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:

### Fall 2017

__Math 894-2 G100 Represetation Theory of Lie Groups and Algebras__

*Nathan Ilten*

__Duration__

Meeting for one two-hour block each week. Due to the amount of prepatation outside of class, the course will be a 2 credit course.

__Tentative outline__

• Lie Groups

• Lie Algebras and Lie Groups

• Initial Classification of Lie Algebras

• Lie Algebras in Dimensions One, Two, and Three

• Representations of sl2C

• Representations of sl3C

• The General Setup: Analyzing the Structure and Representations of an Arbitrary Semisimple Lie Algebra

• sl4C and slnC

• The Classification of Complex Simple Lie Algebras

__Proposed Organization__

Each week, students will be assigned reading and problems to prepare for the next week. One student will summarize the previous week’s reading in a 30 minute presentation. The remaining 90 minutes will be used to discuss the assigned homework. Readings and assignments will be taken from the second half of [Representation Theory: A First Course, Fulton and Harris].

__Method of Evaluation & Grading__

Students will be expected to complete all assigned readings and problems, and to present the reading material at least three times throughout the semester. Presentations and problem solutions will be evaluated by the instructor based on preparation, comprehension, and clarity. The final grade will be combined equally from presentation and problem solution grades.

__Prerequisites__

A reasonable background in algebra, for example, Math 340 and Math 740/440

### Math 895-4 G100

*Alexander Rutherford*

__Meetings__:

Two 2-hour meetings per week.

__Description:__

Complex networks are random graphs with a large number of vertices, whose structure is defined in terms of probability distributions over graphs. They may be generated directly from probability distributions, or by a stochastic process. Common examples are Barabasi-Albert preferential attachment networks, small-world networks, and Erdos-Renyi random graphs. Contact processes are stochastic processes defined on the edges of a random graph. These processes determine the dynamics of the states of the vertices. A prototypical contact process is disease propagation along an edge from an infected vertex to a susceptible vertex, resulting in the state of the susceptible vertex changing to infected. In this course, we will study the mathematics of random graph models and contact processes. Applications include mathematical epidemiology and social dynamics.

__Topics__:

- Random graphs and their properties: degree distribution, clustering coefficient, and the giant component

- Random graph models: Barabasi-Albert, Erdos-Renyi, small-world, and configuration networks

- Special types of random graphs: bipartite graphs, directed graphs

- Contact processes on networks: SIR disease model, SIS disease model, review of compartmental disease models

- Mean field methods

- Percolation on networks: application to SIR model

- Epidemic threshold for diseases processes on networks: Epidemic thresholds for a variety of random graph models and contact processes will be computed. This will include bipartite and directed graph models.

__Student Evaluation:__

1. In class presentation of reading assignments.

2. Written assignments.

3. Written end of term project, which would also be presented orally.

__References:__

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

MEJ Newman, Networks: An Introduction, Oxford University Press (2012).

I Kiss, J Miller, P Simon: Mathematics of Epidemics on Networks: From Exact to Approximate Models, Springer (2017).

R Pastor-Satorras, C Castellano, P Van Mieghem, A Vespignani: Epidemic processes in complex networks. Reviews of Modern Physics, vol 85, 925-979 (2015).

NepidemiX Software Package: http://nepidemix.irmacs.sfu.ca

### Summer 2017

**Math 894-2 G100 Summer School in Probability**

**Math 894-2 G100 Summer School in Probability**

*Marni Mishna*

__Description:__

This reading course will shadow the PIMS-CRM Summer School in Probability, which is a 4 week program at UBC June 5-30, 2017.

It is comprised of 2 main courses (24 hours of instruction each), 3 mini-courses (3 hours each), and presentation sessions.

Students taking this reading course will attend the summer school and then upon its completion will submit written projects, and give a presentation.

The two main courses are:

1. Marek Biskup (UCLA): Level sets and extreme points of the Discrete Gaussian Free Field and related processes.

2. Hugo Duminil-Copin (Institut des Hautes Études Scientifiques): Graphical approach to lattice spin models.

The mini-courses are:

• Sandra Cerrai (University of Maryland, College Park): On some asymptotic problems for stochastic PDEs.

