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# Graduate Courses

## FEATURED COURSES [fall 2023]

APMA 900 • Asymptotic Analysis of Differential Equations

Exact & Approximate Methods for Understanding DEs One aim of this course is to provide an introduction to exact methods for the solution of ordinary and partial differential equations (ODEs & PDEs). Fourier series methods for solving linear DEs are extended to integral solution methods that include the Fourier and Laplace transforms. Investigation of this solution perspective establishes the close connection between complex variable theory and DEs. A different generalization of the Fourier idea leads to the development of Sturm-Liouville eigenfunctions, function (Hilbert) spaces and special function theory. But many ODEs and PDEs encountered in applications are not amenable to exact solution, particularly those involving nonlinearity. Another aim is to introduce a variety of approximation methods that extend our analytical toolbox beyond exact theory. Nonlinear ODEs systems provide many example contexts for the development of these powerful tools. The results can also be useful in benchmarking numericallycomputed solutions, and even decoding exact solutions whose formula complexity defies interpretation. Perturbation theory analyzes problems that are “nearby” to those with known exact properties. This perspective also gives mathematical insight into the consequences of approximations that neglect complicating effects in the reduction of model equations. Finally, yet other types of asymptotic methods address singular situations where small changes to DE problems have a large impact on their solution. Lectures will be based upon a case-study approach of ODE & PDE examples that draw from the interests of course participants. Computational graphics will be an important tool for the lectures and assigned work. Visualization and numerical computing will involve the use and modification of Python & Matlab scripts.

A course on algorithms for algebraic computation and tools for computing with multivariate polynomials, polynomial ideals, matrices and algebraic numbers. Tools include the Fast Fourier Transform, Groebner bases, and the Schwartz-Zippel Lemma. We will use Maple as a calculator and as a programming language to implement algorithms. Instruction in Maple usage and Maple programming will be provided.

__Course Outline
__1. Getting Started (2 weeks)

Programming in Maple tutorial. Analysis of algorithms tutorial. Algorithms for polynomial interpolation. The extended Euclidean algorithm.

2. Algorithms for Linear Algebra (2 weeks)

Solving Ax = b over a field using Gaussian elimination. Fraction-free algorithms for computing det(A) and solving Ax = b. The Berkowitz division free algorithm for computing det(A − λI). Solving Ax = b over Q using p-adic lifting and rational number reconstruction.

3. The Fast Fourier Transform and applications (1.5 weeks)

Fast polynomial multiplication, fast division and fast multi-point evaluation.

4. Computing with multivariate polynomials (1.5 weeks)

Data structures for multivariate polynomials. Term orderings. Multiplication and division using repeated merging and binary heaps.

5. Groebner bases and applications (2 weeks)

Ideals in polynomial rings. Monomial orderings. Ideal membership and polynomial division. The Hilbert basis theorem. Groebner bases. Buchberger’s algorithm. The elimination theorem. Applications of Groebner bases.

6. Computing with algebraic numbers (1.5 weeks)

Computing in Q(α) and Q(α1, α2, ..., αn). Primitive elements, norms and resultants. Factoring polynomials over Q(α).

7. Sparse polynomial interpolation (1.5 weeks)

Sparse polynomials. The Schwartz-Zippel Lemma. Zippel’s sparse interpolation. BenOr/Tiwari sparse interpolation.

## FEATURED COURSES [spring 2023]

**APMA 990 - Mathematics of Evolution**

Much of our understanding of evolution, the process shaping the beautiful biological diversity in our world, is grounded in equally elegant mathematics. In this course we will cover the mathematical description of evolution. Involving a wide range of topics, from the analysis of non-linear dynamics to stochastic processes and partial differential equations, this course will challenge you to take mathematical principals and apply them to the natural world. Throughout this course we will focus particularly on addressing important contemporary existential questions with mathematical models, for example applications of evolution to conservation and public health.

**MATH 495/795 - Differential Geometry: Curves – Surfaces – Manifolds**

Differential geometry is a core subject in mathematics. The tools of differential geometry are routinely used in most subfields of mathematics, while at the same time they have wide applications in various theoretical as well as applied disciplines such as physics (theory of relativity, mechanics), econometrics, computer graphics, signal processing and statistics.

The course is a mathematically rigorous introduction to the fundamental notions in differential geometry: curves, surfaces and abstract differentiable manifolds. About half of the course focuses on curves and surfaces in three-dimensional Euclidean space and introduces key concepts such as first and second fundamental forms, Gauss curvature, covariant derivative and geodesics. The second part of the course centres on Riemannian manifolds, and presents important concepts such as Riemannian connection and the curvature tensor. Towards the very end of the class we will discuss about the General Theory of Relativity, a remarkable application of differential geometry

**MATH 843 - Analytic and Diophantine Number Theory**

More detailed course topics are

- Review of Basic knowledge of undergraduate number theory such as fundamental theorem of arithmetic, prime factorization, modulo arithmetic, primitive roots, arithmetical functions and Dirichlet multiplication (reference: Introduction to Analytic Number Theory by T.M. Apostol)

- Dirichlet series and Dirichlet characters

- Theory of Riemann zeta function and Dirichlet L-function: analytic continuation, Euler product formula, functional equation, zeros free regions, density of zeros and Riemann Hypothesis.
- Analytic proof of Prime Number theorem and prime in Arithmetic Progression
- Hardy-Littlewood Circle method
- Vinogradov three prime theorem
- Other applications.

