Reading Courses

Our regular graduate course offerings are supplemented with reading courses. Topics of reading courses are usually relatively specialized. Faculty members can propose reading courses on virtually any topic, provided they provide a clear description of the material covered, the format of the course and the criteria on which the students will be evaluated.

As examples of the kind of courses we run, we list some below:

If you would like to see reading courses from a specific year, click on the desired year below:

201720162015 | 2014 |2013 | 2012 | 2011 | 2010 | 2009 | 2008
 

Spring 2017

Math 894-G100 Topics in Besov Spaces, Interpolation and Non-linear Approximation

Ben Adcock

Description:

In harmonic analysis and theory of partial differential equations (PDEs) one often works with a function f : Ω → R. In order to quantify the “size” of this function we introduce the notion of a norm that in turn allows us to define a function space. Examples of such spaces are the classical regularity spaces C α(Ω) and the Lebesgue spaces L p (Ω). In PDE theory we are usually concerned with the regularity of a function. For this purpose we often work with the Sobolev spaces Hs (Ω). Intuitively, elements of Hs (Ω) with s ∈ N are functions that have s weak derivatives. In certain applications we need to work with spaces that are more flexible than the Sobolev spaces and give a finer description of regularity. In this course we learn about a generalization known as the Besov spaces. Along the way, we will see some tools from approximation theory such as interpolation spaces that are useful in and of themselves. We consider certain topics in Fourier analysis and theory of wavelets as well as applications involving image processing, signal processing, inverse problems and compressed sensing

Grading:

70% Presenations
30% Final Project

Students will prepare and present each lecture, and distribute notes beforehand.  
Students will be evaluated on preparation, clarity and understanding of the material.

References:

[1] Michael E. Taylor, Partial Differential Equations I: Basic Theory, 2nd Edition, Springer, 2011.
[2] William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000.
[3] Ronald A. DeVore, Nonlinear approximation, Acta Numerica, 7, 51-150, 1998.
[4] Joran Bergh and Jorgen Lofstrom, Interpolation Spaces: An Introduction, Springer, 1970.

Pre-Requisites:

Functional analysis, PDEs. Enrolment by instructors permission only.

Fall 2016

Math 894-G100 Wavelets, Approximation Theory and Signal Processing

Ben Adcock

Description:

We will cover parts of Mallat’s “A Wavelet Tour of Signal Processing: The Sparse Way”, including:

• Chapter 1: Sparse Representations
• Chapter 2: The Fourier Kingdom
• Chapter 3: Discrete Revolution
• Chapter 6: Wavelet Zoom
• Chapter 7: Wavelet Bases
• Chapter 9: Approximations in Bases
• Chapter 10: Compression 

Format:

1 hour lecture every week prepared and presented by students + 30 minutes discussion (90 minutes total).

Grading:

70% Lectures
30% Final Project

Students will prepare and present each lecture, and distribute notes beforehand.  
Students will be evaluated on preparation, clarity and understanding of the material.

References:

A Wavelet Tour of Signal Processing: The Sparse Way - Stephane Mallat

Pre-Requisites:

Basic analysis, Fourier series. Enrolment by instructors permission only.

Math 895-G100 Introduction to Measure-Theoretic Probability

Paul Tupper

Description:

Content: We will be covering the first 13 chapters of Rosenthal's book:
1. Measure Theory
2. Probability Triples
3. Further Probabilistic foundations
4. Expected Values
5. Inequalities and Convergence
6. Distributions of Random Variables
7. Stochastic Processes
8. Discrete Markov Chains
9. More probability theorems.
10. Weak Convergence
11. Characteristic Functions
12. Decomposition of probability laws
13. Conditional Probability and Expectation

Format:

3hr+ meeting every week.  This time will be a combination of lectures, student presentations, and working on problems together.

Grading:

70% Homework
30% Final

References

A First Look at Rigorous Probability Theory, 2nd edition, by Jeffrey Rosenthal

Pre-Requisites:

N/A

Summer 2016

Math 895-G100 Boundary Element Methods

Nilima Nigam

Description:

This will be an advanced graduate course on the numerical analysis of boundary element methods, specifically for elliptic equations.

Detailed Outline:

- Review of Sobolev traces and the well-posedness of elliptic PDE.
- Boundary integral formulations of elliptic PDE. Calderon projections.
- Direct and indirect methods,  wellposedness and regularity theory.
- Boundary element methods: approximation theory of Galerkin methods
- Quadrature and fast solvers for weakly and strongly singular integral equations.
- Implementation of BEM methods via BEM++ (open source)

Grading:

Homework and Project (70%)
Final Presentation (30%)

References

Boundary Element Methods by Sauter and Schwab.

Pre-Requisites:

Permission of Instructor.
APMA 901, APMA 922.
Graduate-level functional analysis and Sobolev spaces will be assumed.

Spring 2016

Math 894-G100 Non-Linear Discrete Optimization

Tamon Stephen

Description:

This course is an introduction to techniques for non-linear discrete optimization. The plan is to cover Graver basis (augmentation) methods; Convex discrete maximization and cutting plane methods. 

Detailed Outline:

For Graver bases, we follow the introduction of [dLHK] (Chapter 3, with material from Chapters 1 and 2 as needed), and optionally Chapter 4. See Also Chapter 3 of [Onn].

For convex discrete maximization, Chapter 2 of [Onn].

For curring plane methods, [GLD] and [LSW].

Grading:

25% written exercises. 75% student presentations of course related material. 

References

[dLHK] Jesus A. De Loera, Raymond Hemmecke, and Matthias Koppe, Algebraic and Geometric Ideas in the Theory of Discrete Optimization. SIAM, 2013. 

[Onn] Shmuel Onn, Nonlinear Discrete Optimization. Zurich Lectures in Advanced Mathematics. European Mathematical Society, 2010. 

[GLS] Martin Grotschel, Laszlo Lovasz, and Alexander Chrijver, Geometric Algorithms and Combinatorial Optimization. Algorithms and Combinatorics: Study and Research Tests, Springer-Verlag, 1988. 

[LSW] Yin Tat Lee, Aaron Sidford, and Sam Chiu-wai Wong, A Faster Cutting Plane Method and it's Implications for Combinatorial and Convex Optimization. arXiv:1508.04874.

Pre-Requisites:

Permission of Instructor. Math 708 Recommended.

Math 894-G200 Singulairty Analysis and Combinatorial Enumuration

Marni Mishna

Description:

The topic of the course is complex analytic methods for asymptomic enumeration.  

Detailed Outline:

IV. Complex Analysis, Rational, and Meromorphic Asymptotics
1. Generating functions as analytic object
2. Analytic functions and meromorphic functions
3. Singularities and exponential growth of coefficients
4. Closure properties and computable bounds
5. Rational and meromorphic functions
6. Localization and singularities
7. Singularities and functional equations
8. Perspective

V. Applications of Rational and Meromorphic Asymptotics
1. A roadmap to rational and meromorphic asymptotics
2. The supercritical sequence schema
3. Regular specifications and languages
4. Nested sequences, lattice paths, and continued fractions
5. Paths in graphs and automata
6. Transfer matrix models

VI. Singularity Analysis of Generating functions
1. A glimpse of basic singularity analysis theory
2. Coefficient asymptotics for the standard scale
3. Transfers
4. The process of singularity analysis
5. Multiple singularities
6. Intermezzo: functions amenable to singularity analysis
7. Inverse functions

VII. Application of Singularity Analysis
1. A roadmap to singularity analysis asymptotics
7. The general analysis of algebraic functions
8. Combinatorial applications of algebraic functions
9. Ordinary differential equations and systems
10. Singularity analysis and probability distribution

Grading:

6 one-hour presentation (50%)
Project (50%)

References

Analytic COmbinatorics, Flajolet and Sedgewick (Cambridge University Press)

Pre-Requisites:

Permission of Instructor.
Knowledge of Complex Analysis.