• Christina Goldschmidt (Oxford): Scaling limits of random graphs.

• Martin Hairer (Warwick): A BPHZ theorem for stochastic PDEs.

__Evaluation:__

Students are expected to faithfully attend the 4 week summer school, including courses, lectures, and problem sessions.

At the completion of the school, the students will prepare:

- a 5 page extended abstract and a 20 page report, both on topics related to the material of the summer school;

- a 50 minute oral presentation on the report material.

__Additional Details:__

http://www.math.ubc.ca/Links/ssprob17/index.php

__Prerequisites:__

Enrolment by instructors permission only.

__Math 895-4 G100 Hilbert Spaces, Approximation, and the Spectral Theory of Compact Operators__

*Nilima Nigam*

__Description__:

Hilbert spaces, the Hahn Banach theorem, the Banach Steinhaus theorem, the open mapping and closed graph theorems. Weak and strong convergence. Approximation theory and orthogonal sets. Compact operators. Spectral theory of compact linear operators and bounded self-adjoint operators.

__Grading__:

100% Assignments & Presenation of Solutions

Reading and Problems will be assigned. Problems will be graded.

Students will present their solutions in detail in class.

Students will prepare and present each lecture, and distribute notes beforehand.

Students will be evaluated on preparation, clarity and understanding of the material.

__References__:

Chapters 3, 4, 6, 8 and 9 of the text by E. Kreyzsig.

A fast review of chapters 1, 2 and 7 will be provided.

__Prerequisites:__

Math 320 & Math 425/725.

Enrolment by instructors permission only.

__MATH 895-4 G200 - Topics in Computer Algebra__

*Michael Monagan*

__Topics:__

1 The Fast Fourier Transform (Lectures May 9, 11)

- The FFT as an affine tranformation.

- The radix 2 and radix 4 FFT.

- The polynomial Newton iteration.

- Application to fast division.

2 Multivariate Polynomial Interpolation (Lectures May 18, 23, 25)

- Browns' algorithm for multivariate polynomial GCDs.

- Zippel's sparse interpolation.

- Ben-Or and Tiwari sparse interpolation.

3 Computational Linear Algebra (Lectures June 6, 8)

- The Bareiss fraction-free algorithm for computing det(A) and solving Ax=b.

- The Berkowitz division free algorithm for computing det(A - lambda I)

- Solving Ax=b over Q using p-adic lifting + rational reconstructin.

4 Algebraic Number Fields (Lectures June 20, 22)

- Representation of elements in Q(alpha) and calculating norms.

- The Trager-Kronecker algorithm for factoring polynomials in Q(alpha)[x].

- A modular gcd algorithm for Q(alpha)[x]

- Cyclotomic fields and solving Ax=b over Q(alpha).

5 Polynomial Data Structures and Algorithms (Lectures July 4, 6)

- Multivariate polynomial representations and term orderings.

- Polynomial multiplication and division using heaps.

- The Kronecker substitution.

Additional Topics

- The Black Box and White Box model of computation.

- Automatic Differentiation.

__Grading:
__ Assignments 60% (five assignments, one per topic, worth 12% each).

- Course Project 40% (presentation as a poster or as a written report).

__References__:

o Modern Computer Algebra by von zur Gathen and Gerhard.

o Algorithms for Computer Algebra by Geddes, Czapor and Labahn.

o Various research papers.

For the Course Project students will choose a research problem either from a list of pre-selected topics provided by the instructor or a topic of their own choice in consultation with the instructor. The project will involve reading selected papers from the literature, implementing one or more algorithms, studying the mathematical basis for the algorithm(s) and comparing algorithms theoretically and/or experimentally. Students may present their work either as a poster at the department's annual Symposium on Mathematics and Computation or as a written report in LaTeX of between 10 and 15 pages in length.

__Prerequisites__:

MACM 401 or MATH 701 or MATH 801. Enrolment with Instructor Permission only.

