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:

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 ANALYTIC 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

 

Spring 2009

 

Fall 2008