |
April 11 |
Normal operators. |
Class starts at 8:10. Last Tutorial is today at 5:30, |
|