Some topics to be covered:
The prerequisites for the course include familiarity with Sobolev and other
function spaces, and in particular with fundamental embedding and compactness theorems.
MAT 1100YY
ALGEBRA
V. Blomer
A selection of topics from:
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In the title of this course the term "real" stands not only as a reference to the field of real numbers but also as a reference to the real (geometric) world. In this course in addition to real varieties we shall also treat complex varieties (and sometimes even varieties defined over some function fields), but we shall primarily focus on geometric aspects of algebraic geometry. We'll pay special attention to topology of real and complex algebraic varieties, in particular to the topics related to Hilbert's 16th problem and to vanishing cycles of complex singularities. We introduce the so-called "patchworking" technique for constructing real algebraic from simpler pieces. The patchworking serves as a link from this course to the second half-course "Tropical Geometry".
Prerequisites: acquaintance with some basic Algebraic
Geometry and Geometric Topology. New Ph.D. students are welcome,
especially if they like Geometry. Please contact the instructor.
MAT 1302HS (APM 461H1S/CSC 2413HS)
COMBINATORIAL METHODS
S. Tanny
We will cover a selection of topics in enumerative combinatorics, such as more advanced methods in recursions, an analysis of some unusual self-referencing recursions, binomial coefficients and their identities, some special combinatorial numbers and their identities (Fibonacci, Stirling, Eulerian), and a general approach to the theory of generating functions.
Prerequisite:
Linear algebra.
Recommended preparation:
an introductory combinatorics course, such as MAT 344H.
MAT 1340HF (MAT 425H1F)
DIFFERENTIAL TOPOLOGY
A. Nabutovsky
Smooth manifolds, Sard's theorem and transversality. Morse theory.
Immersion and embedding theorems. Intersection theory. Borsuk-Ulam theorem.
Vector fields and
Euler characteristic, Hopf degree theorem.
Additional topics may vary.
Textbook:
Victor Guillemin and Alan Pollack, Differential Topology
Prerequisites: MAT 257Y (second year Analysis course), MAT 240H
(second year Algebra course) and MAT 327H (introductory Topology course).
MAT 1342HS (MAT 464H1S)
DIFFERENTIAL GEOMETRY
J. Bland
Riemannian metrics and connections. Geodesics. Exponential map. Complete manifolds.
Hopf-Rinow theorem. Riemannian curvature. Ricci and scalar curvature. Tensors.
Spaces of constant curvature. Isometric immersions. Second fundamental form. Topics from:
Cut and conjugate loci. Variation energy. Cartan-Hadamard theorem. Vector bundles.
References:
Manfredo Perdigao de Carmo: Riemannian Geometry.
J. Cheeger, D. Ebin: Comparison Theorems in Riemannian Geometry, Elsevier, 1975.
Prerequisite: Introductory differential geometry course such as MAT 363H.
MAT 1404HF (MAT 409H1F)
SET THEORY
W. Weiss
We will introduce the basic principles of axiomatic set theory, leading to the undecidability of the continuum hypothesis. We will also explore those aspects of infinitary combinatorics most useful in applications to other branches of mathematics.
Prerequisite: an introductory real analysis course such as MAT 357H
Textbooks:
W. Just and M. Weese: Discovering Modern Set Theory, I and II, AMS.
K. Kunen: Set Theory, Elsevier.
MAT 1508HS (APM 446H1S)
APPLIED NONLINEAR EQUATIONS
A. Burchard
Nonlinear partial differential equations and their physical origin.
Fourier transform; Green's function; variational methods;
symmetries and conservation laws. Special solutions (steady states,
solitary waves, travelling waves, self-similar solutions).
Calculus of maps; bifurcations; stability, dynamics near
equilibrium. Propogation of nonlinear waves; dispersion, modulation,
optical bistability. Global behaviour solutions; asymptotics and blow-up.
Prerequisite:
APM346H1/APM351Y1
MAT 1700HS (APM 426H1S)
GENERAL RELATIVITY
P. Blue
Special relativity. The geometry of Lorentz manifolds. Gravity as a manifestation of spacetime curvature.