Students are only required to have prerequisite of undergraduate level number theory course. Lecture notes will be provided. Course work required is homework for every 2-3 weeks and 3 hours final exam at the end.

How do computer algebra systems like Maple do algebra and calculus?

You will also learn Some algebra e.g. ring homomorphisms. How to use Maple to do algebra and calculus. How Maple represents a mathematical formula on the computer. How to program differentiation and integration in Maple. The Fast Fourier Transform and how to apply it to multiply fast. Hensel's Lemma and how to use it to factor a polynomial. How to analyze the speed of an algorithm

## FEATURED COURSES [FALL 2022]

**MATH 708 - Discrete Optimization** - Surrey Campus

Discrete optimization is a field that has grown almost from scratch in the past 70 years. This development is driven in part from its applicability to a wide range of practical problems, such as scheduling and network design, and its close ties to computer science.

**MATH 795 - G100 Introduction to Topology**

Some of us like coffee. Some of us like donuts. In what sense is a coffee cup similar to a donut? Such a question deals with mathematics in field of topology.

**MATH 800 - G200 Optimal Transportation - remote**

The goal of this course is to teach the students to use mathematical models to improve and optimize public transport networks.

There will be two or three guest lecturers for this course Professor Robert Shorten from Imperial College London, Dr Emanuele Crisostomi from University of Pizza, and/or Professor Tarek Sayed from University of British Columbia.

**MATH 716 - Numerical Analysis II**

Differential Equations (DEs) are the building blocks for mathematical models of physical processes found throughout science and engineering. Since the vast majority of DEs cannot be solved exactly, it is vital to develop algorithms to compute approximate solutions. Suffice to say, many of the things we take for granted in the modern world rely critically on the fast, accurate and robust solution of DEs.

**MATH 795 - G200 Simulation Modelling with Applications **

How many beds do hospitals need to reduce emergency department overcrowding? How many COVID-19 cases will there be in the fall? How can airlines optimize their routes? How can supply chains for manufacturing be made more efficient? These are just some of the questions that simulation modelling is used to answer.

## FEATURED COURSES [SPRING 2022]

**MATH 800 - Introduction to Complex Geometry**

In this introductory course, we will begin by learning the necessary basics of multivariate complex analysis. Armed with a good understanding of the local situation, we will then begin our study of complex manifolds. Topics discussed will include differential forms, (almost) complex structures, sheaves, and vector bundles.

**APMA 905 - Applied Functional Analysis**

This will be an introduction to functional analysis and some of its applications. In a nutshell, we'll investigate the properties of continuous linear mappings of infinite-dimensional vector spaces. See full course promo.

## Featured Course [Summer 2021]

**APMA 940 - Mathematics of Data Science**

Theory and algorithms for problems in data science with an emphasis on mathematical aspects. Topics may include dimension reduction, supervised learning, including regression and classification, unsupervised learning, including clustering and latent variable modeling, deep learning, algorithms for big data, and foundations of learning.

## Featured Courses [Spring 2021]

**APMA 930 - Computational Fluid Dynamics**

Basic equations governing compressible and incompressible fluid mechanics. Finite difference and finite volume schemes for hyperbolic, elliptic, and parabolic partial differential equations. Practical applications in low Reynolds number flow, high-speed gas dynamics, and porous media flow. Software design and use of public-domain codes. Students with credit for MATH 930 may not complete this course for further credit.

How do we represent formulas on a computer? How fast can we multiply integers and polynomials? Can we factor polynomials in polynomial time? This course is about computing with mathematical objects symbolically. This includes numbers, polynomials, and elementary functions.

**MATH 740 - Galois Theory (cross-listed with MATH 440)**

Galois theory studies roots of polynomial equations, combining field theory and group theory to study symmetries of these equations. Famously, these ideas led to a proof that (unlike the quadratic formula for degree-2 polynomials) no formula exists to solve polynomial equations of degree 5 or more. Proving this theorem is one goal of this class.

**APMA 990 - Complex Analysis**

Complex analysis is a subject whose importance has a broad reach within mathematics. This course will revisit the foundations of the study of analytic functions, as well as aim to demonstrate the reach of complex analysis over a wide scope of theoretical, calculational, geometrical and computational questions in mathematics. Special topics may include the Riemann mapping theorem, conformal mapping and Fourier integral theory.