Math 895-4 G100 History of Mathematical Analysis, 1600-1950

Tom Archibald

Topics:

1. Analysis and Synthesis in mathematics to Descartes. 
      Readings: Pappus, Descartes, Bos

2. Analysis and synthesis in Newton. 
      Readings: Newton/Whiteside; Guicciardini

3. Differentials: Leibniz, Bernoullis, Euler, and the emergence of the calculus of one and several variables.
      Readings: Hoffmann, Engelsmann, Peiffer, others

4. Algebraic analysis in the late 18th Century.
      Readings: Lagrange, Jahnke, Fraser

5. Mixed mathematics circa 1800.
      Readings: Lagrange, Laplace, Monge, Poisson, Germain, Fox, Truesdell, Dhombres.

6. Cauchy: rigour and generality.
      Readings: Cauchy, Bottazzini, Belhoste, Lorenat

7. Arithmetic algebraic analysis: Gauss, Dirichlet, Dedekind.
      Readings: Goldstein and Schappacher
 
8. Complex analysis.
      Readings: Bottazzini and Gray; Cauchy, Riemann, Weierstrass

9. Weierstrassian analysis; the “end of the science of quantity.
      Readings: Weierstrass, Dedekind, Siegmund-Schultze, Archibald, Epple

10. Applied analysis in Germany the last half of the nineteenth century.
      Readings: Dirichlet, Riemann, Neumann, Archibald, Siegmund-Schultze

11. Poincaré.
      Readings: Nabonnand, Walter, Gray

12. Set theory and analysis.
      Readings: Cantor, Mittag-Leffler, Schoenflies, etc etc. Turner

13. Hilbert, integral equations and Hilbert spaces. 
      Readings: Sieg, Archibald and Tazzioli, others

14. Metrics and Measures: Fréchet, Lebesgue.
      Readings: Hawkins, Taylor, others

15. Banach and the Polish school.
      Readings: Von Neumann and Stone.

16. Stochastics: von Mises, Kolmogorov, Doob.
      Readings: Bourbaki versus Halmos

17. Numerical methods.
      Readings: Tournès, Goldstine.

Grading

Students will be required to present on and lead discussion of relevant readings and complete a major research paper (~20 pp) on an approved topic.

40% on seminar leadership and 60% on the research paper.

Prerequisites:

Students should have an undergraduate degree in mathematics or the philosophy of mathematics; and permission of the instructor.


Math 895-4 G200 Toric Geometry

Nathan Ilten

Topics:

  • Introduction (by instructor)
  • Background on affine varieties
  • Definition and construction of affine toric varieties
  • Cones and affine toric varieties
  • Properties of affine toric varieties
  • Background on abstract varieties
  • Fans and toric varieties
  • Orbits of toric varieties
  • Completeness of toric varieties
  • Background on projective varieties
  • Projective toric varieties
  • Regular triangulations and initial ideals
  • Linear subspaces of toric varieties

Grading:

Students will be expected to give two lectures, and turn in a written project at the semester’s end. lectures will be evaluated by the instructor based on preparation, comprehension, and clarity. Written project will be evaluated based on originality of presentation, sophistication, and style. Each component will carry a weight of one third in the final grade.

Prerequisities:

A reasonable background in algebra, for example MATH 340. 
Math 740/440 recommended, but not required. 
 

Math 895-G300 - Operational Research Applies to Epidemiology and Health Services

Sandy Rutherford
JF Williams


Topics:
Part I - Qualitative Models

1. Using UML to develop health systems models
   - activity diagrams
   - state machine diagrams

Part II - Continuous Models

1. Compartmental Models
   - SIR model
   - SIS model
   - basic repoductive number
   - herd immunity

2. Systems Dynamics Modelling
   - causal loop diagrams
   - ordinary differential equations models

3. Model Calibration and Validation
   - nonlinear least squares methods
   - heuristic optimisation for model fitting
   - global sensitivity analysis

4. Using Models for Optimal Control of Epidemics & Endemics
   - how to choose objective functions and constraints
   - optimisation methods

Part III - Agent Based Models

1. Network Disease Models
   - introduction to complex networks
   - SIR & SIS models on networks
   - the "friend paradox" in networks

2. Queueing Models for Health Services
   - introduction to queue models
   - queue models for acute care


Student Evaluation:


1. In class presentation of reading assignments. 25%
2. Written assignments. 25%
3. Written end of term project, which would also be presented orally. 50%

References:

Selected chapters would be used from the following books.

F. Brauer, P. van den Driessche, J. Wu: Mathematical Epidemiology,
Lecture Notes in Mathematics, Springer (2008).

W. Hare, K. Vasarhelyi, A. R. Rutherford, & the CSMG: Modelling in
Healthcare, AMS Press (2010).

M. L. Brandeau, F. Sainfort, W. P. Pierskalla (eds.): Operations
Research and Health Care, Kluwer (2004).

A. Saltelli, M. Ratto, T. Andres, F. Campolongo, J. Cariboni,
D. Gatelli, M. Saisana, S. Tarantola: Global Sensitivity Analysis -
The Primer, Wiley (2008).

M. E. J. Newman: Networks - An Introduction, Oxford University Press
(2010).

J. Medhi: Stochastic Models in Queueing Theory, Academic Press (2003).


A selection of papers that would include the following. Additional
papers would be added throughout the course.

A. V. Esensoy, M. W. Carter: Health system modelling for policy
development and evaluation: Using qualitative methods to capture the
whole-system perspective, Operations Research for Health Care, vol. 4
15-26 (2015).

Sarah Kok, A. R. Rutherford, R. Gustafson, R. Barrios, J. S. G. Montaner, K. Vasarhelyi: Optimizing an HIV testing program using a system dynamics model of the continuum of care, Health Care
Management Science, vol. 18, 334-362 (2015).


Fall 2015

MATH 895-4 G100 An Introduction to Compressed Sensing

Ben Adcock

Outline:

This is a course in compressed sensing and its applications.  In the past decade, compressed sensing has emerged as a powerful new theory that overcomes traditional barriers in sampling.  Under appropriate conditions, it states that objects, e.g. signals and images, can be recovered from seemingly highly incomplete data sets.  Moreover, not only is this possible in theory, practical reconstruction can be carried through efficient numerical algorithms.  This has important implications for many real-world applications, not least medical imaging, radar, analog-to-digital conversion, and sensor networks.The goal of this course is to provide a comprehensive introduction to this new field.  Although the course will be primarily mathematical, applications will also be emphasized.

Topics:

We will cover material from Foucart & Rauhut's "A Mathematical Introduction to Compressive Sensing", including:

Chpt 1. An Invitation to Compressive Sensing

Chpt 2. Sparse Solutions of Underdetermined Systems

Chpt 3. Basic Algorithms

Chpt 4. Basis Pursuit

Chpt 5. Coherence

Chpt 6. Restricted Isometry Property

Chpt 9. Sparse Recovery with Random Matrices

Chpt 12. Random Sampling in Bounded Orthonormal Systems

Chpt 14. Recovery of Random Signals using Deterministic Matrices

Grading

Weekly homework problems, class participation and a final project.

Students will present solutions to homework problems each week in class.

Prerequisites:

A good knowledge of linear algebra, analysis, introductory probability and basic programming skills are essential.

Students must get permission from the instructor in order to enroll.

Summer 2015

MATH 892-2 G100 Applied Combinatorics Field School

Karen Yeats & Marni Mishna

This reading course will shadow the Applied Combinatorics summer school at the University of Saskatchewan which is a 2 week program with four 8-hour minicourses and associated problem sessions. Students taking this reading course will attend the summer school (a total of 32 hours of lectures with problem sessions extra) and then upon their return will submit problem solutions and give a presentation.

The minicourses are Random generation of combinatorial structures by Éric Fusy  From Rosenbluth Sampling to PERM - rare event sampling with stochastic growth algorithms by Thomas Prellberg Combinatorial Hopf algebras in particle physics by Erik Panzer Strings, Trees, and RNA Folding by Christine Heitsch

Evaluation
Students will attend the 2 week summer school, attending lectures and problem sessions.  Upon their return they will submit solutions to approximately 3 problems of their choice from each of the speakers' problem sets.  The exact number of problems and any restrictions on which can be used will be determined by Marni and Karen depending on the nature and difficulty of the problems the speakers bring.  Additionally, upon their return the students will each give a 50 minute presentation on a topic of their choice related to the material of the summer school.

