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last updated: April 11, 2007
Page URL: http://www.math.mcgill.ca/goren/courses0607.html
Algebra 1, Topology and Geometry 1, Honors Algebra 2
Instructor: Dr. Eyal Goren
Time of course: MWF 8:359:25 in Arts Building
W120.
Office hours: Friday 10:0012:00, 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_2/70263118094508854
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 10^{th} 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 makeup/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 68 
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 1822 
Rings and Fields. Divisibility, gcd,
Euclidean algorithm for integers. Primes and the sieve of Eratosthenes. 


September 2529 
The Fundamental Thm of
Arithmetic, infinity of primes, 2^(1/2) is irrational, equivalence relations. 


October 26 
Congruences,
Fermat’s little theorem, computing and solving equations in Z_{n}.
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 913 
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 1620 
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^35*x^25*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 CauchyFrobenius 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 10:3511:55 BURN 1205
Office hours: Friday 10:0012:00, BURN 1108,
or by appointment.
Syllabus (Calendar description): Basic
pointset 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, 10:3012:30. 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
makeup/additional work to improve their grade to B.
Academic integrity:
Detailed syllabus (To be updated during the semester)
Date 
Material 
Assignment 
Misc. 
September 68 
Def. Top. spaces, examples, bases 


September 1115 
closed sets, closure, boundary, subspaces, cont’s maps 

September 1822 
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 26 
Topological groups (cont’d). Connected, pathwiseconnected, local properties. 

October 2 there’s no class. A makeup class is scheduled
for October 3 10:00 – 11:30 BURN 920. 
October 913 
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 1620 
StoneCech 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, BorsukUlam, the ham sandwich theorem, fund’t theorem of arithmetic. Free groups, free products,
amalgamated products. 


November 2024 
The SeifertVan 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, MayerVietoris, examples. No retract B^n
> S^{n1}. CW complexes. Cellular homology. Examples. 


December 4 
Functoriality. 

Take home exam will be handed out. 
Instructor: Dr. Eyal Goren
Time of course: MWF 8:359:25 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 selfadjoint operators.
Prerequisites: MATH 235 or permission of the
department.
Text book: Schaum's Outline of
Linear Algebra by Seymour Lipschutz and Marc Lipson (3^{rd}
edition)
Additional
textbooks: Not really necessary. I plan to
have online notes and I think that together with the textbook 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 makeup/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, makeup 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, makeup 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
rowspace 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. CauchySchwartz inequality. GramSchmidt. 
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. CayleyHamilton. 

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, 