## Featured Courses [Fall 2020]

**APMA 920 - Numerical Linear Algebra**

The course is aimed at students interested in scientific computing and modeling. We will cover a variety of topics in numerical linear algebra and its applications with an emphasis on understanding stability (robustness) and speed. We will develop, analyze and implement a range of algorithms and see how they work in practice and theory. We program and test our methods in Matlab – almost no prior knowledge is assumed.

**MATH 708 - Discrete Optimization (cross-listed with MATH 408)**

Discrete optimization is a field that has grown almost from scratch in the past 70 years. This development is driven in part from its applicability to a wide range of practical problems, such as scheduling and network design, and its close ties to computer science. However, it is also a beautiful mathematical topic that connects to diverse areas of mathematics, including classical problems in combinatorics, algebra and geometry.

**MATH 747 - Coding Theory (cross-listed with MATH 447)**

How did the Mariner 9 space probe transmit high-resolution photos of Mars to Earth 50 years ago? How can a compact disc play back music perfectly even after the disc surface is damaged? How do cellphones maintain call quality despite signal reflections from buildings and noise from other calls? Explore the hidden mathematics behind modern communications.

## SFU Calendar Listings

**Mathematics 600-Level Courses**

MATH 603-4: Foundations of Mathematics *

MATH 604-4: Geometry *

** These courses cannot be taken toward Mathematics MSc or PhD program requirements*

**Mathematics 700-Level Courses**

MATH 701-3: Computer Algebra

MATH 708-3: Discrete Optimization

MATH 709-3: Numerical Linear Algebra and Optimization

MATH 716-3: Numerical Analysis II

MATH 718-3: Partial Differential Equations

MATH 719-3: Linear Analysis

MATH 724-3: Applications of Complex Analysis

MATH 725-3: Real Analysis

MATH 739-3: Algebraic Systems

MATH 740-3: Galois Theory

MATH 741-3: Commutative Algebra and Algebraic Geometry

MATH 742-3: Cryptography

MATH 743-3: Combinatorial Theory

MATH 745-3: Graph Theory

MATH 747-3: Coding Theory

MATH 748-3: Network Flows

MATH 761-3: Continuous Mathematical Models

MATH 762-3: Fluid Dynamics

MATH 767-3: Dynamical Systems

MATH 770-3: Variational Calculus

MATH 795-3: Selected Topics in Applied Mathematics

MATH 796-3: Selected Topics in Mathematics

**Mathematics 800-Level Courses**

MATH 800-4: Mathematics: Selected Topics

MATH 801-4: Computer Algebra

MATH 808-4: Advanced Linear Programming

MATH 817-4: Groups and Rings

MATH 818-4: Algebra and Geometry

MATH 819-4: Algebra: Selected Topics

MATH 820-4: Graph Theory

MATH 821-4: Combinatorics

MATH 827-4: Discrete Mathematics: Selected Topics

MATH 831-4: Real Analysis I

MATH 833-4: Analysis: Selected Topics

MATH 841-4: Topology: Selected Topics

MATH 842-4: Algebraic Number Theory

MATH 843-4: Analytic and Diophantine Number Theory

MATH 845-4: Number Theory: Selected Topics

MATH 846-4: Cryptography

MATH 875-0: PhD Preliminary Examination

> *Resources*

MATH 876-0: PhD Comprehensive Examination

> *Resources*

MATH 877-1: Supplementary Reading

MATH 879-0: PhD Thesis Proposal

MATH 880-6: MSc Project

MATH 888-0: Ph.D. Comprehensive Examination: Operations Research

MATH 890-0: Practicum I

MATH 891-0: Practicum II

MATH 894-2: Reading

MATH 895-4: Reading

MATH 898-6: MSc Thesis

MATH 899-12: PhD Thesis

**Mathematics 900-Level Courses: Applied and Computational Mathematics**

APMA 900-4: Asymptotic Analysis of Differential Equations

APMA 901-4: Partial Differential Equations

APMA 905-4: Applied Functional Analysis

APMA 912-4: Advanced Partial Differential Equations

APMA 920-4: Numerical Linear Analysis

APMA 922-4: Numerical Solution of Partial Differential Equations

APMA 923-4: Numerical Methods in Continuous Optimization

APMA 929-4: Selected Topics in Numerical Methods

APMA 930-4: Computational Fluid Dynamics

APMA 934-4: Selected Topics in Fluid Dynamics

APMA 935-4: Analysis and Computation of Models

APMA 939-4: Selected Topics in Mathematical Image Processing

APMA 940-4: Mathematics of Data Science

APMA 981-4: Selected Topics in Continuum Mechanics

APMA 982-4: Selected Topics in Mathematical Physics

APMA 990-4: Selected Topics in Applied Mathematics

APMA 995-0: PhD Oral Candidacy Exam