MATH 895-4 G100 Topics in Computer Algebra

Michael Monagan

Topics

1 The Fast Fourier Transform

- Review of the Radix 2 FFT algorithms
- Review of the fast multiplication using the FFT
- The Newton iteration and fast division
- The fast Euclidean algorithm

2 Polynomial Data Structures and Arithmetic
- Multivariate polynomial representations and term orderings
- Polynomial multiplication and division using heaps

3. Multivariate Polynomial Interpolation
- Browns' algorithm for multivariate polynomial GCDs
- Zippel's sparse interpolation
- Ben-Or and Tiwari sparse interpolation

4. Symbolic Linear Algebra
- The Bareiss fraction-free algorithm for computing det(A) and solving Ax=b
- Rational number reconstruction and solving Ax=b over Q using p-adic lifting

5. Algebraic Number Fields
- Representation of elements and calculating norms
- The Trager-Kronecker algorithm for factoring polynomials in Q(alpha)[x]
- Cyclotomic fields and solving Ax=b over Q(alpha) using a modular method

Grading

Five assignments, one per topic, worth 15% each.  One project worth 25% (presentation as a poster or report). 

References
o Algorithms for Computer Algebra by Geddes, Czapor and Labahn
o Modern Computer Algebra by von zur Gathen and Gerhard

Fall 2014

Math 895-4 G200 Fast Direct Solvers for Integral Equations

MC Kropinski

Fast direct solvers for accelerating the solution to the linear systems arising from the discretization of certain classes of integral equations is a very exciting a recent development in fast integral equations. This reading course will be based on a recently held summer school at Dartmouth College:
https://www.math.dartmouth.edu/~fastdirect/mat.php

The Dartmouth gives a cursory overview of fast direct solvers in the context of solving integral equations, more from an end-user perspective. The aim of the reading course is to fill in the more technical and theoretical work behind these methods. In particular, we will be working through a series of papers (in order):
1. Cheng et al., On the compression of low rank matrices, SISC 2004.
2. Martinsson and Rokhlin, A fast direct solver for boundary integral equations in two dimensions, JCP 2005
3. Greengard et al., Fast direct solvers for integral equations in complex three-dimensional domains, Acta Numerica 2009
4. Greengard and Ho, A Fast Direct Solver for Structured Linear Systems by Recursive Skeletonization, SISC 2012
In addition, we will be writing exploratory matlab codes and implementing black-box solvers for the purposes of solving Laplace's equation in two dimensions.

Participants in this reading course will be responsible for working through all of the technical details in the above papers. They will take turns presenting each paper and preparing useful matlab demonstrations as identified by the class. Evaluation will be based on these presentations and coding demonstrations (70%), as well as in class participation (30%).

Math 894-2 G100 Elliptic Curves

Nils Bruin

Introduction into the arithmetic of elliptic curves.

Content: We will be closely following the standard text: The arithmetic of elliptic curves. J.H. Silverman, Joseph H. Graduate Texts in Mathematics, 106. Springer-Verlag, New York, 1986. xii+400 pp. ISBN: 0-387-96203-4

and, if time permits, the sequel

Advanced Topics in the Arithmetic of Elliptic Curves, J.H. Silverman, Joseph H. Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994. ISBN: 978-0-387-94328-2

The exact material will be determined in consultation with the students during the first meeting. Topics of possible interest include (each of these are covered accessibly in chapters of the books listed above):

- Endomorphism rings of elliptic curves
- Elliptic curves over finite and local fields
- Ellip tic curves over global fields
- Explicit computation of the Mordell-Weil group
- Elliptic surfaces
- Neron models of elliptic curves

Format of the course:
- Participants in the course will be meeting weekly, for 2 hours
- Participants will be lecturing on a rotating schedule
- There will be biweekly assignments (assigned from the text), to be handed in.

Evaluation of performance:
- The students will be graded on: their lectures 60%, general participation 20%, assignments 20%

Math 895-4 G100 Stochastic Processes in Physics and Chemistry

Paul Tupper

1. Course
 4 hours per week

2. We will read the first several chapters of van Kampen’s landmark book Stochastic Processes in Physics and Chemistry
Ch 1. Stochastic Variables
Ch 2. Random Events
Ch 3. Stochastic Processes
Ch 4. Markov Processes
Ch 5. The Master Equation
Ch 6. One-Step Processes
Ch 7. Chemical Reactions (might skip this)
Ch 8. Fokker-Planck Equation
Ch 9. Langevin Approach
Ch 10. The Expansion of the Master Equation

3. Title: Stochastic Processes in Physics and Chemistry

4. Our goal will be to cover approximately a chapter a week. We will meet once a week to discuss the reading. Students will present solutions to problems in the book.

5. Students will be judged on their solutions to the problems.

Spring 2014

M894-2 G100 Further Topics in Complex Analysis

Nilima Nigam

**Course Permission Required

Content:
- Holomorphic functions (including integration over paths, local and global Cauchy theorems, homotopy)
- harmonic functions
- maximum modulus principle (incl. schwarz lemma and phragmen-lindelof)
- Mittag-Leffler
- conformal mapping, normal families
- infinite products
- analytic continuation, monodromy, little and big Picard
- H^p spaces
- linear DE.

This corresponds to roughly to chapters 10-16 of the Green Rudin, and parts of chapters 4, 5, 8 of Ahlfors.

Assessment: Homework and presentations: 100%. Students will present problems each week in class.

Prerequisites: Permission from instructor.

M895-4 G100 Topics in Computer Algebra

Michael Monagan

A second course in computer algebra intended to prepare students for doing research in the field.

Topics: 

1. The Fast Fourier Transform 
- Fast integer multiplication using the FFT
- The Shoenhage-Strassen multiplication algorithm
- The Newton iteration and fast division
- The fast Euclidean algorithm. 

2. Polynomial Data Structures 
- Recursive verses distributed. 
- Sparse polynomial multiplication and division using heaps and geobuckets
- Generic data structures

3. Algebraic Number Fields 
- Representation of elements, norms and resultants
- Cyclotomic fields
- The Trager-Kronecker algorithm for factoring polynomials in Q(alpha)[x]
- Solving Ax=b over Q(alpha) using a modular method

4. Sparse Polynomial Interpolation 
- Zippel's sparse interpolation. 
- Ben-Or and Tiwari interpolation. 
- Application to computing multivariate polynomial greatest common divisors

5. Symbolic Linear Algebra
- The Bareiss fraction-free algorithm for solving Ax=b over an integral domain
- Rational number reconstruction and solving Ax=b over Q using p-adic lifting
- Division free algorithms: the Berkowitz algorithm

 Grading: 
  Five assignments, one per topic, worth 15% each. 
  One project worth 25% (possible presentation as a poster). 

 References 
 Text Algorithms for Computer Algebra by Geddes et. al. 
 Text Modern Computer Algebra by von zur Gathen and Gerhard.

Fall 2013

M895-4 G200 Topics in Kinetic Theory

Weiran Sun

Outline
This course will cover several theoretical topics related to kinetic equations. We will focus on the Boltzmann equations and may also consider other types of kinetic equations at the end. The topics to be covered include: derivation of kinetic equations, basic properties of the linearized Boltzmann operator and the nonlinear operator, well-posedness theory for the Boltzmann equation, formal asymptotic analysis to derive classical fluid equations from the Boltzmann equation, and rigorous justi cations of various hydrodynamic limits. If time permits, we will also discuss transition regime models beyond the classical fluid equations.

Organization
This is a 4-hour 4-credit course. We meet twice a week: MF 4-6pm. The instructor will lecture on Mondays and students will present part of the course material on Fridays. References include two books, one review paper, and various research papers which will be distributed during the class:
1. Cercignani, C.: The Boltzmann equation and its applications, Applied Mathematical Sciences, 67. Springer-Verlag, New York, 1988.
2. Cercignani, C., Illner, R., and Pulvirenti, M,: The mathematical theory of dilute gases, Springer, New York, 1994. 
3. Villani, C.: A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. I, 71-305, North-Holland, Amsterdam, 2002. The URL for the electronic version of this paper is: http://cedricvillani.org/wp-content/uploads/2012/07/B01.Handbook.pdf

Grading
The grade will be based on the in-class presentation and a fi nal project.

