# Courses 2006-07

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last updated: April 11, 2007

## Algebra 1, MATH235

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: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

(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. Solutions to Ass. 8 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.

## Topology and Geometry I, MATH576

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 CHOSEN FROM THE TEXT BOOK OR ASSIGNMENTS. THE MATERIAL IS EVERYTHING UNTIL THE FUNDAMENTAL GROUP. WE COVERED SECTIONS 12 – 29, 31 – 34, 37-38 (THOUGH IN TOPICS CONCERNING METRIC SPACE OUR DISCUSSION WAS LACKING COMPARED TO THE TEXTBOOK, BECAUSE I FIGURED YOU’VE SEEN THIS MATERIAL ELSEWHERE. YOU ARE ONLY “ACCOUNTABLE” TO WHAT WE CONVERED IN CLASS). I WILL NOT EXAMINE YOU ON PROOFS, THE EXAM THOUGH IS WITHOUT BOOKS OR NOTEBOOKS, SO YOU NEED TO MEMORIZE THE DEFINITIONS AND RESULTS.

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: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

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.

Honors Algebra 2 – MATH 251

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.

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: McGill University values academic integrity. Therefore, all students must understand the meaning and consequences of cheating, plagiarism and other academic offences under the Code of Student Conduct and Disciplinary Procedures (see www.mcgill.ca/integrity for more information).

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 Jordan canonical form. ExampleB March 26 – 30 The Jordan canonical form.  Symmetric, self adjoint and normal operators. 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,