Academic year 2019-2020

Fall 2019

MATH369 Abstract Algebra I: Group theory

SCHEDULE: W-F 11h45-13h00
OFFICE HOURS: W-F 14h00-15h00, LB 927.09

TEXT: Abstract Algebra, by David S. Dummit and Richard M. Foote.

GRADING: The grading will be the higher of:
(a) Assignments 20%, Midterm 30%, Final 50%
(b) Assignments 20%, Final 80%
For the assignments, I am counting the best 7 ones out of 9.
The assignments and their solutions will be available on this WEB page.

OUTLINE: This is a standard first course in group theory, assuming no background in the subject, but some mathematical maturity, as the ability to work formally and prove results. We will cover (in parts) Chapter 1,2,3,4 in Dummit and Foote.

MIDTERM: Friday October 25
Solutions to Midterm

Practice Final Exam
Solutions to Practice Final Exam

Assignement 1. Due Friday September 20. Solutions 1.
Assignement 2. Due Friday September 27. Solutions 2.
Assignement 3. Due Friday October 4. Solutions 3.
Assignement 4. Due Friday October 11. Solutions 4.
Assignement 5. Due Friday October 18. Solutions 5.
Assignement 6. Due Friday November 8. Solutions 6.
Assignement 7. Due Friday November 15. Solutions 7.
Assignement 8. Due Friday November 22. Solutions 8.
Assignement 9. Due Monday December 2. Solutions 9.

The final exam will cover all the material that we saw in class during the term. The material that is in the textbook but that we did not covered in class is not included. Here are the sections of the textbook that we covered (some of them only partially).
Section 0.3
Chapter 1, all sections.
Chapter 2, all sections except 2.4.
Chapter 3, all sections except 3.4.
Chapter 4, 1.2, 4.2, 4.2 and part of 4.5.

Winter 2020

MATH470/4 Abstract Algebra II: Ring theory

SCHEDULE: Wednesday-Friday 10h15-11h30


TOPICS: Introduction to the theory of rings and modules: rings, ideals, quotients, ring of fractions, euclidean domains, principal ideals domain, unique factorisation domains, polynomial rings; modules and vector spaces.

TEXTBOOK: Abstract Algebra by David S. Dummit and Richard M. Foote.

EVALUATION: The grading will be the higher of:
(a) Assignments 20%, Midterm 30%, Final 50%
(b) Assignments 20%, Final 80%

DATE OF MIDTERM: Friday March 6
Practice Midterm
Practice Midterm Solutions
Solutions to Midterm

Assignment 1 Due Friday January 24. Solutions to Assignement 1.
Assignment 2 Due Friday February 7. Solutions to Assignement 2.
Assignment 3 Due Friday February 21. Solutions to Assignement 3.
Assignment 4 Due Wednesday April 8. Solutions to Assignment 4.

Analytic Number Theory (MATH394, MAST 699/4, MAST 833)

SCHEDULE: W-F 16h15-17h30, Room LB 759-6 (access by the 6th floor)


TOPICS: This is an introduction to analytic number theory and L-functions. We will cover some elementary methods, and their generalisations to Dirichlet's proof that there are infinitely many primes in arithmetic progressions. We will then proceed to prove the prime number theorem providing an aymptotic for the number of primes, with an explicit error term.

TEXTBOOK: The distribution of prime numbers by D. Koukoulopoulos

EVALUATION: Assignments and presentation by the students.

Assignment 1 Due Friday January 31. Solutions 1
Assignment 2 Due Friday February 21.
Assignment 3 Due Wednesday April 1.

Students presentations:
Wednesday April 1 16h15-17h00: David Ayotte "Modular symbols"
Wednesday April 1 17h00-17h45: Sivasankar Nair "Distribution of discriminants of Abelian number fields"
Friday April 3 16h15-17h00: Habid Alizadeh "Vinogradov's Theorem on sums of three primes"
Friday April 3 17h00-17h45: Arghya Datta "Some Tauberian Theorems related to coin tossing"
Wednesday April 8 16h15-17h00: Danny Shu, "Dirichlet's Class Number Formula"
Wednesday April 8 17h00-17h45: Andy Ramirez-Cote "Effective bounds for class numbers"
Wednesday April 15 16h15-17h00: Arihant Jain "Artin's Conjecture on primitive roots"
Wednesday April 15 17h00-17h45: Neha Nanda "Equidistribution"
Friday April 17 16h15-17h00: Subham Roy "Zeroes of L-functions and random matrix theory"
Friday April 17 17h00-17h45:

Academic year 2018-2019

Fall 2018

MAST 699/2, E (MAST 833/2) Elliptic Curves        This course is intended to be an introduction to the topic of elliptic curves, covering the geometry of algebraic curves and elliptic curves, elliptic curves over the rational, over finite fields, over the complex numbers and local fields, and the Mordell-Weil Theorem.

SCHEDULE: Tuesday 16h00-17h30, Wednesday 13h30-15h00, Room LB-759-6 (access by the 6th floor)

OFFICE HOURS: Tuesday-Wednesday 15h00-16h00, Room LB-759-6 (access by the 6th floor)

TEXTBOOK: The Arithmetic of Elliptic Curves, by Joseph Silverman.


Due Wednesday September 19.
Exercises (1.1 or 1.2), 1.3, 1.6, 1.7, 1.8, 1.10, 1.11, 1.12 in Silverman.
Exercise on infinite Galois groups, Section 14.9, no 19, Dummit and Foote
B.SC. students: Do 2 of them.
M.SC. students: Do 3 of them.
Ph. D. students: Do 4 of them.
Partial solutions to Assignement 1.

Due Wednesday October 3.
Assignement 2.
Partial solutions to Assignement 2.

Due Wednesday October 24.
Assignement 3.
Partial solutions to Assignement 3.

Due Wednesday November 14.
Assignement 4.

Due Wednesday November 28.
Exercises 8.1, 8.2, 8.4, 8.12 (do 6 of them), 8.15, 8.19, 10.1 10.13, 10.14 in Silverman.
B.SC. students: Do 2 of them.
M.SC. students: Do 3 of them.
Ph. D. students: Do 4 of them.

Winter 2019

MATH 203/4 G       

SCHEDULE: Wednesday-Friday 11:45-13:00, MB 2.210

OFFICE HOURS: Wednesday-Friday 14:00-15:00, LB 927.09

Sunday March 10h00 AM
Section G (A to K) Room H-521
Section G (L to Z) Room H-561
Sample for Midterm Exam
Solutions for Midterm Exam

WEDNESDAY APRIL 24, 14h00-16h00
FRIDAY APRIL 26, 14h00-16h00

Fall 2018
Partial Solutions to Fall 2018 midterm
Winter 2018

MATH472/4 Galois Theory       

SCHEDULE: Monday 14:00-15:30, LB 928

Assignment 1
Solutions to assignment 1
Assignment 2
Solutions to assignment 2
Assignment 3
Solutions to assignment 3
Assignment 4
Solutions to assignment 4
Assignment 5
Solutions to assignment 5

Start date: Sunday April 21, 9h00 (AM)
End date: Tuesday April 23, 9h00 (AM)

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