Sep

14

Please note that Ragnar Buchweitz's course on Symmetries I: Finite Groups, MAT 1103HF, has shifted to Thursdays 11-2 from the original time slot of Thursdays 10-1. The room is the same: HU 1018, 215 Huron St.

Sep

14

Sep

13

Please be advised that Lisa Jeffrey's graduate course, MAT 1312H - Hamiltonian Group Actions, has been moved to the second semester. Scheduling will be decided later.

Aug

26

Our new DCS faculty member Vinod Vaikuntanathan will be teaching the following grad course this Fall. It should be of interest to people who want to learn about cryptography and/or the combinatorics of lattices. COURSE NUMBER: CSC 2414F (Fall 2011) ** NEW TIME: Tuesday 3-5 (room BA B025) ** COURSE TITLE: Topics in Applied Discrete Math: Lattices in Computer Science Course Description: Integer lattices are powerful mathematical objects that have found applications in many diverse facets of computer science, most notably in the areas of cryptography and combinatorial optimization. This course gives an introduction to the theory of integer lattices -- their algorithms and applications to combinatorial optimization, their recent use in cryptography culminating in the first construction of a fully homomorphic encryption scheme, and the fascinating complexity landscape associated with lattice problems. This course will touch several related areas and applications: â€¢ Algorithms and Combinatorial Optimization (1/4): The asymptotically fastest Integer Programming algorithm known to date is based on lattices. We will study lattice algorithms and their applications to combinatorial optimization. â€¢ Cryptography (3/4): Lattices have proven themselves to be a double-edged sword in cryptography. While they were first used to break cryptosystems, they have more recently been instrumental in designing a wide range of secure cryptographic primitives, including public key encryption, digital signatures, encryption resistant to key leakage attacks, identity based encryption, and most notably, the first fully homomorphic encryption scheme. Prerequisites: We will assume knowledge of basic math (linear algebra and probability) and introductory level algorithms (analysis of algorithms, polynomial time and NP-hardness). We will NOT assume any prior knowledge of cryptography or advanced complexity theory.

Aug

23

MAT 1195HF ELLIPTIC CURVES AND CRYPTOGRAPHY: MATHEMATICAL ASPECTS OF CRYPTOGRAPHY R. Venkatesan Mondays and Tuesdays, 10-11:30 a.m., in HU 1018, 215 Huron St. We will study a number of papers related to design, algorithms and security analysis of cryptographic primitives based on hard problems in number theory, elliptic curves, and other domains such as codes and lattices. Dixons algorithm, Number field sieve, Pollard Rho, Bit security of some primitives. Attacks on Knapsacks and RSA variants, Authentication protocols and use of Zero-Knowledge primitives, Schemes for cloud scenarios. Brief look at complexity issues and the construction of hash functions, MACS, and Ciphers, and attacks on them. Prerequisites: Students should have some introduction to number theory, and elliptic curves. Useful references: http://www.amazon.com/Introduction-Modern-Cryptography-Principles-Protocols/dp/1584885513/ref=sr_1_1?ie=UTF8&qid=1314023370&sr=8-1 http://www.amazon.com/Elliptic-Curves-Cryptography-Mathematics-Applications/dp/1420071467/ref=sr_1_3?s=books&ie=UTF8&qid=1314023661&sr=1-3 http://www.amazon.com/Introduction-Cryptography-Discrete-Mathematics-Applications/dp/1584886188/ref=sr_1_3?s=books&ie=UTF8&qid=1314023761&sr=1-3#_ --------------------------------------------------------------

Aug

10

MAT 1011HF (MAT495H1F) INTRODUCTION TO LINEAR OPERATORS G. A. Elliott MWF 2 in HU 1018, 215 Huron St. **Topics (and cross-listed)****: **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). *Prerequisites: * Elementary analysis and linear algebra (including the spectral theorem for self-adjoint matrices). *Textbook: * Gert K. Pedersen, Analysis Now *Recommended references: * Paul R. Halmos, A Hilbert Space Problem Book Mikael Rordam, Flemming Larsen, and Niels J. Laustsen, An Introduction to K-Theory for C*-Algebras Bruce E. Blackadar, Operator Algebras: Theory of C*-Algebras and von Neumann Algebras *MAT1016HS TOPICS IN OPERATOR ALGEBRAS: K-THEORY AND C*-ALGEBRAS G. A. Elliott MWF 2 in Room 1018, 215 Huron St. 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, this theory became increasingly important in other branches 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 (undergraduate students are welcome) will be encouraged to prepare oral or written reports on various subjects related to the course, including both 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

Apr

08

The absolute deadline to add summer reading/research courses and/or graduate course from other departments to your program is Friday, May 13th For the master's students who have not yet enrolled in their supervised research project, I will add the course MAT 4000YY on ROSI. What still is required is the completion of the following two forms: 1. add/drop form http://www.sgs.utoronto.ca/Assets/SGS+Digital+Assets/current/Student+Forms/Add-Drop+Course.pdf please note: enrolment in courses from other departments requires the other department's approval. 2. reading and/or research course form http://www.sgs.utoronto.ca/Assets/SGS+Digital+Assets/current/Student+Forms/Reading_and_Research.pdf Hard copies of the forms above are available in the mailroom. The second form requires the signature of the course/project supervisor. This form will also provide the subtitle info which I will then add to your course listing on ROSI. If you are having any difficulty securing a supervisor for the master's project or a reading course, please let me know.

Mar

02

Jeremy Quastel will be teaching a new graduate course "Kardar-Parisi-Zhang equation from particle systems" Mondays 1-3 in Rm 210, Fields Institute. Starting Mar 7. The course is a part of the activities related to Fields Institute Thematic Program "Dynamics and Transport in Disordered Systems".