M895-4 G100 Introduction to Measure-Theoretic Probability

Paul Tupper

Text: A First Look at Rigorous Probability Theory, 2nd edition, by Jeffrey Rosenthal

Format: 2hr+ meeting every week.  This time will be a combination of
lectures, student presentations, and working on problems together.

Assessment: There will be weekly homework assignments and an in-class final exam. The homework assignments will be taken from problems in the book.

Content: We will be doing the first 13 chapters of Rosenthal's book:
1. Measure Theory
2. Probability Triples
3. Further Probabilistic foundations
4. Expected Values
5. Inequalities and Convergence
6. Distributions of Random Variables
7. Stochastic Processes
8. Discrete Markov Chains
9. More probability theorems.
10. Weak Convergence
11. Characteristic Functions
12. Decomposition of probability laws
13. Conditional Probability and Expectation

Summer 2013

M895-4 G100 Topics in Computer Algebra

Michael Monagan

A second course in computer algebra intended to prepare students for doing research in the field. 

 Topics: 

 1 The Fast Fourier Transform 
   Fast integer multiplication using the FFT 
   The Shoenhage-Strassen multiplication algorithm. 
   The Newton iteration and fast division. 
   The fast Euclidean algorithm. 

 2 Polynomial Data Structures 
   Recursive verses distributed. 
   Sparse polynomial multiplication and division using heaps and geobuckets. 
   Generic data structures. 

 3 Algebraic Number Fields 
   Representation of elements, norms and resultants. 
   Cyclotomic fields. 
   The Trager-Kronecker algorithm for factoring polynomials in Q(alpha)[x]. 
   Solving Ax=b over Q(alpha) using a modular method. 

 4 Sparse Polynomial Interpolation 
   Zippel's sparse interpolation. 
   Ben-Or and Tiwari interpolation. 
   Application to computing multivariate polynomial greatest common divisors. 

 5 Symbolic Linear Algebra 
   The Bareiss fraction-free algorithm for solving Ax=b over an integral domain. 
   Rational number reconstruction and solving Ax=b over Q using p-adic lifting. 
   Division free algorithms: the Berkowitz algorithm. 

 Grading: 
  Five assignments, one per topic, worth 15% each. 
  One project worth 25% (possible presentation as a poster). 

 References 
 Text Algorithms for Computer Algebra by Geddes et. al. 
 Text Modern Computer Algebra by von zur Gathen and Gerhard. 

Spring 2013

Spectral Methods

Manfred Trummer

This course will touch on a number of mathematical and computational topics arising in
spectral methods (and possibly other high-order methods) for numerically solving
differential equations. We will cover the first ten chapters of the book below.

Outline
1. Differentiation Matrices
2. Fourier Transform – continuous and semi-discrete
3. Periodic grids: Discrete Fourier transform and FFT
4. Smoothness and spectral accuracy
5. Polynomial interpolation
6. Boundary Value problems
7. Chebyshev series
8. Eigenvalues and pseudospectra
9. Time-stepping and stability regions
10. Robustness, conditioning, round-off errors

Text: L.N. Trefethen
Spectral Methods in Matlab
SIAM Books ISBN: 0898714656

Grading: 80% Homework, 20% project.
Weekly meetings, assignments for each chapter.

Analytic Combinatorics in Several Variables

Karen Yeats

The recent release of the book [1], currently available in draft form from http://www.cs.auckland.ac.nz/~mcw/Research/mvGF/asymultseq/ACSVbook/
ACSV121108submitted.pdf makes it rather easy to plan a reading course on this topic. The book is organised to be a largely self contained reference, split into 4 parts. We plan to follow Part 3, the core material on multivariate enumeration, with asides into parts 2 and 4 as necessary to give essential background exposition. 

Part 1 is introductory, and should be understood by most participants. The following list is of interesting chapter titles in [1] given in numerical order.

Part II Mathematical background
4 Saddle integrals in one variable
5 Saddle integrals in more than one variable
7 Cones, Laurent series and amoebas

Part III Multivariate enumeration
8 Overview of analytic methods for multivariate generating functions
9 Smooth point asymptotics
10 Multiple point asymptotics
11 Cone point asymptotics
12 Worked examples
13 Extensions

Part 4 is omitted in the list above as it contains appendices and exposition can be included as needed in the topics above.

References
[1] R. Pemantle and M. C. Wilson. Analytic combinatorics in several variables. Cambridge University Press, 2013.

p-adic analysis

Nils Bruin

Outline:
An important step in the proof of the Weil conjectures is to establish that the zeta-function of a hypersurface over a finite field is a rational function. Dwork gave a p-adic analytic proof of this fact. The proof is nicely described in Koblitz, Neal, p-adic numbers, p-adic analysis and zeta functions, GTM 58, Springer 1977.

The participating students will work through the book with the aim of arriving at Dworks's proof. In the process, the students will get familiar with the concepts on which p-adic analysis is built.

Organization:
Since there is a group of students who have already expressed interest in the course, we will run the course in a seminar format:

At a first organizational meeting, we will divide up the book in several lectures, allocated to the participants. Every week, one student will lecture on the allocated material during a 2 hour meeting. The lectures  are aimed at one hour each, leaving ample room for discussion and questions.

Evaluation:
The grade will be based on the participation in the lectures and on the assignments.

Fall 2012

Computational Aspects of Medical Imaging

Manfred Trummer

This course will touch on a number of mathematical and computational topics arising in medical imaging. We will concentrate on problems related to image reconstruction. The reading course is a compressed version of the special topics course I taught in 2007.

Recommended text: Charles L Epstein
Mathematics of Medical Imaging
Prentice Hall; 1st edition (February 24, 2003), ISBN: 0130675482

Outline
1) Introduction
i) Image Modalities (CT, MRI, PET, SPECT)
ii) The Reconstruction Problem
iii) Radon Transform, X-Ray Transform
2) Analysis and Signal Processing Background
i) Fourier Transform
ii) Convolution
iii) Radon Transform
iv) Fourier Series and Discrete Fourier transform
3) Reconstruction
i) Fourier Based Reconstruction
ii) Algebraic Reconstruction Techniques
iii) Probabilities and the ML-EM Algorithm
4) Ill-posed problems. Regularization. Dynamic Imaging.
5) Optimization.
i) Quasi-Newton Methods
ii) Simulated Annealing

Grading: 80% homework, 20% participation/reading.

Weekly meetings, biweekly assignments.

Several Complex Variables and Analytical Combinatorics

Karen Yeats

We will introduce the basic elements of analysis with several complex variables illustrating all the notions with examples coming from analytic combinatorics. We will conclude by discussing in more detail applications to asymptotic analysis for multivariate generating functions
Outline.
[I] From One to Several Complex Variables
- Residue theorem and consequences (maximum principle, Cauchy's inequality,...);
- What cannot be generalized (Riemann's mapping theorem, Picard's theorem,...);
[II] Analytic Continuation and Singularities
- Analytic continuation and domain of holomorphy;
- Hartog's theorem;
- Monodromy principle and some sheaf theory (connection with local systems and   differential equations);
- Weierstrass theorem and analytic nullstellensatz;
- Meromorphic functions and Cousin problems;
[III] Geometric Perspective
- Multidimensional residues and geometry;
- Complex analytic varieties;
[IV] Asymptotic analysis
- Asymptotic analysis for multivariate generating functions
References:
- Several Complex Variables with Connections to Algebraic Geometry and Lie Groups , J.L.Taylor, AMS Graduate Studies in Mathematics.
- Twenty combinatorial examples of asymptotics derived from multivariate generating functions, R. Permantle and M. Wilson, to appear in SIAM Review.

We will meet once a week for approximately 2 hours.