Einstein's equation. Cosmological consequences: the big bang and inflationary universe.
Schwarschild stars: the bending of light and perihelion procession of Mercury.
Black hole dynamics. Gravitational waves.
Prerequisites:
Thorough knowledge of linear algebra and multivariable
calculus. Some familiarity with partial differential equations, topology, differential
geometry, and/or physics will prove helpful.
Reference:
R. Wald, General Relativity, University of Chicago Press
MAT 1723HF (APM 421H1F)
MATHEMATICAL CONCEPTS OF QUANTUM MECHANINCS AND QUANTUM INFORMATION
I. M. Sigal
The goal of this course is to explain key concepts of Quantum Mechanics and
to arrive quickly to some topics which are at the forefront of active
research, such as Bose-Einstein condensation, control of chemical reactions
and quantum information. We will try to be as self-contained as possible and
rigorous whenever the rigour is instructive. Whenever the rigorous treatment
is prohibitively time-consuming we give an idea of the proof, if such
exists, and/or explain the mathematics involved without providing all the
details.
Prerequisites for this course: some familiarity with elementary
ordinary and partial differential equations and elementary theory of
functions and operators.
Syllabus:
References:
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics,
Springer
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory,
Amsterdam, The Netherlands: North Holland.
MAT 1856HS (APM 466H1S)
MATHEMATICAL THEORY OF FINANCE
Instructor: TBA
Introduction to the basic mathematical techniques in pricing theory and
risk management: Stochastic calculus, single-period finance, financial
derivatives (tree-approximation and Black-Scholes model for equity derivatives,
American derivatives, numerical methods, lattice models for interest-rate
derivatives), value at risk, credit risk, portfolio theory.
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Topics will be chosen from among the following:
Other reference:
"Magnetic resonance imaging: Physical Principles & Sequence Design".
EM Haake, BW Brown, MR Thompson, R Venkatesan, J Wiley 1999.
Marking scheme:
75% project, 25% homework.
MAT 1035HF
INTRODUCTION TO C*-ALGEBRAS
M.-D. Choi
This course is concerned with the basic aspects of C*-algebras.
During the first half of the semester, the lectures will be
devoted to a systematic investigation of the finite-dimensional case---namely,
the theory of n \times n complex matrices. Through many simple concrete
examples we may describe various phenomena arising from the interplay
of normed structure, order structure and algebraic structure. Later
in the semester, we will continue to explore more striking facts
pertinent to the infinite-dimensional case. Based on the background
of students, various topics of related interest will be pursued.
MAT 1045HS
INTRODUCTION TO (SMOOTH) ERGODIC THEORY
G. Forni
The course is an introduction to some of the basic notions and
methods of
the theory of dynamical systems. It covers notions of topological
dynamics and
ergodic theory such as minimality, topological transitivity, ergodicity,
unique ergodicity, weak mixing and mixing. These notions will be
explained
by examining simple concrete examples of dynamical systems such as
translations and automorphisms of tori, expanding maps of the interval,
topological Markov chains, etc. Fundamental theorems of ergodic theory
such as the Poincare recurrence theorem, the Von Neumann mean ergodic
theorem and the Birkhoff ergodic theorem will be presented. Time
permitting
we will outline either entropy theory (topological entropy and
Kolmogorov-Sinai entropy) and/or the subadditive ergodic theorem, the
theory of Lyapunov exponents and the Oseledets theorem (with examples).
Prerequisites: Knowledge of real analysis, basic topology and measure
theory. Some knowledge of functional analysis would be useful.
MAT 1052HF
HARMONIC ANALYSIS, DIOPHANTINE EQUATIONS AND INEQUALITIES
M. Goldstein
Waring's Problem. Weyl's Inequality. The Hardy-Littlewood Circle
Method. Continued Fractions and Approximation by Rationals.
Roth's Theorem.
