An instructor-supervised group project in an off-campus setting. Details at https://www.artsci.utoronto.ca/current/academics/research-opportunities…. Not eligible for CR/NCR option.

An instructor-supervised group project in an off-campus setting. Details at https://www.artsci.utoronto.ca/current/academics/research-opportunities…. Not eligible for CR/NCR option.

Credit course for supervised participation in faculty research project. Details at https://www.artsci.utoronto.ca/current/academics/research-opportunities…. Not eligible for CR/NCR option.

Commutative rings; quotient rings. Construction of the rationals. Polynomial algebra. Fields and Galois theory: Field extensions, adjunction of roots of a polynomial. Constructibility, trisection of angles, construction of regular polygons. Galois groups of polynomials, in particular cubics, quartics. Insolvability of quintics by radicals.

Euclidean and non-Euclidean plane and space geometries. Real and complex projective space. Models of the hyperbolic plane. Connections with the geometry of surfaces.

This course is the second part of the "Classical Geometries" MAT402H1 course. It is mainly dedicated to detailed study of classical real projective geometry and projective geometry over other fields. It is also devoted to the study of spherical and elliptic geometry.

Set theory and its relations with other branches of mathematics. ZFC axioms. Ordinal and cardinal numbers. Reflection principle. Constructible sets and the continuum hypothesis. Introduction to independence proofs. Topics from large cardinals, infinitary combinatorics and descriptive set theory.

Joint undergraduate/graduate course - MAT409H1/MAT1404H

A selection from the following: finite fields; global and local fields; valuation theory; ideals and divisors; differents and discriminants; ramification and inertia; class numbers and units; cyclotomic fields; Diophantine equations.

A selection from the following: distribution of primes, especially in arithmetic progressions and short intervals; exponential sums; Hardy-Littlewood and dispersion methods; character sums and L-functions; the Riemann zeta-function; sieve methods, large and small; Diophantine approximation, modular forms.

Joint undergraduate/graduate course - MAT417H1/MAT1202H

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.

The course will survey the branch of mathematics developed (in its abstract form) primarily in the twentieth century and referred to variously as functional analysis, linear operators in Hilbert space, and operator algebras, among other names (for instance, more recently, to reflect the rapidly increasing scope of the subject, the phrase non-commutative geometry has been introduced). The intention will be to discuss a number of the topics in Pedersen's textbook Analysis Now. Students will be encouraged to lecture on some of the material, and also to work through some of the exercises in the textbook (or in the suggested reference books).

Joint undergraduate/graduate course - MAT436H1/MAT1011H

The theory of operator algebras was begun by John von Neumann eighty years ago. In one of the most important innovations of this theory, von Neumann and Murray introduced a notion of equivalence of projections in a self-adjoint algebra (*-algebra) of Hilbert space operators that was compatible with addition of orthogonal projections (also in matrix algebras over the algebra), and so gave rise to an abelian semigroup, now referred to as the Murray-von Neumann semigroup.

Later, Grothendieck in geometry, Atiyah and Hirzebruch in topology, and Serre in the setting of arbitrary rings (pertinent for instance for number theory), considered similar constructions. The enveloping group of the semigroup considered in each of these settings is now referred to as the K-group (Grothendieck's terminology), or as the Grothendieck group.

Among the many indications of the depth of this construction was the discovery of Atiyah and Hirzebruch that Bott periodicity could be expressed in a simple way using the K-group. Also, Atiyah and Singer famously showed that K-theory was important in connection with the Fredholm index. Partly because of these developments, K-theory very soon became important again in the theory of operator algebras. (And in turn, operator algebras became increasingly important in other branches of mathematics.)

The purpose of this course is to give a general, elementary, introduction to the ideas of K-theory in the operator algebra context. (Very briefly, K-theory generalizes the notion of dimension of a vector space.)

The course will begin with a description of the method (K-theoretical in spirit) used by Murray and von Neumann to give a rough initial classification of von Neumann algebras (into types I, II, and III). It will centre around the relatively recent use of K-theory to study Bratteli's approximately finite-dimensional C*-algebras---both to classify them (a result that can be formulated and proved purely algebraically), and to prove that the class of these C*-algebras---what Bratteli called AF algebras---is closed under passing to extensions (a result that uses the Bott periodicity feature of K-theory).

Students will be encouraged to prepare oral or written reports on various subjects related to the course, including basic theory and applications.