Lectures will rotate among the participants (both for credit and not for credit participants, including myself), following the references indicated in the outline.

Students taking it for credit will be asked, in addition to taking their turn lecturing, to write short summaries of the lectures indicating the key points.  These will be posted on the learning seminar's webpage, http://people.math.sfu.ca/~kyeats/seminars/sem_cur.html

Students taking it for credit will be evaluated on their lectures and their summaries. 

Symbolic Dynamics

Bojan Mohar

Course Objectives
Symbolic dynamics is the study of dynamical systems with discrete time and
discrete space. We plan to study this area following the book of Kitchens, titled
Symbolic Dynamics. We will focus on theoretical results and mathematical
methods of a discrete  flavour.

Method of Evaluation
Students will be evaluated based on written and oral reports.

Reference Texts
Main text:
B. P. Kitchens, Symbolic Dynamics
Other references:
D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding
M Brin and G. Stuck, Introduction to Dynamical Systems

Matroid Theory

Matt DeVos

Outline: We begin with the central examples, and numerous equivalent definitions of a matroid. Then we will turn to the use of matroids in combinatorial optimization, including the greedy algorithm, matroid intersection, matroid union and algorithms.  Following this we will study connectivity and minors, proving Tutte's characterization of binary matroids, Seymour's wheels and whirls theorem and Tuttes classification of graphic matroids.  If any time remains, we shall look to some extremal properties of matroids.

Organization: We will meet once a week for 2 hours.

Evaluation: Homework 50%, Presentations 50%

Modelling of Complex Social Systems

Vahid Dabbaghian

This is a seminar course that reviews theory and research in complex social systems.  In particular we will focus on the impact of social interactions on the dynamic of urban transformations such as crime and infectious diseases in municipal environments.  The seminars incorporate conceptual modelling, mathematical modelling and computer simulations.  This course is suitable for students who are interested in interdisciplinary problems without necessarily strong mathematics and computer science background.

Exact concepts and modelling techniques covered will vary with class size and interest, but in general the following topics will be covered.

 - Good Modelling Practices: Simplicity, Adaptability, Reproducibility, Validation
 - Complex Social Networks: What are they, why model them, examples
 - Operational Management Models: System Dynamics, Scheduling, Queuing Models
 - Forecasting Models: Regression Analysis, Markov Models, Discrete Event Models
 - Pattern Reconstruction Simulation Models: Cellular Automata, Network Models, Agent Based Models
 - Fuzzy Model: Fuzzy Systems, Fuzzy Cognitive Maps.

Class Discussion:        10%
Class Presentations:    30%
Research Project:         60%

There is no specific textbook for the class.  The course will draw on material from a wide range of sources and will provide students with book excerpts and journal papers as appropriate to supplement lecture notes.

No specific courses are required, however students should be in a graduate program. The 4th year students of an honours undergraduate program can register in this course only with permission from the department.

Summer 2012

PDE and Pseudo-Differential Operators

Nilima Nigam

Material covered: A fast introduction to distributions and Sobolev spaces.Local existence theorems, the properties of the  Laplace operator, subharmonic functions and barriers, layer potentials, elliptic BVP, interior regularity and smoothness up to the boundary, the wave and the heat equations, constant-coefficient hypoelliptic operators, pseudodifferential operators.

Prerequisites for course: Metric spaces (Math 320), Measure and Integration (Math 425)

Texts:
The PDE book by Folland, with a focus on the main theorems in the beginning 6 chapters, and harmonic analysis in 7-8:  http://www.amazon.ca/Introduction-Partial-Differential-Equations-Second/dp/0691043612

(Vol. II) by Hormander on the  analysis of linear differential operators, probably chapters X and XI:  http://www.amazon.com/Analysis-Linear-Partial-Differential-Operators/dp/3540225161/ref=pd_bxgy_b_text_b

Grading: The students will be assessed on weekly homeworks which they will present in class, and a final (worth 20%).

Dynamical Systems, Chaos, and Noise

Sandy Rutherford

4 unit reading course

Meetings: two 2-hour meetings per week

Prerequisites

1. Introductory dynamical systems at the level of
   S.H. Strogatz, Nonlinear Dynamics and Chaos.

2. Introductory stochastic processes at the level of G. Grimmett and
   D. Stirzaker, Probability and Random Processes.

Core References:

L. Arnold, Random Dynamical Systems, Springer (2003).
A. Lasota and M.C. Mackey, Chaos, Fractals, and Noise, Springer (1994).
J.D. Meiss, Differential Dynamical Systems, SIAM (2007).

Supplementary References:

C. Gardiner, Stochastic Methods: A Handbook, Springer (2010).
J. Jost, Dynamical Systems: Examples of Complex Behaviour, Springer (2005).
K. Xu, Stochastic pitchfork bifurcation: numerical simulations and
  symbolic calculations using MAPLE, Mathematics and Computers in
  Simulation, vol. 38, 199-209 (1995).

Outline:

1. Review of Differential Dynamical Systems
   - flows, linearization, stability, Lyapunov functions, LaSalle's
     Invariance Principle, Hartman-Grobman Theorem, attractors, basins,
     periodic orbits
   (Meiss, Chapt. 4)

2. Invariant Manifolds
   - stable manifolds, unstable manifolds, heteroclinic orbits, Local
     Stable Manifold Theorem, global stable manifolds, center
     manifolds
   (Meiss, Chapt. 5)

3. Phase Plane Analysis
   - index theory, Poincare-Bendixson Theorem
   (Meiss, Chapt. 6)

4. Chaotic Dynamics
   - Lyapunov exponents, strange attractors
   (Meiss, Chapt. 7)

5. Bifurcation Theory
   - unfolding vector fields, normal forms, saddle node bifurcation,
     Andronov-Hopf bifurcation, cusp bifurcation, Takens-Bogdanov
     bifurcation, homoclinic bifurcation
   (Meiss, Chapt. 8)

Supplementary reading for 1-5: (Jost, Chapt. 2) and (Arnold, App. B)

6. Review of Stochastic Processes
   - measure theory, Markov operators, Frobenius-Perron operator
   (Lasota, Chapt. 1-3)
   Supp. reading: (Arnold, App. A) and (Gardiner, Chapt. 1-3)

7. A Measure Theoretic Approach to Chaos
   (Lasota, Chapt. 4)

8. Entropy
   (Lasota, Chapt. 9)
   Supp. reading: (Jost, sect. 6.1-6.5)

9. Stochastic Perturbation of Continuous Time Systems
   - Wiener processes, Ito integral, stochastic differential
     equations, Fokker-Planck equation
   (Lasota, Chapt. 11)
   Supp. reading: (Gardiner, Chapt. 4)

10. Invariant Manifolds for Smooth Random Dynamical Systems
    - unstable manifold, stable manifold, center manifold, Global
      Invariant Manifold Theorem, Hartman-Grobman Theorem, local
      invariant manifolds
    (Arnold, Chapt. 7)

11. Normal Forms for Stochastic Differential Equations
    - nonresonant case, small noise case
    (Arnold, sect. 8.5)

12. Stochastic Bifurcation Theory
    - transcritical bifurcation, pitchfork bifurcation, saddle node
      bifurcation
    (Arnold, sect. 9.1-9.1)
    Supp. reading: (Xu).

As much of the above material as time permits will be covered.

Students will be graded on:
  1. Ability to present reading material in class.
  2. Homework excercises.
  3. Term project.

Topics in Magnetohydrodynamics (MHD) and Computational MHD

Steven Pearce, Computing Science Department

Resources:

  • Jackson (Classical Electrodynamics)
  • Canuto et al (Spectral Methods in Fluid Dynamics)
  • Pearce, et al
  • Pearce, personal notes

Rationale:

In this course we will briefly review the fundamentals of hydromagnetic theory and plasma physics and then focus on specific problems in the generation of planetary dynamos.  Computational methods will be explored in spherical geometries utilizing pseudospectral methods.  Attention will be focussed on the avoidance of aliasing errors in the solution of the complex coupled set of nonlinear PDEs that describe dynamo action, specifically in the Earth’s outer core.