References:
1. Natanson, M., Additive Number Theory,
2. Montgomery, H., Ten Lectures on the Interface between
Analytic Number Theory and Harmonic Analysis
3. Schmidt, W., Diophantine Approximations
Prerequisites: Introductory courses in complex analysis (e.g. MAT 354H)
and real analysis (e.g. MAT 357H).
MAT 1062HS
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
C. Sulem
GOAL: The goal is to introduce some basic numerical techniques
for solving ordinary and partial differential equations.
The course is intended to students in Applied Mathematics and Engineering
who wish to learn numerical methods useful for their research.
Galois Theory belongs to algebra. It is very understandable and has a lot of applications. For example it explains why algebraic equations are usually not solvable by means of radicals.
About 30 years ago I constructed a topological version of Galois Theory for functions in one complex variable. According to it, there are topological restrictions on the way the Riemann surface of a function representable by radicals covers the complex plane. If the function does not satisfy these restrictions, then it is not representable by radicals. Beside its geometric clarity the topological results on nonsolvability are stronger than the algebraic results. They have a lot of generalizations.
In the course I plan to present Galois Theory in details, to discuss topological results on nonsolvability by radical and their generalizations, to give an introduction to a multidimensional version of the theory.
Prerequisite:Prerequisites:
The spectral theorem
References:
M. Rordam, K-Theory for C*-Algebras
H. Lin, An Introduction to the Classification of Amenable C*-Algebras
A. Connes, Noncommutative Geometry
MAT 1126HF
INTRODUCTION TO NONHOLONOMIC MECHANICS AND GEOMETRY
B. Khesin
Nonholonomic mechanics describes the geometry of systems subordinated to
nonholonomic constraints, i.e systems whose restrictions on velocities do
not arise from the constraints on the configuration space. The best known
examples of such systems are a sliding skate and a rolling ball, as well
as their numerous generalizations.
We start with an introduction to the Euler-Lagrange equation and the
Lagrange-d'Alambert principle. After defining Hamiltonian systems and
their integrability we go through
This course will provide a basic introduction to algebraic geometry and the underlying commutative algebra. Topics will include:
This course is intended to introduce students to a recent technique in Algebraic Geometry based on application of the moment map and toric degenerations. One of the simplest examples of the moment map is the logarithm map that takes a point of the complex torus C*^n to the point in Rn obtained by taking the logarithm of the absolute value coordinatewise . The images of holomorphic subvarieties of C*n under this map are called amoebas.
If one modifies this moment map by taking the logarithm with base t and lets t to go to infinity then the amoebas tend to some piecewise- linear polyhedral complexes in R^n. The dimension of these limiting complexes is equal to the dimension of the original varities. It turns out that such complexes can be considered as algebraic varieties over the so-called tropical semifield. The term "tropical semifield" appeared in Computer Science and, in the current context, refers to the real numbers augmented with the negative infinity and equipped with two operations, taking the maximum for addition and addition for multiplication.
Polynomials over the tropical semifield are convex piecewise-linear functions and geometric objects associated to these polynomials are certain piecewise-linear complexes in Rn.In the course we consider applications of both the amoebas themselves and the resulting tropical geometry. One area where amoebas turn out to be useful is Topology of Real Algebraic Varieties, in particular, problems related to Hilbert's 16th problem. Using amoebas we show topological uniqueness of a homologically maximal curve in the real torus R*2 and deduce a partial topological description for hypersurfaces in R*n for n>2. Applications of tropical geometry include construction of real algebraic varieties with prescribed topology (patchworking) as well as enumerative geometry.
A typical problem in enumerative algebraic geometry is to compute the number of curves of given degree and genus and with a given set of geometric constraints (e.g. passing through a point or another algebraic cycle, being tangent to such cycle, etc.). For a proper number of geometric constraints one expects a finite number of such curves. Even in the cases when this number is not finite there exists a way to interpret the answer to such problem as a (perhaps fractional or negative) Gromov-Witten number. Tropical geometry can be used for computation of these numbers. In this course we'll compute such numbers for arbitrary genus and degree when the ambient space is a toric surface and for genus 0 (and arbitrary degree) if the ambient space is a higher-dimensional toric variety. In addition we consider real counterparts of the enumerative problems, in particular, the Welschinger invariant, and do some computations for them.