Joint undergraduate/graduate course - MAT437H1/MAT1016H

A selection of topics from: Representation theory of finite groups, topological groups and compact groups. Group algebras. Character theory and orthogonality relations. Weyl's character formula for compact semisimple Lie groups. Induced representations. Structure theory and representations of semisimple Lie algebras. Determination of the complex Lie algebras.

Joint undergraduate/graduate - MAT445H1/MAT1196H

Basic notions of algebraic geometry, with emphasis on commutative algebra or geometry according to the interests of the instructor. Algebraic topics: localization, integral dependence and Hilbert's Nullstellensatz, valuation theory, power series rings and completion, dimension theory. Geometric topics: affine and projective varieties, dimension and intersection theory, curves and surfaces, varieties over the complex numbers. This course will be offered in alternating years.

Joint undergraduate/graduate course - MAT448H1/MAT1155H

Projective geometry. Curves and Riemann surfaces. Algebraic methods. Intersection of curves; linear systems; Bezout's theorem. Cubics and elliptic curves. Riemann-Roch theorem. Newton polygon and Puiseux expansion; resolution of singularities. This course will be offered in alternating years.

Harmonic functions, Harnack's principle, Poisson's integral formula and Dirichlet's problem. Infinite products and the gamma function. Normal families and the Riemann mapping theorem. Analytic continuation, monodromy theorem and elementary Riemann surfaces. Elliptic functions, the modular function and the little Picard theorem.

Joint undergraduate/graduate course - MAT454H1/MAT1002H

Lebesgue measure and integration; convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem. Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p spaces, Hölder and Minkowski inequalities.

Joint undergraduate/graduate course - MAT457H1/MAT1000H

Fourier series and transform, convergence results, Fourier inversion theorem, L^2 theory, estimates, convolutions. Banach spaces, duals, weak topology, weak compactness, Hahn-Banach theorem, open mapping theorem, uniform boundedness theorem.

Joint undergraduate/graduate course - MAT458H1/MAT1001H

This course focuses on key notions of classical mechanics: Newton equations, variational principles, Lagrangian formulation and Euler-Lagrange equations, the motion in a central force, the motion of a rigid body, small oscillations, Hamiltonian formulation, canonical transformations, Hamilton-Jacobi theory, action-angle variables, and integrable systems.

Riemannian metrics. Levi-Civita connection. Geodesics. Exponential map. Second fundamental form. Complete manifolds and Hopf-Rinow theorem. Curvature tensors. Ricci curvature and scalar curvature. Spaces of constant curvature.

Joint undergraduate/graduate course - MAT464H1/MAT1342H

This course addresses the question: How do you attack a problem the likes of which you have never seen before? Students will apply Polya's principles of mathematical problem solving, draw upon their previous mathematical knowledge, and explore the creative side of mathematics in solving a variety of interesting problems and explaining those solutions to others.

Seminar in an advanced topic. Content will generally vary from semester to semester. Student presentations are required.

Seminar in an advanced topic. Content will generally vary from semester to semester. Student presentations are required.

A course in mathematics on a topic outside the current undergraduate offerings. For information on the specific topic to be studied and possible additional prerequisites, go to http://www.math.toronto.edu/cms/current-students-ug/.

Joint undergraduate/graduate course - MAT482H1/MAT1901H

A course in mathematics on a topic outside the current undergraduate offerings. For information on the specific topic to be studied and possible additional prerequisites, go to http://www.math.toronto.edu/cms/current-students-ug/.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

Completed applications for this course are due to the Math Undergraduate Program Office no later than the third day of the term that the reading course will start.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 36L course. Not eligible for CR/NCR option.

Completed applications for this course are due to the Math Undergraduate Program Office no later than the third day of the term that the reading course will start.

Independent research under the direction of a faculty member. Not eligible for CR/NCR option. Similar workload to a 72L course.

Completed applications for this course are due to the Math Undergraduate Program Office no later than the third day of the term that the reading course will start.

Independent study under the direction of a faculty member. Topic must be outside undergraduate offerings. Workload equivalent to a 72L course. Not eligible for CR/NCR option.

Completed applications for this course are due to the Math Undergraduate Program Office no later than the third day of the term that the reading course will start.

This course will introduce the principles of semiotic thought, applying them to the study of language, social organization, myth, and material culture. Examples may be drawn from everyday life as well as from classical and popular art and music, and from screen culture.