Core Graduate Courses | Cross-Listed Courses | Topics Courses
CORE COURSES
MAT1000HF/MAT457H1F
REAL ANALYSIS I
Measure Theory: Lebesgue measure and integration, convergence theorems, Fubini's theorem, Lebesgue differentiation theorem, abstract measures, Caratheodory theorem, Radon-Nikodym theorem.
Functional Analysis: Hilbert spaces, orthonormal bases, Riesz representation theorem, compact operators, L^p-spaces, Holder and Minkowski inequalities.
Textbook:
Gerald Folland, Real Analysis: Modern Techniques and their Applications, Wiley 2nd edition, 1999
References:
Elias Stein and Rami Shakarchi: Measure Theory, Integration, and Hilbert Spaces
Eliott H. Lieb and Michael Loss: Analysis AMS Graduate Texts in Mathematics, 14 (either edition)
H.L. Royden: Real Analysis, Macmillan, 1988.
A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis, 1975.
MAT1001HS/MAT458H1S
REAL ANALYSIS II
Fourier Analysis: Fourier series and transforms, Fourier inversion and Plancherel formula, estimates and convergence results, topological vector spaces, Schwartz space, distributions.
Functional Analysis: The main topic here will be the spectral theorem for bounded self-adjoint operators, possibly together with its extensions to unbounded and differential operators.
Textbook:
G. Folland, Real Analysis: Modern Techniques and their Applications, Wiley.
Reference:
E. Lorch, Spectral Theory.
W. Rudin, Functional Analysis, Second Edition, Indian Edition (if available; the book is hard to get, although there is a pdf on line).
MAT1002HS/MAT454H1S
COMPLEX ANALYSIS
I. Uriarte-Tuero
(View Timetable)
- Review of holomorphic and harmonic functions (Chapters 1-4 in Ahlfors).
- Topology of a space of holomorphic functions: Series and infinite products, Weierstrass p-function, Weierstrass and Mittag-Leffler theorems.
- Normal families: Normal families and equicontinuity, theorems of Montel and Picard.
- Conformal mappings: Riemann mapping theorem, Schwarz-Christoffel formula.
- Riemann surfaces: Riemann surface associated with an elliptic curve, inversion of an elliptic integral, Abel’s theorem.
- Further topics possible; e.g., analytic continuation, monodromy theorem.
Recommended prerequisites:
Undergraduate courses in real and complex analysis.
Textbook:
L. Ahlfors, Compex Analysis, third edition, McGraw-Hill
Recommended references:
H. Cartan, Elementary Theory of Analytic Functions of One or Several Complex Variables, Dover
D. Marshall, Complex Analysis, Cambridge Math. Textbooks
M.F. Taylor, Introduction to Complex Analysis, American Math. Soc., Graduate Studies in Math. 202
MAT1060HF
PARTIAL DIFFERENTIAL EQUATIONS I
This is a basic introduction to partial differential equations as they arise in physics, geometry and optimization. It is meant to be accessible to beginners with little or no prior knowledge of the field. It is also meant to introduce beautiful ideas and techniques which are part of most analysts' basic bag of tools.
Textbook:
L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN-13 978-0821849743
References:
- R. McOwen, Partial Differential Equations, (2nd ed),
Hardcover: 2003 Prentice Hall ISBN 0-13-009335-1,
Paperback: 2002 Pearson ISBN-13 978-0130093356 - Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 978-1-4614-4808-2
MAT1061HS
PARTIAL DIFFERENTIAL EQUATIONS II
This course will consider a range of mostly nonlinear partial differential equations, including elliptic and parabolic PDE, as well as hyperbolic and other nonlinear wave equations. In order to study these equations, we will develop a variety of methods, including variational techniques, and fixed point theorems. One important theme will be the relationship between variational questions, such as critical Sobolev exponents, and issues related to nonlinear evolution equations, such as finite-time blowup of solutions and/or long-time asymptotics.
The prerequisites for the course include familiarity with Sobolev and other function spaces, and in particular with fundamental embedding and compactness theorems.
Other topics in PDE will also be discussed.