Prerequisites: MAT 1194HF (MAT 449H1F) or some basic
knowlege of Topology and Geometry. Contact the instructor for
permission.
MAT 1197HF
GEOMETRY OF FLAG VARIETIES FOR SEMI-SIMPLE ALGEBRAIC GROUPS AND REPRESENTATION THEORY
S. Arkhipov
Below are the main topics to be covered in the course.
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Topics to be covered: Fundamentals of design; existence of triple systems; enumeration; computational methods; isomorphism and invariants; configuration theory; chromatic invariants; resolvability; directed, Mendelsohn, and mixed triples sytems.
Prerequisite(s):
Linear Algebra and an introductory course in combinatorics such as MAT 344HF
are recommended.
MAT 1344HS
INTRODUCTION TO SYMPLECTIC GEOMETRY
Y. Karshon
We will discuss a variety of concepts, examples, and theorems of
symplectic geometry and topology. These may include, but are not
restricted to, these topics: review of differential forms and
cohomology; symplectomorphisms; local normal forms; Hamiltonian
mechanics; group actions and moment maps; geometric quantization;
a glimpse of holomorphic techniques.
Prerequisites: manifolds, differential forms, cohomology.
Main recommended book:
Ana Cannas da Silva, "Lectures on Symplectic
Geometry", Lecture Notes in Mathematics 1764, Springer-Verlag 2001.
(A second edition should come out at some point but it's not out yet.)
Other relevant books:
Dusa McDuff and Dietmar Salamon, "Introduction to Symplectic Topology",
Oxford Science Publications.
V.I.Arnold, "Mathematical Methods of Classical Mechanics", Second Edition.
Helmut Hofer and Eduard Zehnder, "Symplectic Invariants and Hamiltonian
Dynamics", Birkhäuser, 1994.
Viktor Ginzburg, Victor Guillemin, and Yael Karshon, "Moment maps,
cobordisms, and Hamiltonian group actions", American Mathematical
Society Mathematical Surveys and Monographs 98.
MAT 1350HF
THE JONES POLYNOMIAL
D. Bar-Natan
The Jones polynomial is perhaps the simplest knot invariant to
define; it can be defined (and will be defined in the first class) in
about 5 minutes, invariance can be proven in about 15 minutes, it can be
programmed in another 10 minutes, and then it can be evaluated for the
first few hundred knots in some 10 minutes or so. In the rest of the
semester we will see that the Jones polynomial has some knot theoretic
implications, has lovely generalizations and fits within some nice
pictures, and is a wonderful excuse and unifying centre for the study of
several other deep subjects, including but not limited to combinatorics,
homological algebra, Lie algebras, quantum algebra, category theory and
even quantum field theory. Some of these subjects we will cover in great
detail; others, for the luck of time, will only be briefly touched.
Prerequisites: Graduate core courses in Topology (MAT 1300Y)
and in Algebra (MAT 1100Y), or their equivalents.
MAT 1352HF
INDEX THEORY
E. Meinrenken
This course will be an introduction to the Atiyah-Singer index theorem
for elliptic operators. We will cover the following topics:
Prerequisites: Knowledge of graph theory and harmonic analysis.
MAT 1450HS
SET THEORY: STRUCTURAL RAMSEY THEORY AND DYNAMICS OF GROUPS OF AUTOMORPHISMS
S. Todorcevic
This course is a natural continuation of the 2005-06 MAT 1450HF course,
but will not be bounded to those who took it as the subject matter is
related but not too much dependent. The stress this time will be on
finite Ramsey theory, or more precisely finite structural Ramsey
theory, and its relationship to Fraisse theory of homogeneous structure and topological
dynamics of groups of automorphisms.
