New time for Symmetries I – Finite Groups course

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.
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MAT 1312H – Hamiltonian Group Actions

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.
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Recently announced CSC course has a new time

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:

• 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.

• 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.

New Graduate Course update

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#_

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Updated descriptions for George Elliott’s Graduate Courses

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
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.

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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".
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