 Grading Scheme:

30%: Written preliminary report, midterm.

30%: Written preliminary manuscript.

40%: Oral examination/presentation at end of course.

Meeting Schedule:

Two one-hour meetings per week for 13 weeks.

Spring 2012

Fundamentals of Arithmetic Geometry

Nils Bruin

Description: 
Arithmetic geometry studies the interplay between geometry and number theoretic properties. In this course the students will learn the geometric fundamentals, with a view towards the number theoretic applications.

We will follow the book:
 Marc Hindry, Joseph H. Silverman, Diophantine Geometry: An Introduction, 
 GTM 201, Springer Verlag (2000), ISBN: 0-387-98975-7; 0-387-98981-1 

Students will be meet for two hours per week, where they will present the material assigned the week before and discuss solutions to assigned problems. 

The performance of the student will be evaluated based on the presentations and assignments.

Introduction to Measure-Theoretic Probability

Paul Tupper

Format: 3hr+ meeting every week.  This time will be a combination of lectures, student presentations, and working on problems together.

Content: We will be doing the first 13 chapters of Rosenthal's book:
1. Measure Theory
2. Probability Triples
3. Further Probabilistic foundations
4. Expected Values
5. Inequalities and Convergence
6. Distributions of Random Variables
7. Stochastic Processes
8. Discrete Markov Chains
9. More probability theorems.
10. Weak Convergence
11. Characteristic Functions
12. Decomposition of probability laws
13. Conditional Probability and Expectation

Text: A First Look at Rigorous Probability Theory, 2nd edition, by
Jeffrey Rosenthal


Assessment: There will be bi-weekly homework assignments and a final
exam (possibly a take-home exam).
The homework assignments will be taken from problems in the book.

Fall 2011

Historiography of Modern Mathematics

Tom Archibald

A collection of readings illustrating different issues in the writing of the history of modern mathematics. Responding to the norms of the historical profession more generally, historians of mathematics increasingly use a variety of tools and methods to analyze and depict the past. History of mathematics is also frequently tinged with philosophical issues of various kinds. Selections will include samples of biographical writing (Parshall), historical prefaces and commentary on edited mathematical texts (Nabonnand on Poincaré), adaptation of tools from the history of science (Epple on epistemic configurations and school formation in topology), reception studies (Goldstein and Schappacher), work from social history (Turner on the research imperative in mathematics, Fabiani on Disciplinarity), the understanding of mathematics in a given time as related to a sets of practices (Mancosu, Høyrup),  and microhistorical studies (Rowe, Ginzburg). 

Prerequisites: Graduate level work in history of mathematics or permission. 

Evaluation: 50% through weekly discussion, 50% on a research paper showing a nuanced appreciation of historical method. 

Summer  2011

Modified Generalized Laguerre Functions Tau Method for Solving the Lane-Emden Equation

Steven Pearce, Computing Science Department

Resources:

  • Canuto et al
  • Pearce, et al
  • Pearce, personal notes

Rationale: In this course we will study Galerkin, collocation and Tau methods which are some spectral methods for solving nonlinear partial differential equations. Then we will apply the Tau method for solving the singular Lane-Emden equation; the operational matrices of the derivative and product of the Modified generalized Laguerre functions will be studied.  Then we will apply these matrices together with the Tau method for solving the Lane-Emden equation which is an important model in the study of stellar structure.  Finally, we will compare our results with the literature.

Grading Scheme:

30%: Written preliminary report, midterm.

30%: Written preliminary manuscript.

40%: Oral examination/presentation at end of course.

Meeting Schedule:

Two one-hour meetings per week for 13 weeks.

Topics in Computer Algebra

Michael Monagan

Topics:
1 The Fast Fourier Transform, fast integer multiplication (Shoenhage-Strassen), and the fast Euclidean algorithm.

2 Sparse polynomial multiplication and division using heaps and geobuckets.

3 Sparser polynomial interpolation over finite fields and it's application to computing multivariate polynomial GCDs.
 
4 Introduction to computational symbolic linear algebra.The Bareiss fraction-free algorithm for solving Ax=b over an integral domain.Rational number reconstruction and solving Ax=b over Q using p-adic lifting.

5 Algebraic number fields: some theory and solving Ax=b over Q(alpha). The Trager-Kronecker algorithm for factoring polynomials in Q(alpha)[x].

Grading: Five assignments, one per topic, worth 20% each.

This is a 4-hour course.

References/Additional Reading:
 o Text Algorithms for Computer Algebra by Geddes et. al.
 o Text Modern Computer Algebra by von zur Gathen and Gerhard.
 o Paper Sparse Polynomial Arithmetic by Johnson.
 o Paper Sparse Polynomial Division using Heaps by Monagan and Pearce.
 o Paper Sparse Polynomial Interpolation over Finite Fields by Javadi and Monagan.
 o Paper Maximal Quotient Rational Reconstruction by Monagan.
 o Paper Solving Linear Systems over Cyclotomic Fields by Chen and Monagan.

Geometry and Optimization

Luis Goddyn and Matt DeVos 

Syllabus
Topics in Geometry and Optimization 

Resources
 - J. Conway and N. Sloane, Sphere Packiongs, Lattices and Groups 
 - L. Schrijver, "Theory of Linear and Integer Programming" 
 - Luis Goddyn, Personal Notes on geometric optiimization. 
 - Several papers on the topic. 

Rationale: 
The topic of geometric optimization fits well with Ms. Taghipour's research area, 
but is not well covered in any of the standard courses. 

Grading Scheme: 
30% each:  Two short written reports, one for each instructor. 
40%:  Oral exam at the end of the course. 

Meeting Schedule: 
Two 1-hr meetings per week for 10 weeks.

The Numerical Solution of Integral Equations

Mary-Catherine Kropinski

We will be covering topics in the book "The Numerical Solution of Integral Equations of the Second Kind" by Kendall Atkinson (or a similar book and/or collection of papers). Topics will possibly include Projection Methods, the Nystrom method, solving multivariable integral equations, iterative methods and boundary integral equations on smooth planar curves. Presentations of material will rotate through students and instructor. Meetings will be biweekly for approximately 1.5-2 hours. 

In addition, students in the course would complete a computing project focusing on either a particular method or methods or a particular problem of scientific interest that involves solving an integral equation. 

Evaluation: Students will be graded on participation, their presentations and their project.

Prerequisites: Math 495 from Spring 2011 (An Introduction to Integral Equations) or permission by instructor. 

Mathematical Models of Whole Genomes Analysis

Cedric Chauve

This course will describe mathematical models used to analyse whole genomes.

The analysis of whole genomes problems we will consider are:

1. the assembly and mapping (mostly physical) of genomes, and
    their modelling using graphs and binary matrices (3/4 weeks),
2. the detection of genomic features conserved in two or more genomes,
    based on the model of common intervals and its variants (2 weeks),
3. the computation of genomic distances from the breakpoint graph (3 weeks),
4. the computation of median and ancestral genomes (2 weeks),
5. analysis of next-generation sequencing data (2 weeks).

The emphasis will be on understanding the subtle balance between the relevance
of the mathematical models with regard to the motivating biological problems
(results on real datasets will be studied) and  the tractability of the models
(algorithms will be studied).

The references include the book "The combinatorics of genome rearrangements"
(Fertin  et al., MIT Press 2009) and recent research surveys and papers.  

There will be both lectures (roughly 3 hours every two weeks) and seminar or
discussions (one hour every two weeks).

The evaluation will be based on

- participation during the lectures (15%)
- bi-weekly assignments (focusing on mathematical aspects, to ensure
  understanding of the technical aspects of the models, 30%)
- regular presentations (at least 2 per students 15%)
- final project (report+presentation, 40%)

Prerequisite: MATH 445/745 or MATH 443/743 or MATH 408/708 or equivalent.

Students with credit for MATH 496/796 taught in Summer 2010 may not take this course for further credit.

Spring  2011

Mathematical Epidemiology

Ralf Wittenberg and Sandy Rutherford

This will be a reading course on mathematical models for epidemics, with particular emphasis on network models. The first part of the course will cover compartmental (ODE) models, while the latter part considers epidemic models on networks, following an introduction to networks.

