Third-Year Mathematics Major

Semester 1

1) Algebra (4 hours/week)

Group Theory
1.    Definitions (subgroup, distinguished subgroup, quotient, factorization of a morphism...) Order of a finite group. Lagrange’s theorem. Examples of cyclic groups.
2.    Group operation on a set: orbits, stabilizers. Examples of common operations. Conjugacy classes in a group, conjugacy classes of subgroups... Applications: p-groups, Sylow theorems,...
3.    Generator sets of groups.
4. Decomposition of a group: direct and semidirect products.
5. Classical groups (linear, orthogonal, etc.)
6.    Examples of simple groups.

Ring theory
1. Definitions (rings, ideals, quotients, factorization of a morphism)
2.    Principal rings, Euclidean rings.
3. k[X]. Irreducible polynomials

Field theory
1.    Fields of fractions of an integral ring. Q. K(X).
2. Characteristic of a field. Prime subfields. Frobénius homomorphism in characteristic p.
3.    Finite fields.

2) Fourier and Hilbert analysis (5 h/week)

The Lebesgue integral (axiomatic treatment)
Lp spaces
Hilbert spaces: algebra and geometry
Hilbert spaces: analysis
Fourier series (concrete examples of Hilbert bases)
Other Hilbert bases
Discrete Fourier transform
Fourier transform
Interlude: Overview of Applications of Fourier Analysis
o Applications in Other Branches of Mathematics:
o Applications to Image Processing:
Distributions
Periodic Sobolev spaces in 1 dimension
Periodic Sobolev spaces in 2 dimensions
Linear PDEs in periodic Sobolev spaces
Variational interpretation of these PDEs

3) Numerical Analysis of ODE (4 hrs/week)

Objectives: When systems of ordinary differential equations do not play a direct role in modeling physical, chemical, biological, or even financial models, these systems emerge following spatial semi-discretization of PDEs; these are then systems that can involve up to several hundred million unknowns. The questions that naturally arise concern the existence, uniqueness, and lifetime of solutions, as well as their behavior for large time scales... Systems with solutions that can be easily expressed explicitly are rather rare, particularly for large systems; qualitative and/or numerical analysis proves to be an essential mathematical tool. The questions that need to be addressed concern the quality and accuracy of numerical solutions, the preservation of physical properties, and the behavior of transient or long-term solutions.
This course addresses these questions and also covers methods of integration, interpolation, and the solution of nonlinear problems, which are highly useful for the construction of numerical schemes. The study of delay ODE, ODE with integro-differential terms, or ODE with boundary conditions may be briefly covered.

Course Outline

1. Existence and Uniqueness Theorems for Solutions
2. Linear and Nonlinear Differential Systems. Qualitative analysis. Study of critical points. Lyapunov stability theorem.
3. Numerical methods for solving ODE.
4. Applications to boundary value problems.
5. Representation of real numbers and study of rounding errors
6. Approximate interpolation and integration
7. Solving nonlinear equations

Course Structure
Course concepts are illustrated through weekly lab sessions in Python. Assessment consists of a two-part exam (theoretical and numerical) and a continuous assessment grade (homework and text analysis).

4) Integration and Probability (5 hours/week)

Measure theory

1. Tribes, measures, Dynkin’s theorem. Abstract integration: integration of positive measurable functions, the monotone convergence theorem, and Fatou’s theorem; integrable functions, dominated convergence, and differentiation under the sum sign.
2. Construction of measures: Carathéodory’s extension theorem. Case of the Lebesgue measure on R^d. Change-of-variables formula. Stieljes integral, Radon measures, Riesz representation theorem.
3. Product measures, Fubini–Tonelli and Fubini–Lebesgue theorems. Integration by parts.
4. Lp spaces, Jensen’s, Hölder’s, and Minkowski’s inequalities, density theorems (no proofs here; the general theory is not covered in this course). Radon-Nykodyn derivative.

Introduction to Probability

1. Historical introduction. Probabilistic language and modeling.
2. Random variables, mathematical expectation. Distribution of a random variable and classical distributions. Skorokhod’s representation theorem. Concept of a tribe generated by a random variable. Moments, characteristic functions. Generating functions.
3. Independence: definitions and characterizations. Coalition theorem. Asymptotic tribe, all-or-nothing distribution, Borel-Cantelli lemmas. Strong laws of large numbers. Applications.
4. Convergence in distribution: definitions and characterizations, Levy’s theorem. Comparison of different types of convergence. Central limit theorem. Applications.
5. Topology and Differential Calculus (4.5 hours/week)

General Topology

1. Fixed-Point Theorem
2. Cauchy–Lipschitz Theorem
3. Differentiability, Mean Value Inequality, and Taylor Formulas
4. Local inversion theorems and implicit function theorems, submersion, immersion, constant rank theorem
5. Subvarieties of Rn
6. Optimization

6) An introductory module (3 to 5 hours/week)

It is chosen from among the following modules (subject to change):

• Quantum Mechanics (Department of Physics)
• Computability and Complexity (Department of Computer Science)

Semester 2

7) Complex Analysis (4 hours/week)

1. Holomorphic functions on an open set of C. Operations on holomorphic functions. Cauchy-Riemann equations. Principal value of the logarithm.
2. Integral of a function along a path in C. Index of a path with respect to a point.
3. Cauchy’s formula for holomorphic functions on a convex open set of C. Homotopy.
4. Cauchy’s formula for holomorphic functions on a simply connected open set of C.
5. Analytic functions. Morera’s theorem. Liouville’s theorem and the d’Alembert–Gauss theorem. Zeros of holomorphic functions. Elementary principles of analytic extension. The maximum principle and the Phragmen–Lindelöf method.
6. Application to interpolation. Topology of H(C). Series of holomorphic functions
7. Isolated singularity of a holomorphic function. Meromorphic functions. The residue theorem and its applications.
8. Conformal mappings, Riemann’s theorem. Applications to fluid mechanics.
9. Special functions and their applications: Riemann’s gamma and zeta functions and Weierstrass’s p-function.

8) Linear Algebra ((4 hours/week)

Summary: to be completed

Syllabus

1. To be completed

9) Series of 3 mini-courses (3 weeks each)

1. Introduction to complex systems
2. Geometric measure theory
3. Analysis of PDEs

10) Introductory Research Internship at the Borelli Center

Minimum duration of 5 months, generally in groups of 2, part-time until mid-May (at least 2 days/week) and full-time thereafter.

11) Reading Group

12) “Research Panorama” Seminar

A “Research Panorama” seminar provides an opportunity to explore research in mathematics and current topics. Attendance at the seminar counts toward completion of the bachelor’s degree.

13) English Courses

English courses round out the program.