Prerequisites: Basic core courses in mathematics.
Textbooks:
Graham-Rothschild-Spencer, Ramsey Theory, 1990
My textbook "Introduction to Ramsey Spaces", which at the moment
is available as draft but at that time may be even in press.
MAT 1502HS
STOCHASTIC CALCULUS
J. Quastel
Random walk, Markov chains, Martingales, Brownian motion, Stochastic
Integrals, Markov processes and associated partial
differential equations, Ito's formula, Stochastic differential
equations, Cameron-Martin-Girsanov formula, relation between discrete
and continuous time models, applications including Black-Scholes
formula.
Prerequisite:
Probabilty or Real analysis.
MAT 1739HF
INTRODUCTION TO SUPERSYMMETRIC QUANTUM FIELD THEORIES
K. Hori
This course will give an accessible introduction to supersymmetric
quantum field theories in low dimensions. Supersymmetry is relevant
for many recent developments in various fields in mathemetics, including
Gromov-Witten invariants, Donaldson invariants, Floer homology, various
fixed point theorems, etc. No special background is assumed.
The second group of models deals with mechanisms through which networks of interacting biomolecules (proteins or genes) carry out the essential functions in living cells. Among the questions which are addressed here is how the genetic and biochemical networks withstand considerable variations and random perturbations of biochemical parameters. The complexity and high inter-connectedness of these networks makes the question of the stability in their functioning of special importance.
Finally we will discuss mathematical models of the dynamics of HIV-1 and of cancer growth.
The models above are expressed in terms of Markov chains and stochastic ordinary differential equations. In addition, in the first case, reaction-diffusion equations (e.g. Keller-Segel equations) and stochastic particle dynamics are used. This mathematical background together with its biological interpretation will be developed in the course.
Prerequisites for this course: some familiarity with elementary
ordinary and partial differential equations and elementary probability
theory. No knowledge of biology is required.
MAT 1900Y/1901H/1902H
READINGS IN PURE MATHEMATICS
Numbers assigned for students wishing individual instruction
in an area of pure mathematics.
MAT 1950Y/1951H/1952H
READINGS IN APPLIED MATHEMATICS
Numbers assigned for students wishing individual instruction
in an area of applied mathematics.
STA 2111HF
GRADUATE PROBABILITY I
B. Virag
Random variables, expected value, independence, laws of large numbers, random walks, martingales, Markov chains.
Prerequisite: measure theory (may be taken at the same time) or permission of the instructor.
Textbook:
Durrett, Probability: Theory and Examples
STA 2211HS
GRADUATE PROBABILITY II
J. Quastel
Weak convergence, central limit theorems, stable laws, infinitely divisible laws, ergodic theorems, Brownian motion.
Textbook:
Durrett, Probability: Theory and Examples
The following course is offered to help train students to become effective tutorial leaders and eventually lecturers. It is not for degree credit and is not to be offered every year.
MAT 1499HSThe goals of the course include techniques for teaching large classes, sensitivity to possible problems, and developing an ability to criticize one's own teaching and correct problems.
Assignments will include such things as preparing sample classes, tests, assignments, course outlines, designs for new courses, instructions for teaching assistants, identifying and dealing with various types of problems, dealing with administrative requirements, etc.
The course will also include teaching a few classes in a large course under the supervision of the instructor. A video camera will be available to enable students to tape their teaching for later (private) assessment.
(Math graduate students cannot take the following courses for graduate credit.)
MAT 2000Y READINGS IN THEORETICAL MATHEMATICS(These courses are used as reading courses for engineering and science students in need of instruction in special topics in theoretical mathematics. These course numbers can also be used as dual numbers for some third and fourth year undergraduate mathematics courses if the instructor agrees to adapt the courses to the special needs of graduate students. A listing of such courses is available in the 2006-2007 Faculty of Arts and Science Calendar. Students taking these courses should get an enrolment form from the graduate studies office of the Mathematics Department. Permission from the instructor is required.)
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