Textbook:
L. C. Evans, Partial Differential Equations, AMS 2010 (2nd revised ed) ISBN-13 978-0821849743
Reference:
- R. McOwen, Partial Differential Equations, (2nd ed),
Hardcover: 2003 Prentice Hall ISBN 0-13-009335-1,
Paperback: 2002 Pearson ISBN-13 978-0130093356 - Jurgen Jost, Partial Differential Equations. 3rd Ed. New York: Springer, 2013. ISBN 978-1-4614-4808-2
MAT1100HF
ALGEBRA I
Basic notions of linear algebra: brief recollection. The language of Hom spaces and the corresponding canonical isomorphisms. Tensor product of vector spaces.
Group Theory: Isomorphism theorems, group actions, Jordan-Hölder theorem, Sylow theorems, direct and semidirect products, finitely generated abelian groups, simple groups, symmetric groups, linear groups, nilpotent and solvable groups, generators and relations.
Ring Theory: Rings, ideals, Euclidean domains, principal ideal domains, and unique factorization domains.
Modules: Modules and algebras over a ring, tensor products, modules over a principal ideal domain
Recommended prerequisites are a full year undergraduate course in Linear Algebra and one term of an introductory undergraduate course in higher algebra, covering, at least, basic group theory. While this material will be reviewed in the course, it will be done at "high speed", assuming that you have already some familiarity with the basics. You will be very well prepared indeed, if you have no difficulties reading and understanding the book, listed here under "Other References", M. Artin: Algebra that the author wrote for his undergraduate algebra courses at MIT.
Textbooks:
Lang: Algebra, 3rd edition
Dummit and Foote: Abstract Algebra, 2nd Edition
Other References:
Jacobson: Basic Algebra, Volumes I and II.
Cohn: Basic Algebra
M. Artin: Algebra.
MAT1101HS
ALGEBRA II
Fields: Algebraic and transcendental extensions, normal and separable extensions, fundamental theorem of Galois theory, solution of equations by radicals.
Commutative Rings: Noetherian rings, Hilbert basis theorem, invariant theory, Hilbert Nullstellensatz, primary decomposition, affine algebraic varieties. structure of semisimple algebras, application to representation theory of finite groups.
Recommended textbooks:
Grillet: Abstract Algebra (2nd ed.)
Dummit and Foote: Abstract Algebra, 3rd Edition
Jacobson: Basic Algebra, Volumes I and II.
Lang: Algebra 3rd Edition
MAT1300HF
DIFFERENTIAL TOPOLOGY
Local differential geometry: the differential, the inverse function theorem, smooth manifolds, the tangent space, immersions and submersions, regular points, transversality, Sard's theorem, the Whitney embedding theorem, smooth approximation, tubular neighborhoods, the Brouwer fixed point theorem.
Differential forms: exterior algebra, forms, pullbacks, integration, Stokes' theorem, div grad curl and all, Lagrange's equation and Maxwell's equations, homotopies and Poincare's lemma, linking numbers.
Prerequisites: linear algebra; vector calculus; point set topology
Textbook:
Differential Topology, Victor Guillemin and Alan Pollack,
American Mathematical Society ISBN-10: 0821851934, ISBN-13: 978-0821851937
MAT1301HS
ALGEBRAIC TOPOLOGY
Fundamental groups: paths and homotopies, the fundamental group, coverings and the fundamental group of the circle, Van-Kampen's theorem, the general theory of covering spaces.
Homology: simplices and boundaries, prisms and homotopies, abstract nonsense and diagram chasing, axiomatics, degrees, CW and cellular homology, subdivision and excision, the generalized Jordan curve theorem, salad bowls and Borsuk-Ulam, cohomology and de-Rham's theorem, products.
Textbook:
Allen Hatcher, Algebraic Topology
Recommended Textbooks:
Munkres, Topology
Munkres, Algebraic Topology
MAT1600HF
MATHEMATICAL PROBABILITY I
The class will cover classical limit theorems for sums of independent random variables, such as the Law of Large Numbers and Central Limit Theorem, conditional distributions and martingales, metrics on probability measures.
References:
Lecture notes and a list of recommended books will be provided.
Recommended prerequisite:
Real Analysis I.
Textbook:
Durrett's "Probability: Theory and Examples", 4th edition
MAT1601HS
MATHEMATICAL PROBABILITY II
The class will cover some of the following topics: Brownian motion and examples of functional central limit theorems, Gaussian processes, Poisson processes, Markov chains, exchangeability.
References:
A list of recommended books will be provided.
Recommended prerequisites:
Real Analysis I and Probability I.
