Academic year 2021-2022

Fall 2021

MAST 699/2, A (MATH 494 & MAST 833) Algebraic number theory
The course is entirely on-line, but some office hours are in presence.
SCHEDULE: WEDNESDAY-FRIDAY 16:15-17:30
OFFICE HOURS: Monday 13:30 - 14:30 (LB 927)

OUTLINE: This is a first course in the study of algebraic number fields. In the first part of the course, we will concentrate on proving the two main basic results in the subject: the ideal class group is finite and the unit group is finitely generated. Other topics will include: the distribution of ideals, the Dedekind zeta function and the class number formula, and the Artin map.

TEXT: Number Fields (Marcus), Algebraic Number Theory (Janusz) or your favorite text on the subject.

EVALUATION: Assignments and course presentations.

ASSIGNMENTS:
Assignement 1. Due Friday September 24.
Assignement 2. Due Friday October 8.

Academic year 2020-2021

Fall 2020

MATH369 Abstract Algebra I: Group theory
The course is entirely on-line, including the exams.

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.
Course Outline

SCHEDULE: W-F 11h45-13h00
OFFICE HOURS: W-F 13h15-14h15

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%

MIDTERM: Friday October 30

For the assignments, I am counting the best 8 ones out of 10.
The assignments have to be sent by email AS A PDF FILE to the marker at the due date BEFORE 14h00. Late assignments will NOT be accepted.
The assignments and their solutions will be available on this WEB page.

ASSIGNMENTS:
Assignement 1. Due Friday September 25. Solutions 1.
Assignement 2. Due Friday October 2. Solutions 2.
Assignement 3. Due Friday October 9. Solutions 3.
Assignement 4. Due Friday October 16. Solutions 4.
Assignement 5. Due Friday October 23. Solutions 5.
Assignement 6. Due Friday November 6. Solutions 6.
Assignement 7. Due Friday November 13. Solutions 7.
Assignement 8. Due Friday November 20. Solutions 8.
Assignement 9. Due Friday November 27. Solutions 9.
Assignement 10. Due Friday December 4. Solutions 10.

FINAL EXAM:
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.

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

FINAL EXAM:
Practice Final Exam
Solutions to Practice Final Exam

ASSIGNMENTS:
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.

FINAL EXAM:
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

OFFICE HOURS: TBA, Room LB-927.9

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)

OFFICE HOURS: TBA, Room LB-927.9

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.

ASSIGNMENTS:

ASSIGMENT 1:
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.

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

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

ASSIGMENT 4:
Due Wednesday November 14.
Assignement 4.

ASSIGMENT 5:
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

MIDTERM EXAM:
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

OFFICE HOURS FOR FINAL EXAM (LB927-9):
WEDNESDAY APRIL 24, 14h00-16h00
FRIDAY APRIL 26, 14h00-16h00

SAMPLES FOR FINAL EXAM:
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

TAKE HOME FINAL EXAM:
Start date: Sunday April 21, 9h00 (AM)
End date: Tuesday April 23, 9h00 (AM)

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