## PIMS Calgary CRG Launch 2010

- Date: 04/06/2010

##### Expanders, Group Theory, Arithmetic Geometry, Cryptography and much much more....

Dr. Eyal Goren,McGill University

###### Abstract:

The theory of expander graphs is undergoing intensive development. It finds more and more applications to diverse areas of mathematics. In this talk, aimed at a general audience, I will introduce the concept of expander graphs and discuss some interesting connections to arithmetic geometry, group theory and cryptography, including some very recent breakthroughs.

###### Biography:

Eyal Goren is a professor of mathematics at McGill University, specializing in arithmetic geometry. He did his Ph.D. in the Hebrew university in Jerusalem, graduating in January 1997, under the supervision of E. De Shalit. After a postdoctoral period at Harvard University and Utrecht University, he arrived as a postdoctoral fellow at McGill University and became an assistant professor there in June 1999, and a full professor in June 2009. His research interests include moduli spaces of abelian varieties and curves, modular forms, complex multiplication and expander graphs.

##### Quantum magic in secret communication

Dr. Gilles Brassard, Université de Montréal

###### Abstract:

Quantum Cryptography is the first near-term practical application of the emerging field of quantum information processing. It allows two parties who share only a short prior secret to exchange messages with provably perfect confidentiality under the nose of an eavesdropper whose computational power is unlimited and whose technology is restricted only by the accepted laws of physics. In this talk, we shall tell the tale of the origin of Quantum Cryptography from the birth of the first idea by Wiesner in 1970 to the invention of Quantum Key Distribution in 1983, to the first prototypes and ensuing commercial ventures, to exciting prospects for the future. No prior knowledge in quantum mechanics or cryptography will be expected.

###### Biography:

Professor of computer science since 1979 and Canada Research Chair at the Université de Montréal, Gilles Brassard laid the foundations of quantum cryptography at a time when only a handful of people worldwide were interested in quantum information processing. Among his main other achievements are the invention of privacy amplification, quantum teleportation, quantum entanglement distillation and amplitude amplification. Editor-in-Chief for Journal of Cryptology from 1991 until 1997, he is the author of three books that have been translated into eight languages. He is a Fellow of the Royal Society of Canada (Academy of Science), of the Canadian Institute for Advanced Research and of the International Association for Cryptologic Research. Among his many awards, we note the E.W.R. Steacie Memorial Fellowship, the Killam Research Fellowship, the Prix Marie-Victorin, the Rank Prize in Opto-Electronics and the NSERC Award of Excellence.

University of Calgary

###### 6 April 2010

Luncheon for CRG Participants - 12:30-14:30

Evans Room, Rozsa Centre, UC

Public Lectures - 15:00-16:30

15:00:

###### Expanders, Group Theory, Arithmetic Geometry, Cryptography and much

much more...., Eyal Goran

15:45:

###### Quantum magic in secret communication, Gilles Brassard

Husky Oil Great Hall, Rozsa Centre, UC

Reception - 16:30-17:30

Evans Room, Rozsa Centre, UC

University of Calgary

###### 7 April 2010, 10:00

###### Special Number Theory Lecture - 10:00 (MS431)

###### Class invariants in genus 2.

Speaker: Eyal Goren

Abstract: The main technique to construct class invariants, namely a set of distinguished elements in class fields of a given CM field, is by evaluating modular functions at CM points corresponding to that field. The case of quadratic imaginary fields is classic and well understood;it draws on the existence of modular units and good reduction for CM elliptic curves. The case of quartic fields is much less understood, but has recently seen some new results. Similar to the case of elliptic curves, the reduction of abelian surfaces with CM, and special divisors on the moduli space, play a key role. In this talk, I will report on recent work with Kristin Lauter that provides information about the prime factorization of certain class invariants (first studied by the speaker with DeShalit) and bounds the denominator of the so-called absolute Igusa invariants of genus 2 curves with CM. As will be explained, this last fact has applications to cryptography.

Mathematics

of Quantum Information CRG

Barry Sanders

Gilad Gour

L-Functions

and Number Theory CRG

Matthew Greenberg

Clifton Cunningham