Textbook:
Durrett's "Probability: Theory and Examples", 4th edition
MAT1850HS
LINEAR ALGEBRA AND OPTIMIZATION
A. Stinchcombe
(View Timetable)
This course will develop advanced methods in linear algebra and introduce the theory of optimization. On the linear algebra side, we will study important matrix factorizations (e.g. LU, QR, SVD), matrix approximations (both deterministic and randomized), convergence of iterative methods, and spectral theorems. On the optimization side, we will introduce the finite element method, linear programming, gradient methods, and basic convex optimization. The course will be focused on fundamental theory, but appropriate illustrative applications may be chosen by the instructor.
CROSS-LISTED
MAT1016HF/MAT437H1F
TOPICS IN OPERATOR ALGEBRAS: K-THEORY AND C*-ALGEBRAS
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 classication 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.
Prerequisites:
An attempt will be made to supply the necessary prerequisites when needed (rather few, beyond just elementary algebra and analysis).
Textbook:
Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras
Recommended References:
Edward G. Effros, Dimensions and C*-algebras
Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras
Kenneth R. Davidson, C*-Algebras by Example
MAT1200HS/MAT415H1S
ALGEBRIC NUMBER THEORY
Dedekind domains, ideal class group, splitting of prime ideals, finiteness of class number, Dirichlet unit theorem, further topics such as counting number fields as time allows.
Prerequisites:
Solid knowledge of abstract algebra is essential (e.g. Dummit and Foote, MAT347, MAT1100-1101)
Reference(s):
The main reference will be Problems in Algebraic Number Theory by J. Esmonde and Ram Murty Milne’s course notes.
MAT1202HF/MAT417H1F
ANALYTIC NUMBER THEORY
- Introduction to some of the famous problems and theorems of the subject
- Simple tools from elementary number theory, algebra and analysis
- Dirichlet’s theorem on primes in arithmetic progressions
- Prime Number Theorem
- Prime number theorem for arithmetic progressions
- An introduction to sieve methods
- A selection, if time permits, of some subset of the following topics:
- further zeta-function theory
- L-functions and character sums
- exponential sums and uniform distribution
- Hardy-Littlewood-Ramanujan method
- further theory of prime distribution
Prerequisites:
- A half-year course in complex variables such as MAT 334. (This is the most important prerequisite.)
- A course in groups, rings, fields, such as MAT 347.
- A half year course in introductory number theory such as MAT 315.
- A commitment to attend all lectures.
Texts: There is no formal text. The following books are useful references.
A) General Analytic Number Theory:
-
-
- H. Davenport, Multiplicative number theory, 3rd ed. (revised by H.L. Montgomery) Graduate Texts in Mathematics, Vol. 74 Springer-Verlag 2000
- H. Iwaniec and E. Kowalski, Analytic number theory, American Mathematical Society Colloquium Publications, Vol. 53. 2004
- H. L. Montgomery and R. C. Vaughan, Multiplicative number theory I.
-
Classical theory, Cambridge Studies in Advanced Math, 97, Cambridge 2007.
B) More specialized texts:
- J. Friedlander and H. Iwaniec, Opera de cribro, American Mathematical Society Colloquium Publications, Vol 57, 2010.
- E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed, (revised by D. R. Heath-Brown) Clarendon Press, Oxford 1986.
- R. C. Vaughan, The Hardy-Littlewood method, 2nd ed. Cambridge Tracts in Mathematics,Vol. 125, Cambridge 1997
MAT1302HS/APM461H1S
COMBINATORIAL THEORY
A selection of topics from such areas as graph theory, combinatorial algorithms, enumeration, construction of combinatorial identities.
Prerequisites:
Linear algebra, elementary number theory, elementary group and field theory, elementary analysis.
MAT1304HF/CSC2429HF
Topics in Combinatorics: Finite Fields
This course will cover some important classical and modern themes in the study of finite fields. These will include:
- Solutions of equations
- Pseudorandomness
- Exponential sums and Fourier techniques
- Algebraic curves over finite fields, the Weil theorems
- Additive combinatorics and the sum-product phenomenon
- Applications to combinatorics, theoretical computer science and number theory
MAT1340HS/MAT425H1S
DIFFERENTIAL TOPOLOGY
Sard's theorem and transversality. Immersion and embedding theorems. Morse theory. Intersection theory. Borsuk-Ulam theorem. Euler characteristic, Poincare-Hopf theorem and Hopf degree theorem. Additional topics may vary.
Prerequisites:
Introduction to Topology course (MAT327H) and Analysis (MAT257Y).