### Spring 2017

__Math 894-2 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.

__Prerequisites:__

Functional analysis, PDEs. Enrolment by instructors permission only.

### Fall 2016

__Math 894-2 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

__Prerequisites:__

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

__Math 895-4 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

__Prerequisites:__

N/A

### Summer 2016

__Math 895-4 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.

__Prerequisites:__

Permission of Instructor.

APMA 901, APMA 922.

Graduate-level functional analysis and Sobolev spaces will be assumed.

### Spring 2016

__Math 894-2 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.

__Prerequisites:__

Permission of Instructor. Math 708 Recommended.

__Math 894-2 G200 Singularity Analysis and Combinatorial Enumeration__

*Marni Mishna*

__Description__:

The topic of the course is complex analytic methods for asymptotic 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)

__Prerequisites:__

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 895-4 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 894-2 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__

__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

__Math 894-2 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.

**Permission from instructor.**

__Prerequisites:__**Math 895-4 G100 Topics in Computer Algebra**

**Math 895-4 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

**Math 895-4 G200 Topics in Kinetic Theory**

**Math 895-4 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 justications 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 final project.

**Math 895-4 G100 Introduction to Measure-Theoretic Probability**

**Math 895-4 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

**Math 895-4 G100 Topics in Computer Algebra**

**Math 895-4 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**

**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**

**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__

__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**

**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**

**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**

**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**

**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__

__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__

__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__

__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__

__Topics in Magnetohydrodynamics (MHD) and Computational MHD__

*Steven Pearce, Computing Science Department
*

Resources:

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__

__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__

__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__

__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__

__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__

__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__

__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__

__The Numerical Solution of Integral Equations__

*Mary-Catherine Kropinsk*i

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__

__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__

__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__

__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.

- Probabilities and Events
- Random Variables
- Conditional Probability
- Markov Chains
- Exponential Distribution and Poisson Process
- Continuous-Time Markov Chains

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

- 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.
- 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__

__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__

__Discrete optimization with Applications__

*Tamon Stephen*

### Summer 2010

__An applications-driven introduction to finite elements__

__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__

__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:

- Classical dynamics
- Dynamics over local fields: good reduction
- Dynamics over global fields
- Families of dynamical systems
- Dynamics over local fields: bad reduction
- Dynamics associated to algebraic groups
- 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__

__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

*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__

__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__

__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:**

- Review of relevant topics from probability
- Random Walks and the Central Limit Theorem
- random walks
- diffusion & Fokker-Planck eqn
- central limit theorem

- Large Deviations
- cumulant expansion
- large deviation theorem
- large deviations with constraints

- Power Law Distributions
- stable laws (Gaussian & Levy laws)
- intuitive calculation tools for power law distributions
- Fox function, Mittag-Leffler function, and Levy distributions

- 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

- Long-Range Correlations
- criterion for relevance of correlations
- statistical interpretation
- correlation and dependence

- Phase Transitions: Critical Phenomena and First-Order Transitions
- spin models
- first-order versus critical transitions

- Transitions, Bifurcations, and Percursors
- supercritical bifurcation
- critical percursor fluctuations
- scaling and percursors near spinodals
- selection of an attractor

- The Renormalization Group
- general framework
- example: the hierarchical model
- criticality and the renormalization group

- Applications to Financial Modelling
- calculation of the critical exponents for the 1929 crash
- self-similarity in financial time-series data

### Fall 2009

__Combinatorial Optimization__

__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__

__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 Algeb____r__a

__Topics in Computer Algeb__

*Michael Monagan*

Topics:

- The Fast Fourier Transform, fast integer multiplication (Shoenhage-Strassen), and the fast Euclidean algorithm.
- Sparse polynomial multiplication and division using heaps and geobuckets.
- Computing multivariate polynomial GCDs over Z: Brown's dense interpolation algorithm and Zippel's sparse interpolation algorithm.
- 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.
- 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 Probab**ility

**Advanced Probab**ility

*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.