|Algebraic Geometry Math 5065|
1. Linear Algebra (a one-semester undergraduate course like Math 2135)
2. Abstract Algebra (one year at the undergraduate level like Math 3030, the students should know what rings and fields are in particular, and basics about ideals). They should have seen the example of a polynomial ring.
3. Complex numbers (not a complex analysis course, but some familiarity with complex numbers, how they work, and the fact that every polynomial has a complex root.
Introduction to Algebraic Geometry:
Algebraic Geometry is the study of curves, surfaces, and higher-dimensional geometrical objects that are defined by *polynomial* equations. Examples are lines in the plane (indeed, lines in any dimension); conics in the plane (parabolas, ellipses, hyperbolas); quadric surfaces in 3-space (e.g., the parabolic hyperboloids!); linear subspaces in any dimension (in British English these are called 'flats'). Some of these objects (like conics) have been studied intensely for millenia; quadric surfaces for centuries (since the Renaissance); and yet the subject is quite alive with progress. Applications include interpolation theory, cryptography (see for example Prof. Bauer's lectures), coding theory, and splines, and computer graphics. However this course will focus on the basic theory and examples more than on any systematic development of applications.
The primary feature of algebraic geometry that distinguishes itself from other geometry subjects is the intimate interplay of the geometry with the algebra. Virtually all objects in algebraic geometry come with an algebraic structure associated to them (usually a ring), and it is a major theme of the subject that the geometry can be derived from the algebra as well as the reverse, to a large extent. That will also be a major theme of this course.
1. Affine space and affine algebraic sets (varieties)
[affine coordinate rings, polynomial maps, rational functions]
2. Projective space and projective varieties
[homogeneous coordinates and rings, rational functions, infinity]
3. Examples of projective varieties
[linear spaces, conics and quadrics, plane curves of higher degree]
4. Smoothness and Dimension
[Jacobian condition, Krull Dimension]
5. Cubic Curves
[Classification, the group law, birational models, elliptic curves]
6. Cubic Surfaces
[The 27 lines on a smooth cubic surface]
"Elementary Algebraic Geometry", by Klaus Hulek. Published by the American Mathematical Society;
Student Mathematical Library Volume 20 (2003) ISBN 0-8218-2952-1
I will also supplement a bit with my own notes. Other supplemental texts (not required though) could be:
W. Fulton: "Algebraic Curves: and introduction to algebraic geometry", reprint, Addison Wesley 1989
J. Harris: "Algebraic Geometry, a first course", Springer-Verlag 1992
R. Miranda: "Algebraic Curves and Riemann Surfaces", AMS, Graduate Studies in Mathematics Volume 5 (1995)
M. Reid: "Undergraduate Algebraic Geometry", LMS Student Texts 12, Cambridge University Press 1988
I. Shafarevich: "Basic Algebraic Geometry I", 2nd Edition, Springer-Verlag 1994
Regular Exercises: 40%
Two Exams: 30% each
|Elliptic Curve Cryptography Math 5166|
Course in undergraduate abstract algebra (for instance MATH 3030X/Y.06 offered by Dalhousie University). Topics that should have been covered include: groups, sub-groups, homomorphisms, rings, ideals, polynomial rings, fields, and finite fields. Some minimal Galois theory would also be useful, but not required.
Elliptic Curves - Number Theory and Cryptography by Lawrence C. Washington
The grade for this course will be based solely on 3 assignments, one due every week starting with the second week.
This course will provide an introduction to elliptic curves with an emphasis on their use in cryptography. We will begin with some basic foundations of cryptology and the problems that are of interest to the modern cryptographer. Particular emphasis will be given to public-key cryptography. This will provide the framework and motivation for our study of elliptic curves.
Special attention will be given to elliptic curves over finite fields,
although some time will be given to other global fields.
Topics covered will include the group law, torsion points, group structure,
Hasse-Weil bound, Weil pairing, affine and projective spaces, and
supersingular curves. Additional material may be covered if there is time.
In regards to elliptic curve cryptography, we will study the efficient
implementation of point addition, key exchange protocols, encryption
schemes and digital signature schemes. Finally, we will discuss the
elliptic curve discrete logarithm problem and the associated complexity
for generic elliptic curves and special classes of elliptic curves.
|Theory of Wavelets Math 5054|
The purpose of this course is to present the theory of wavelets, illustrate why they provide us with a particularly powerful tool in mathematical analysis and indicate how they can be used in applications.
Measure theory (e.g. Math 4010);
(Elementary) Functional analysis (e.g.Math 4140);
Fourier transform and its basic properties.
Numerical analysis of wavelet methods.
Studies in Mathematics and its Applications, 32. North-Holland Publishing Co., Amsterdam, 2003
Hernandez, Eugenio; Weiss, Guido
A first course on wavelets.
Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1996
Wavelets in numerical simulation. Problem adapted construction and applications.
Lecture Notes in Computational Science and Engineering, 22. Springer-Verlag, Berlin, 2002
Canuto, Claudio and Tabacco Anita
Ondine Biortogonali: teoria e applicazioni. Quaderni UMI, 46. Pitagora Editrice, Bologna, 1999
|Massive Networks and Internet Mathematics Math 5056|
1. Introductory Discrete Mathematics and/or Graph Theory (e.g. Math 2112/2113)
2. Introductory Probability (e.g. Math 3360)
3. Introductory Linear Algebra (e.g. Math 2030/2040)
Massive real-world networks have recently attracted much research attention. An important example of such a network is the web graph, where the nodes are HTML web pages, and there is an edge between two pages if there is a hyperlink from one to the other. Other such networks are collaboration graphs, call graphs, and protein graphs. These networks exhibit the small world property (only a few edges are needed to travel from one node to another), and they have power law degree distributions (while very many nodes have few edges incident with them, a few nodes, called hubs, have a large number of edges incident with them). The course will focus on the graph theoretical structure of such massive self-organizing networks, and introduce the stochastic and deterministic models used to study them. Our emphasis will be on the web graph, although other real-world networks will be discussed. Linear algebraic techniques commonly used in information retrieval in the World Wide Web, such as Kleinberg's HITS algorithm, or Google's Pagerank, will be discussed.
Please see below for a tentative weekly schedule.
Text: There is no required text for the course.
A final mark out of 100 will be calculated as follows:
The presentation will consist of a 45 minute lecture given in the last week of the course. The topic of the presentation will be chosen in consultation with the instructor, and will be the same topic chosen for the project (see below). A list of suggested topics will be circulated on Week 1.
The project will consist of a survey paper written on the topic chosen for the presentation. The paper will be roughly 10 pages long (single-spaced), and is due the last day of lectures.
Tentative Weekly schedule
Introduction to massive real-world graphs and random graphs
Models of massive networks: degree distributions
Searching networks: HITS and Pagerank