MAT1342HF/MAT464H1F
RIEMANNIAN GEOMETRY
The topics include: Riemannian metrics, Levi-Civita connection, geodesics, isometric embeddings and the Gauss formula, complete manifolds, variation of energy.
It will cover chapters 0-9 of the "Riemannian Geometry" book by Do Carmo.
MAT1404HF/MAT409H1F
INTRODUCTION TO MODEL THEORY AND SET THEORY
S. Todorcevic
(View Timetable)
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.
Prerequisite:
MAT357H1
Textbooks:
Set Theory by William Weiss
MAT1700HS/APM426H1S
GENERAL RELATIVITY
This course is an introduction to general relativity theory for students in mathematics and physics alike. Beginning from basic principles this course aims to discuss the modern theory of gravitation and the geometry of space and time, and will explore many of its consequences ranging from black holes to gravitational waves.
More specifically the course covers a discussion of the equivalence principle and its consequences, the geometry of curved spaces, and Einstein's field equations in the presence of matter; we will explore the geometry of the simplest spherically symmetric black hole spacetimes, and proceed to the dynamical formulation of general relativity, and the prediction of gravitational waves. If time permits we shall also discuss in some detail the post-Newtonian approximation.
There will be student projects/presentations on selected topics, such as astrophysical applications, and Penrose's incompleteness theorem.
Prerequisites:
Some prior knowledge of special relativity, and elementary Riemannian geometry will be assumed, but are not strictly required, as relevant concepts will be introduced in the course.
References:
Robert Wald, General Relativity, UCP 1984 Norbert Straumann
General Relativity, Springer 2012 Stephen Hawking and George Ellis
The large scale structure of the space-time, CUP 1973 Charles Misner, Kip Thorne, and John Wheeler, Gravitation, Freeman 1973
MAT1723HF/APM421H1F
MATHEMATICAL FOUNDATIONS OF QUANTUM MECHANICS AND QUANTUM INFORMATION THEORY
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. In particular we will present an introduction to quantum information theory, which has witnessed an explosion of research in the last decade and which involves some nice mathematics.
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:
* Some familiarity with elementary ordinary and partial differential equations
* Knowledge of elementary theory of functions and operators would be helpful
References:
S. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 2nd edition, Springer, 2011
L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008
For material not contained in this book, e.g. quantum information theory, we will try to provide handouts and refer to on-line sources.
Useful, but optional, books on the subject are:
Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information (Paperback - Sep 2000), Cambridge University Press, ISBN 0 521 63503 9 (paperback)
A. S. Holevo, Statistical Structure of Quantum Theory, Springer, 2001
A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory, Amsterdam, The Netherlands: North Holland
MAT1856HS/APM466H1S
MATHEMATICAL THEORY OF FINANCE
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.
Prerequisites:
APM 346H1, STA 347H1
MAT1901HF/MAT482H1F
TOPICS IN MATHEMATICS: DYNAMICS OF TRANSFORMATION GROUPS AND THE CORRESPONDING STRUCTURAL RAMSEY THEORY
S. Todorcevic
(View Timetable)
In the last twenty years there is a growing interest in the connection between these two fields analogous to the one found in the 1970's by Furstenberg for the Szemeredi theorem but now with the roles reversed. The course will start with the basic Model theory of Fraisse structures and their limits and then continue with study of Logic actions of these limits. The goal is to reach a level where structural Ramsey theory could be used to study these actions.
This should be accessible to students familiar with basic concepts in mathematics who will surely profit from just being exposed to the constructions of Fraisse limits such as, for example, the Urysohn metric space or the Gurarij Banach space.
TOPICS COURSES
MAT1045HF
Topics in Ergodic Theory: Introduction to random walks on groups
This class will focus on properties of group actions from a probabilistic point of view, investigating the relations between the dynamics, measure theory and geometry of groups. The class is in preparation to the Winter 2024 program on "Randomness and Geometry" at the Fields Institute.
We will start with a brief introduction to ergodic theory, discussing measurable transformations and the basic ergodic theorems. Then we will approach random walks on matrix groups and lattices in Lie groups, following the work of Furstenberg. Topics of discussion will be: positivity of drift and Lyapunov exponents. Stationary measures. Geodesic tracking. Entropy of random walks. The Poisson-Furstenberg boundary. Applications to rigidity. We will then turn to a similar study of group actions which do not arise from homogeneous spaces, but which display some features of negatively curved spaces: for instance, hyperbolic groups (in the sense of Gromov) and groups acting on hyperbolic spaces. This will lead us to applications to geometric topology: in particular, to the study of mapping class groups and Out(F_N).