Evaluation: Registered students will be assessed on class participation, and on regular homework assignments incorporating both mathematical analysis and computational investigations (Matlab/Python) of various models; there will also be an end-of-semester project with a written paper and presentation.

Meetings: This will be a 4-hour course; we will meet twice a week (provisionally on Monday and Wednesday afternoons).

References:

I: Compartmental Models

  • J.D. Murray, "Mathematical Biology", 3rd edition, Springer- Verlag (2002) - available online [Vol.I, Ch. 10; Vol.II, Ch. 13]
  • F. Brauer, P. van den Driessche and J. Wu (eds.) "Mathematical Epidemiology", Lecture Notes in Mathematics vol. 1945, Springer-Verlag (2008) - available online [selected chapters]

II: Network Models

  • A. Barrat, M. Barthelemy, and A. Vespignani, "Dynamical Processes on Complex Networks", Cambridge University Press (2008)

Fall 2010

Intensive Introduction to Probabilistic Modeling

Paul Tupper

  • Text: Introduction to Probability Models 9th edition, by Sheldon M. Ross.

Outline: First 6 chapters of the text.

  1. Probabilities and Events
  2. Random Variables
  3. Conditional Probability
  4. Markov Chains
  5. Exponential Distribution and Poisson Process
  6. Continuous-Time Markov Chains

Organization: Two methods will be used to guarantee the high level of difficulty of the course.

  1. Meetings twice per week. These will consist of lectures on the material, discussions of challenging problems, and feedback on students problem solutions. The duration of these meetings will depend on the student(s). If the student(s) benefit from full lectures, four hours a week will be devoted to me lecturing. If the student(s) are relatively independent the time will be used more flexibly.
  2. Assignments. There will be a schedule of reading from the text. Every two weeks students will hand in solutions to questions selected from the text. I will carefully read the student's answers and grade the questions. Students can arrange to meet with me to discuss the readings and the questions on an ad hoc basis.

Evaluation: Students will be evaluated purely on the solutions to homework questions that they hand in.

Rigour and Proof in Mathematics after 1800

Tom Archibald

Beginning with work of Gauss, Lagrange and Cauchy, the course will look at original source material and appropriate historical literature in order to examine aspects of the process of proof, concentrating on algebra and analysis. Evaluation: leading seminars and participation in discussion (40%); research paper (60%).

Prerequisite: previous grad training in history of mathematics or a closely related field.

Discrete optimization with Applications

Tamon Stephen

Summer 2010

An applications-driven introduction to finite elements

Nilima Nigam

The course will assume students are familiar with basic finite difference theory, C++, and have some prior exposure to finite elements. The goal of the course is to get students able to implement simple finite element applications, without getting bogged down with the computer-science details of mesh generation and assembly.

The course will familiarize students with the Wilkinson-prize winning finite element development software Deal II. We'll go through the basics of finite element codes - setting up meshes for simple geometries, setting up stiffness and mass matrices, and then using the extensive ARPACK-based linear algebra routines to solve sample problems. By the end of the course students will be expected to code up a sample application of their own interest.

Spring 2010

Special topics in Number Theory: Arithmetic Dynamics

Nils Bruin

Arithmetic Dynamics is a relatively new area of research. It may be viewed as the transposition of classical results in the theory of Diophantine equations to the setting of discrete dynamical systems, such as iterated polynomial maps.

The interplay between arithmetic and dynamics yields a particularly rich structure, which resembles the much older and established field of arithmetic algebraic geometry. The course text gives a particularly accessible introduction to the field.

We will follow 

  • Silverman, Joseph H. The arithmetic of dynamical systems. Graduate Texts in Mathematics, 241. Springer-Verlag, New York, 2007. x+511 pp. ISBN: 978-0-387-69903-5.

quite closely.  The chapters are:

  1. Classical dynamics
  2. Dynamics over local fields: good reduction
  3. Dynamics over global fields
  4. Families of dynamical systems
  5. Dynamics over local fields: bad reduction
  6. Dynamics associated to algebraic groups
  7. Dynamics in dimension greater than one

Format: Given the advanced nature of the material, the course will be run in a mixed seminar/workgroup style. Every week, there will be 2 hours of seminar and an additional 2 hours of workgroup, where the lecturer and the students can look at problems and further delve into the theory.

Grading:

  • Participating students are expected to prepare at least one seminar contribution, which will be assessed and count towards the grade.
  • Students will regularly hand in assignments, which will be marked.
  • Students will be graded on participation in the problem sessions.

See also the course webpage.

Differential Geometry

Karen Yeats

We will cover differential geometry up to de Rham cohomology and then investigate Chen's iterated integrals as a de Rham theory of P^1 - {0,1,infinity} leading, time permitting, to multiple zeta values from a differential topology perspective. The course is designed to provide the student with a topological and geometric background, and also link in to multiple zeta values, which are an interest of mine.

This course begin by following Spivak's Differential Geometry Vol 1 up to chapter 8, then it will proceed to Richard Hain's notes from the 2005 Arizona Winter School, Lectures on the Hodge-de Rham theory of the fundamental group of P^1 - {0, 1, infinity}.

We will meet for two hours on Fridays, discussing problems when the material is straightforward enough for independent reading, and rotating presenting the material otherwise. A smaller set of problems will be selected for handing in.

Evaluation will be based on presentations of the course material and handed in problems.

p-Adic Analysis

Nils Bruin

An important step in the proof of the Weil conjectures is to establish that the zeta-function of a hypersurface over a finite field is a rational function. Dwork gave a p-adic analytic proof of this fact. The proof is nicely described in

  • Koblitz, Neal, p-adic numbers, p-adic analysis and zeta functions, GTM 58, Springer 1977.

In the course we will work through the book to arrive at Dworks proof. Per week the student studies a portion of the book and present the material during the weekly meetings. The student will also do some of the exercises that are part of the book.

The student will be evaluated based on the presentations and exercises.

Historiography of Mathematics

Tom Archibald

Description:  This reading course surveys major developments in historical method in the study of the history of mathematics and the sciences. Readings will include work of H. Butterfield, T. S. Kuhn, I. Lakatos, P. Feyerabend, I. Hacking, B. Latour, M. Foucault, P. Bourdieu, D. Mackenzie, and selected historical articles influenced by the methodological approaches they espouse.

Evaluation will be based on oral and written reports on readings and on a research paper, 50-50.

Renormalization Group Analysis of Critical Phenomena

Sandy Rutherford

Meetings: 2 x 2-hours meetings/week.

Evaluation: Work through a recent paper on the subject giving an oral presentation and a written report.

References:

  • D. Sornette, Critical Phenomena in Natural Sciences, 2nd edn, Springer, 2006.
  • D. Sornette and R. Woodard, Financial Bubbles, Real Estate Bubbles, Derivative Bubbles, and the Financial and Economic Crisis, arXiv:q-fin/0905, 2009.
  • G. Arcioni, Using self-similarity and renormalization group to analyze time series, arXiv:q-fin/0805, 2008.
  • D. Sornette and A. Johansen, Large financial crashes, Physica A 245, 411-422, 1997.

