A course on algorithms for algebraic computation and tools for computing with long integers, multivariate polynomials, elementary functions. Tools include the Fast Fourier Transform, the Schwartz-Zippel Lemma and Hensel’s Lemma. We will use the computer algebra system Maple as a calculator and as a programming language to implement algorithms. The course will answer these questions.
How does one represent a mathematical formula like f(x) = xsin(2x) on the computer ?
How can we program differentiation e.g. f'(x) = sin(2x) + 2 cos(2x) ?
How can we simplify a mathematical formula?
How fast can we multiply long integers on a computer?
Is there an algorithm for calculating an antiderivative like Z ((2/x + (lnx/x))/ (1 + lnx)2 )dx?
How can we decide which algorithm is best?
What is the Fast Fourier Transform and why do many consider it the most important algorithm of all time?
How can we factor a polynomial like x5 − 5x4 + 15x3 − 23x2 + 23x − 5 ?
Exact & Approximate Methods for Understanding DEs One aim of this course is to provide an Math 708 (Discrete Optimization) will be offered in the Fall 2024 term at the 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. However, it is also a beautiful mathematical topic, that connects to diverse areas of mathematics, including classical problems in combinatorics, algebra and geometry.
Arguably the most famous discrete optimization problem is the Travelling Salesman Problem of finding the shortest tour through a given set of cities. For instance, the circuit below is the shortest tour through the 13509 cities in the mainland U.S.A. with a population of at least 500.
Math 708 is an introductory graduate course that focuses on formulating integer programs, and solving them with polyhedral methods. The main background required is some familiarity with linear programming.
The course starts with a review of linear ODE systems and of some basic results in ODE theory (existence and uniqueness, smooth dependence on initial conditions, maximal interval of dependence). The course then moves into nonlinear systems and covers general concepts such as phase portraits, stable, unstable and centre manifolds, nonlinear stability and Lyapunov functions, Hartman-Grobman Theorem, periodic orbits and limit cycles, Poincare-Bendixson theory. We will also look into various computational aspects that relate to the theoretical material. Computational explorations of phase portraits, invariant manifolds, limit cycles, etc will be performed using Matlab..
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.
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.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