MAT1062HF
Topics in Partial Differential Equations I: Hyperbolic Differential Equations
New methods and results in hyperbolic PDEs will be presented including the stability of black holes, the long-time existence of quasi-linear equations, and the formation of singularities in finite time.
MAT1103HS
TOPICS IN ALGEBRA I: Algebraic Geometry and Smooth Topology
http://www.fields.utoronto.ca/activities/23-24/SGC-algebraic-geometry
A. Khovanskii
(View Timetable)
In the course I will discuss relations between algebra and geometry which are useful in both directions. Newton polyhedron is a geometric generalization of the degree of a polynomial.
Newton polyhedra connects the theory of convex polyhedra with algebraic geometry of toric varieties. For example, Dehn–Zommerville duality for simple polyhedra leads to the computation of the cohomology ring of smooth toric varieties. Riemann–Roch theorem for toric varieties provides valuable information on the number of integral points in convex polyhedra and unexpected multidimensional generalization of the classical Euler–Maclaurin summation formula. Newton polyhedra allow to compute many discrete invariants of generic complete intersections. Newton–Okounkov bodies connect the theory of convex bodies (not necessary polyhedra) with algebraic geometry. These bodies provide a simple proof of the classical Alexandrov–Fenchel inequality (generalizing the isoperimetric inequality) and suggest analogues of these inequality in algebraic geometry.
Tropical geometry and the theory of Gröbner bases relate piecewise linear geometry and geometry of lattice with algebraic geometry. All needed facts from algebraic geometry and convex geometry will be discussed in details during the course.
References
1. G.Kempf, F.Knudsen, D.Mamford, B.Saint-Donat. Toroidal Embeddings, Springer Lecture Notes 339, 1973. 2
2. K.Kaveh, A.Khovanskii. Newton–Okounkov bodies, semigroups of integral points, graded algebras and intersection theory, Annals of Mathematics, V. 176, No 2, 925–978, 2012.
3. Some extra papers will be handout during the course.
Grading: one final presentation or written report.
MAT1120HS
Lie Groups and Lie Algebras I
E. Meinrenken
(View Timetable)
The theory of Lie groups and Lie algebras is a classical and well-established subject of mathematics. The plan for this course is to give an introduction to the foundations of this theory, with emphasis on compact Lie groups and semi-simple Lie algebras.
Topics to be discussed include:
- The classical Lie groups
- Abstract Lie groups
- Lie algebra of a Lie group
- Actions of Lie groups and Lie algebras
- Exponential map
- Enveloping algebras, PBW theorem
- Semi-simple Lie algebras, relation with compact Lie groups
- maximal torus, Cartan subalgebras
- representations of sl(2,C)
- roots and weights
- Weyl group, chambers
- Coxeter-Dynkin diagrams, classification
- Finite-dimensional representations of semi-simple Lie algebras
- Weyl character formula
Prerequisites:
Manifolds, algebraic topology
MAT1128HS
TOPICS IN PROBABILITY: Random Matrix
This course will serve as an introduction to the universality theory of random matrices. No prior exposure to random matrix theory is required or assumed. The main topics that will be covered are:
- The local semicircle law for Wigner matrices and recent generalizations to more complicated matrix ensembles via the quadratic vector equation
- Local ergodicity of Dyson Brownian motion and its applications to local eigenvalue statistics
- The Tao-Vu four moment theorem
References:
Lectures on the local semicircle law for Wigner matrices", Benaych-Georges and Knowles; "A dynamical approach to random matrix theory" Erdos and Yau; "Topics in random matrix theory" by Tao-Vu.
MAT1191HF
TOPICS IN ALGEBRAIC GEOMETRY: Polytopes, Matroids, and Algebraic Geometry
H. Spink
(View Timetable)
TBD
MAT1191HS
TOPICS IN ALGEBRAIC GEOMETRY: Curves and Surfaces
This course serves as an introduction to the concrete aspects of algebraic geometry in low dimensions, loosely following Hartshorne Chapters IV and V (with additional outside topics). The goal is to serve as a transition from the technical foundations of algebraic geometry to questions closer to actual research in the field. Topics will include the geometry of canonical curves, automorphisms of curves, equations for curves of low genus, birational geometry of surfaces with a view towards the Enriques classification, as well as more advanced topics dictated by student interest, for example the geometry of the moduli space of curves.