Outline:

  1. Review of relevant topics from probability
  2. Random Walks and the Central Limit Theorem
    • random walks
    • diffusion & Fokker-Planck eqn
    • central limit theorem
  3. Large Deviations
    • cumulant expansion
    • large deviation theorem
    • large deviations with constraints
  4. Power Law Distributions
    • stable laws (Gaussian & Levy laws)
    • intuitive calculation tools for power law distributions
    • Fox function, Mittag-Leffler function, and Levy distributions
  5. Statistical Mechanics and the Concept of Temperature
    • statistical derivation of temperature
    • statistical mechanics as probability theory with constraints
    • generalising the concept of temperature to non-thermal systems
  6. Long-Range Correlations
    • criterion for relevance of correlations
    • statistical interpretation
    • correlation and dependence
  7. Phase Transitions: Critical Phenomena and First-Order Transitions
    • spin models
    • first-order versus critical transitions
  8. Transitions, Bifurcations, and Percursors
    • supercritical bifurcation
    • critical percursor fluctuations
    • scaling and percursors near spinodals
    • selection of an attractor
  9. The Renormalization Group
    • general framework
    • example: the hierarchical model
    • criticality and the renormalization group
  10. Applications to Financial Modelling
    • calculation of the critical exponents for the 1929 crash
    • self-similarity in financial time-series data

Fall 2009

Combinatorial Optimization

Matthew DeVos

Numerous deep theorems in combinatorics have been achieved by first associating a continuous space (polytope, manifold, etc) with a combinatorial object, then establishing properties of this new space, and then translating these properties back to the combinatorial setting. In this course, we will explore this approach. We will first prove some of the classical theorems of this type (Edmond's matching polytope, Seymour's cone of cycles, etc) and then move on to some more exotic ones (semidefinite programming and the Lovasz Theta Function).

Grading: students will be responsible for presenting material to the (mini) class on 2-3 occasions, and will be given homework problems (roughly) biweekly. Each of these will make up 1/2 of the student's grade.

Methods in enumerative combinatorics

Marni Mishna

Main textbook: Analytic Combinatorics by Flajolet and Sedgewick, Cambridge University Press, 2009.

  • PART I: COMBINATORIAL STRUCTURES
  • PART II: INTRODUCTION TO ANALYTIC METHODS
  • PART III: RANDOM STRUCTURES

Grading: 60% 4 written assignments, 20% final written project, 20% one hour-long presentation

Summer 2009

Topics in Computer Algebra

Michael Monagan

Topics:

  1. The Fast Fourier Transform, fast integer multiplication (Shoenhage-Strassen), and the fast Euclidean algorithm.
  2. Sparse polynomial multiplication and division using heaps and geobuckets.
  3. Computing multivariate polynomial GCDs over Z: Brown's dense interpolation algorithm and Zippel's sparse interpolation algorithm.
  4. The Bareiss fraction-free algorithm for solving Ax=b over an integral domain. Rational number reconstruction and solving Ax=b over Q using p-adic lifting.
  5. Algebraic number fields: some theory and solving Ax=b over Q(alpha). The Trager-Kronecker algorithm for factoring polynomials in Q(alpha)[x].

Grading: Five assignments, one per topic, worth 20% each.

Spring 2009

History of Mathematics: Analysis from Antiquity to the Present

Tom Archibald

This course will provide the kind of background that would ordinarily be expected in comprehensive exams on history of mathematics by tracing several topics from remote antiquity to the mid-twentieth century. We will focus on the concept of analysis. The course is given in conjunction with the undergraduate survey, Math 380, though attendance in that course is not required.

Course requirements consist in reading and presenting discussions of a combination of primary materials (original texts) and secondary materials (historical commentary). A research paper of roughly 20 pp is required.

Evaluation is 60% paper and 40% participation in weekly discussions.

Advanced Probability

Paul Tupper

The course will cover the essentials of measure-theoretic probability together with some of its more interesting applications.

Topics in order are: Probability Triples, Random Variables, Expected Values, Inequalities, Distributions, Basic Stochastic Processes, Discrete Markov Chains, Limit Thoerems, Weak Convergence, Characteristic Functions, Conditional Probability, Martingales, Brownian Motion

The student will be expected to read a chapter a week of Rosenthal and do all the exercises in the chapter. (There are not that many in each.) The student will present solutions in one two-hour meeting each week.

The student will be evaluated on his presented solutions each week, and on his response to oral questions during these meetings.

Approximation Algorithms

Abraham Punnen

Topics to be covered: Approximation Algorithms for symmetric and asymmetric traveling salesman problem. Reading 6 papers on the topic, meet once a week for discussions. A final report on the topic with implementation and testing of some of the algorithms, and a discussion of related open problems with critical analysis is required.

Grading will be based on weekly discussions (20%) and final report (80%).

Elliptic curves and the Mordell-Weil theorem

Nils Bruin

We will be studying elliptic curves, with the goal of understanding the Mordell-Weil theorem, following The Arithmetic of Elliptic Curves, by Joseph H. Silverman published by Springer, 1986 (ISBN 0387962034).

The course will be run in seminar format: the meetings will consist of presentations by the participants,

Grades will be assigned based on the presentation and participation in the course.

See the webpage.

Fall 2008

History of Mathematics in National Context

Tom Archibald

While mathematics has an international character, the readings in this course focus on aspects of the subject that vary according to national and regional variation. Topics will include: the development of international communication in mathematics and science in the seventeenth and eighteenth centuries; regional variations in the institutionalization of mathematical research and education, 1750 onward; the professionalization of mathematics in the universities over the course of the nineteenth century; schools versus research groups and research programs, and their relation to national settings; mathematics and the state in various contexts (military included); differential participation in the nascent international community 1890-1945. The subject will be examined in the context of specific mathematical developments

Evaluation: Leading seminars and participation in discussion: 40%; Research paper 60%.

Topics in Algebraic Geometry

Michael Monagan

References:

  • IVA, "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea. Springer-Verlag Undergraduate Texts in Mathematics.
  • UAG, "Using Algebraic Geometry" by Cox, Little, and O'Shea. Springer-Verlag Graduate Texts in Mathematics.
  • LLMM, "Hilbert's Nullstellensatz and an Algorithm for Proving Combinatorial Infeasibility" by Susan Margulies et. al. Proceedings of ISSAC '08, ACM, 2008.

Part I. Buchberger's algorithm and the FGLM Groebner basis conversion algorithm. Study, implement, and try out Buchberger's S-pair critera for improving his algorithm. Study, implement, and try out the FGLM Groebner basis conversion algorithm of Faugere, Gianni, Lazard, and Mora for converting a Grobner basis for a zero-dimensional ideal from one monomial to another. Reference CLO-UAG sections 2.3 and 2.4. Reference CLO-IVA section 2.9, and 3.1 exercises #5,6(b),7.

Part II An unexpected application of Hilbert's Nullstellensatz. Study the recent (2008) paper of LLMM. Given a simple graph G the paper shows how to convert the question "Is G k-colorable" into the question "Does the linear system Ax=b" have a solution over GF(2). It uses the algebraic formulation of graph k-colorability and applies Hilbert's Nullstellensatz.

Part III An application of Groebner bases. Study the material on Automatic Geometry Theorem Proving from CLO-IVA section 6.4. In particular the examples and exercises which illustrate that a theorem might hold one component but not all components of an ideal. Reference CLO-IVA exercises 6.4 #7, 11, 13, 17. And an example from Tomas Recio from a talk he in July.

Assessment: I will set an assignment on each part.

Convergence of Probability Measures

Richard Lockhart

Course syllabus:

  • Probability Theory by Laha and Rohatgi Chapters, 1, 2, 3, 5 section 1, and possibly chapter B6
  • Convergence of Probability Measures, 2nd edition by Patrick Billingsley As much as possible of Chapters 1, 2 and 4.

Meeting every 2 weeks for 2 hours to discuss the material the student has been reading. He will do problems to be chosen along the way in terms of their utility.

Fall 2016

Math 894-G100 Wavelets, Approximation Theory and Signal Processing

Ben Adcock

Description:

We will cover parts of Mallat’s “A Wavelet Tour of Signal Processing: The Sparse Way”, including:

• Chapter 1: Sparse Representations
• Chapter 2: The Fourier Kingdom
• Chapter 3: Discrete Revolution
• Chapter 6: Wavelet Zoom
• Chapter 7: Wavelet Bases
• Chapter 9: Approximations in Bases
• Chapter 10: Compression 

Format:

1 hour lecture every week prepared and presented by students + 30 minutes discussion (90 minutes total).

Grading:

70% Lectures
30% Final Project

Students will prepare and present each lecture, and distribute notes beforehand.  
Students will be evaluated on preparation, clarity and understanding of the material.

References:

A Wavelet Tour of Signal Processing: The Sparse Way - Stephane Mallat

Pre-Requisites:

Basic analysis, Fourier series. Enrolment by instructors permission only.