I. Riemann-Roch
II. Equations for low-genus curves, automorphisms of curves
III. Jacobians, the Torelli theorem
IV. The canonical embedding
V. Curves in P^3
VI. Intersection theory on surfaces, Riemann-Roch for surfaces
VII. Rational and ruled surfaces, cubic surfaces
VIII. Enriques-Kodaira classification of surfaces and examples
IX. Hodge index theorem for surfaces and the Weil conjectures for curves
X. Advanced topicsending on interest of participants.
MAT1192HF
ADVANCED TOPICS IN ALGEBRAIC GEOMETRY: Derived Algebraic Geometry
N. Rozenblyum
(View Timetable)
Derived algebraic geometry is a generalization of algebraic geometry modeled on derived commutative rings, which introduces homotopy theory into the subject. This generalization is particularly well suited for studying singularities on moduli spaces and has come to play an important role in geometric representation theory and mathematical physics. This course will be an introduction to the subject with a focus of deformation theory and applications to moduli spaces. It will be mostly self-contained, assuming only some familiarity with algebraic geometry and homological algebra.
References:
Bertrand Toën, Derived algebraic geometry, EMS Surv. Math. Sci. 1 (2014), 153–240
J. Lurie, Spectral algebraic geometry
D. Gaitsgory and N. Rozenblyum, A study in derived algebraic geometry
MAT1192HS
ADVANCED TOPICS IN ALGEBRAIC GEOMETRY: Moduli Spaces in Algebraic Geometry
A key feature of algebraic geometry is that the spaces which parametrize objects of interest can themselves be given the structure of an algebraic scheme. This course will develop the machinery to make this idea precise. We shall precisely define what it is we want from a moduli space and then, following Grothendieck, proceed to costruct Hilbert Scheme. This will be our big hammer, which we shall use to construct lots of examples of specific moduli spaces: The Picard Scheme (moduli space of line bundles), (Stable) Curves, Abelian Varieties, (Stable) sheaves etc.. Depending on time and interest, we will explore how such a uniform definition and construction allows us to give applications in finite characteristic, and perhaps how we can use finite characteristic to prove characteristic 0 results.
MAT1210HF
TOPICS IN NUMBER THEORY: Class Field Theory
This course will be a continuation of MAT 415/1200. It will give an introduction to class field theory, the study of abelian extensions of number fields and p-adic fields, focusing on statements and examples such as the Kronecker-Weber Theorem. At the beginning, we will review inertia groups and decomposition groups, and we will use that foundation to introduce the Galois theory of local fields.
Main Textbook: Janusz - Algebraic Number Fields
MAT1305HF
TOPICS IN GEOMETRIC TOPOLOGY: An introduction to Teichmüller space and Mapping Class Group
We will discuss a number of aspects of the geometry of the Teichmüller space of Riemann surfaces: its asymptotic geometry, as seen through Thurston's compactification by measured laminations; its coarse geometry, as captured by the hyperbolicity properties of the complex of curves; and the dynamical properties of its geodesic flows. The goal is to survey what is known and explore some open problem and areas of potential research.
MAT1312HF
TOPICS IN GEOMETRY: Hamiltonian Group Actions and Equivariant Cohomology
Material:
1. Symplectic vector spaces
2. Hamiltonian group actions
3. The Darboux theorem
4. Elementary properties of moment maps
5. Coadjoint orbits
6. Symplectic reduction
7. Convexity
8. Toric manifolds
9. Equivariant cohomology
10. The Duistermaat-Heckman theorem
11. Geometric quantization
12. Flat connections on 2-manifolds
Text: S. Dwivedi, J. Herman, L. Jerey, and T. van den Hurk, Hamiltonian Group Actions and Equivariant Cohomology
MAT1312HS
TOPICS IN GEOMETRY: Meromorphic connections and the Stokes phenomenon
Meromorphic connections on Riemann surfaces appear in many parts of mathematics: they played a major role in the 19th century, in the form of ordinary differential equations with singularities in the complex domain. The Stokes phenomenon was discovered in this context, as an asymptotic discontinuity in the behaviour of solutions, such as the Airy functions in geometric optics or the Gauss confluent hypergeometric functions. One highlight of this work was the discovery in 1900 of the Painlevé transcendants and the statement of Hilbert’s 16th problem concerning the monodromy map.
In the 20th century, there was a focus on the appearance of these objects in completely integrable systems: the isomonodromic deformation theory provides a multitude of integrable systems which underlie a vast number of previously-known systems, such as reductions of the Yang-Mills equations. Most recently, meromorphic connections have appeared in two new contexts: first in understanding the behaviour of stability conditions on Abelian categories in algebraic geometry, and second in providing a new interpretation of the Feynman integral of quantum field theory.
In this course, we shall explore a small part of the vast literature on meromorphic connections, leading to an investigation of the most recent work of Bridgland-(Toledano-Laredo) and Witten which provides spectacular applications of the subject.
We hope to cover the following
-
- ODEs on Riemann surfaces
- Meromorphic connections
- The Riemann-Hilbert correspondence
- The Stokes Phenomenon and Stokes matrices, solution asymptotics
- The Stokes sheaf
- The Isomonodromic integrable system and Frobenius manifolds
- The Hitchin moduli space and the Geometric Langlands programme
This will be essentially a reading course + seminar for students. Evaluation will be based on general participation as well as a final project based on reading a paper in the field. Also, there is a possibility that a group of students may assist in the taking of notes for the class, as a substitute for the final project.
References for the course will include Wasow’s textbook on asymptotic methods in ODE, Deligne’s monograph on differential equations, Olver’s book on Asymptotics, the monograph by Zoladek, Deligne-Malgrange’s book on irregular connections, and Boalch’s thesis.
MAT1341HS
Topics in Differential Geometry
Y. Liokumovich
(View Timetable)
We will study manifolds with positive scalar curvature and related topics with an emphasis on minimal surface approach. This is currently a very active area of research attracting a broad range of mathematicians and physicists. The course will cover the Schoen-Yau proof of Geroch conjecture, mu-bubbles, classification results for manifolds with positive scalar curvature in low dimensions, synthetic definitions of scalar curvature, convergence results and counterexamples for manifolds with scalar curvature bounded from below.
MAT1347HS
TOPICS IN SYMPLECTIC GEOMETRY AND TOPOLOGY: Introduction to Contact Geometry
E. Murphy
(View Timetable)
Contact geometry is a type of geometry existing in all odd dimensions, closely tied to symplectic geometry, and with numerous relationships to 3-manifold theory, complex geometry, and mathematical physics. The first portion of the course will be on the basic geometric properties of general contact manifolds and symplectic fillings. From there we'll focus on contact 3-manifold theory, including Legendrian knots, overtwistedness and the Thurston-Bennequin inequality, open book decompositions. The final part of the course (as time and popular opinion demand) will discuss Legendrian contact homology, confoliation theory, high-dimensional contact geometry, or other topics.
MAT1351HF
Topics in Homotopy Theory: Seiberg–Witten Theory and Equivariant Stable Homotopy Theory
This course is about two seemingly disparate topics: gauge-theoretic invariants of smooth 4-manifolds and equivariant stable homotopy theory. They meet in Bauer and Furuta’s construction of the Seiberg–Witten invariants. This construction reformulates the invariants as a homotopy class, in fact, an equivariant one between spectra with Pin(2)-actions. This homotopy-theoretic reformulation streamlines the analysis and allows one to use the computational tools of equivariant stable homotopy theory to prove many surprising results about smooth 4-manifolds. It also seems to a good setting for an extension to families of 4-manifolds, and 4-manifolds with boundary. The intention is to cover the following topics:
- Basic theory and examples of smooth 4-manifolds.
- The Bauer–Furuta construction of the Seiberg–Witten invariants.
- Equivariant stable homotopy theory.
- Furuta’s 10/8 theorem and Hopkins–Lin–Shi–Xu’s improvement to 10/8 + 4.
- Versions for families and boundary, if time permits.
We will not require any background not covered in the core courses. That is, we will attempt to explain the algebraic topology so that students with a background in analysis can follow, and explain the analysis so that students with a background in algebraic topology can follow.
MAT1360HS
Complex Manifolds: Abelian Varieties and Theta Functions
Abelian varieties are higher dimensional generalizations of elliptic curves. They have a rich and fascinating geometry and play a fundamental role in many of the most important recent developments in number theory.
This course will provide a basic introduction to their theory followed by a sketch of more advanced aspects.
We will mostly follow the classic treatment in Mumford, Abelian varieties. However, we will work mostly over C and will discuss additional topics from the analytic theory of theta functions and Siegel modular forms.
Topics may include:
- complex tori
- line bundles on complex tori
- theta functions
- algebraizability, Appel-Humbert theorem
- Siegel space and Ag
- abelian varieties
- cohomology and base change
- the theorem of the cube
- the dual abelian variety
- the Poincare bundle
- polarizations
- structure of End(A), Rosati involution
- moduli spaces
- Siegel modular forms
- complex multiplication
- Shimura varieties of PEL type
Prerequisites:
Basic algebraic geometry, e.g. the first chapter of Hartshorne. Basic differential geometry, e.g., differential forms, complex manifolds, deRham cohomology.
Familiarity with sheaves and sheaf cohomology would be helpful.
Textbooks:
- Mumford, Abelian Varieties, AMS 2012. ISBN-13: 978-8185931869
- C. Birkenhake and H. Lange, Complex abelian varieties, second edition,
- Grundlehren der math. Wiss. 302, Springer 2004. ISBN-13: 978-3540204886
MAT1435HF
TOPICS IN SET THEORY: Measurable combinatorics and local algorithms
Measurable combinatorics is the study of combinatorial problems on spaces with additional measure theoretic or topological restrictions. These problems appear naturally in ergodic theory, probability and operator algebras among other areas. In this course, we will explore a recent connection between measurable combinatorics and variations of the LOCAL model of computation of Linial following work of Bernshteyn and others.
MAT1435HS
TOPICS IN SET THEORY: Forcing and its Applications
S. Todorcevic
(View Timetable)
TBD
MAT1502HS
TOPICS IN GEOMETRIC ANALYSIS: Algebraic and Geometric Aspects of Kahler-Einstein Manifolds
T. Collins
(View Timetable)
This course will discuss aspects of Kahler-Einstein metrics and their connections to algebraic geometry. Roughly speaking, the theme of the course is to investigate the relationship between differential geometry and algebraic geometry in Kahler geometry. The course will begin with some background material in complex algebraic geometry and Riemannian geometry. We will discuss Yau's proof of the existence of Kahler-Einstein metrics with negative, or zero Ricci curvature. We will then focus on how these manifolds can "degenerate". We will discuss Donaldson-Sun's proof that geometric limits of polarized Kahler-Einstein manifolds (that is "Gromov-Hausdorff" limits) can be identified with algebraic varieties, and explain how this result implies the uniqueness of tangent cones for these manifolds. Possible further topics include aspects of collapsing degenerations, connections to real Monge Ampere equations and mirror symmetry, and complete, non-compact Calabi-Yau manifolds.
MAT1510HF
DEEP LEARNING: THEORY & DATA SCIENCE
Deep learning systems have revolutionized field after another, leading to unprecedented empirical performance. Yet, their intricate structure led most practitioners and researchers to regard them as blackboxes, with little that could be understood. In this course, we will review experimental and theoretical works aiming to improve our understanding of modern deep learning systems.
Link to course website: https://sites.google.com/view/mat1510
Prerequisite:
Undergraduate Linear Algebra
MAT1739HF
Topics in Mathematical Physics: Introduction to Waves and Black Holes
Y. Shlapentokh-Rothman
(View Timetable)
The goal of this course is to provide an introduction to the mathematical theory of General Relativity through the lens of the study of wave propagation on black holes. In particular, we will introduce the Schwarzschild black hole solution and then study the various techniques developed over the last 20 years for establishing uniform boundedness and decay statements for solutions to the corresponding wave equation. Time-permitting we may also discuss either applications to nonlinear problems or the case of other black hole solutions such as the Kerr spacetime.
No prior knowledge of PDE's or General Relativity will be necessary!
Prerequisites:
Comfort with elementary aspects of Riemannian geometry will be assumed.
MAT1845HF
Dynamical Systems: Renormalization in one-dimensional dynamics: an introduction
Renormalization has become a central philosophy in low-dimensional dynamics. Among its many applications are Kolmogorov-Arnold-Moser (KAM) theory, interval exchange transformations (where it is known as Rauzy-Veech induction), Feigenbaum-type universality, and even fluid dynamics. This course will serve as an introduction to renormalization magic, using the key example of analytic homeomorphisms of the circle. The prerequisites are a standard course in complex analysis, and some basic familiarity with the notions of discrete-time dynamics.
Students requiring individual instruction in mathematical topics should consult with the Mathematics Graduate Office.