Number Theory

The power and weakness of randomness (when you are short on time)

Speaker: 
Avi Wigderson
Date: 
Fri, Mar 8, 2013
Location: 
Department of Mathematics, UBC
Conference: 
PIMS/UBC Distinguished Colloquium
Abstract: 
Avi Wigderson is a widely recognized authority in theoretical computer science. His main research area is computational complexity theory. This field studies the power and limits of efficient computation and is motivated by such fundamental scientific problems as: Does P=NP? Can every efficient process be efficiently reversed? Can randomness enhance efficient computation? Can quantum mechanics enhance efficient computation? He has received, among other awards, both the Nevanlinna Prize and the Gödel Prize.

Cryptography: Secrets and Lies, Knowledge and Trust

Speaker: 
Avi Wigderson
Date: 
Thu, Mar 7, 2013
Location: 
PIMS, University of British Columbia
Conference: 
PIMS Public Lecture
Abstract: 
What protects your computer password when you log on, or your credit card number when you shop on-line, from hackers listening on the communication lines? Can two people who never met create a secret language in the presence of others, which no one but them can understand? Is it possible for a group of people to play a (card-less) game of Poker on the telephone, without anyone being able to cheat? Can you convince others that you can solve a tough math (or SudoKu) puzzle, without giving them the slightest hint of your solution?These questions (and their remarkable answers) are in the realm of modern cryptography. In this talk I plan to survey some of the mathematical and computational ideas, definitions and assumptions which underlie privacy and security of the Internet and electronic commerce. We shall see how these lead to solutions of the questions above and many others. I will also explain the fragility of the current foundations of modern cryptography, and the need for stronger ones.No special background will be assumed.

Numbers and Shapes

Speaker: 
Henri Darmon
Date: 
Thu, Nov 1, 2012
Location: 
PIMS, University of Calgary
Conference: 
Hugh C. Morris Lecture
Abstract: 
Number theory is concerned with Diophantine equations and their solutions, encoded in discrete structures involving integers, rational numbers or algebraic quantities. Topology studies the properties of shapes that are unchanged under continuous or smooth deformations, a technique of choice being the construction of appropriate homological invariants. It turns out--perhaps surprisingly to the uninitiated--that these invariants can be endowed with sufficient structure to capture a tremendous amount of arithmetic information. The powerful interplay between arithmetic and topological ideas underlies the most important breakthroughs in the study of Diophantine equations, such as Faltings’ proof of the Mordell Conjecture and Wiles’ proof of Fermat’s Last Theorem. It is also at the heart of more recent and still very fragmentary attempts to construct algebraic points on elliptic curves when their existence is predicted by the Birch and Swinnerton-Dyer conjecture. This lecture will attempt to give a non-technical sampler of some of the rich, fascinating interactions between arithmetic questions and topological insights.

Ranks of elliptic curves

Speaker: 
Brian Conrey
Date: 
Wed, Jun 1, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 
We show how to use conjectures for moments of L-functions to get insight into the frequency of rank 2 elliptic curves within a family of quadratic twists.

Moments of zeta and L-functions on the critical Line II (3 of 3)

Speaker: 
K. Soundararajan
Date: 
Fri, Jun 3, 2011
Location: 
University of Calgary, Calgary, Canada
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

I will discuss techniques to get upper and lower bounds for moments of zeta and L-functions. The lower bounds are unconditional and the upper bounds in general rely on the Riemann Hypothesis. In several cases of low moments, one can obtain asymptotics, and I may discuss a couple of such recent cases.

This lecture is part of a series of 3
  1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
  2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
  3. Lecture 3: Moments of zeta and L-functions on the critical line, II

Moments of zeta and L-functions on the critical Line I (2 of 3)

Speaker: 
K. Soundararajan
Date: 
Thu, Jun 2, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

I will discuss techniques to get upper and lower bounds for moments of zeta and L-functions. The lower bounds are unconditional and the upper bounds in general rely on the Riemann Hypothesis. In several cases of low moments, one can obtain asymptotics, and I may discuss a couple of such recent cases.

This lecture is part of a series of 3
  1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
  2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
  3. Lecture 3: Moments of zeta and L-functions on the critical line, II

Distribution of Values of zeta and L-functions (1 of 3)

Speaker: 
K. Soundararajan
Date: 
Thu, Jun 2, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

I will discuss the distribution of values of zeta and L-functions when restricted to the right of the critical line. Here the values are well understood by probabilistic models involving “random Euler products”. This fails on the critical line, and the L-values here have a different flavor here with Selberg’s theorem on log normality being a representative result.

This lecture is part of a series of 3
  1. Lecture 1: distribution-values-zeta-and-l-functions-1-3
  2. Lecture 2: Moments of zeta and L-functions on the Critical Line, I
  3. Lecture 3: Moments of zeta and L-functions on the critical line, II

Special values of Artin L-series (3 of 3)

Speaker: 
Ram Murty
Date: 
Wed, Jun 1, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

Dirichlet’s class number formula has a nice conjectural generalization in the form of Stark’s conjectures. These conjectures pertain to the value of Artin L-series at s = 1. However, the special values at other integer points also are interesting and in this context, there is a famous conjecture of Zagier. We will give a brief outline of this and display some recent results.

This lecture is part of a series of 3.

Artin’s holomorphy conjecture and recent progress (2 of 3)

Speaker: 
Ram Murty
Date: 
Tue, May 31, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

Artin conjectured that each of his non-abelian L-series extends to an entire function if the associated Galois representation is nontrivial and irreducible. We will discuss the status of this conjecture and discuss briefly its relation to the Langlands program.

This lecture is part of a series of 3.

Introduction to Artin L-series (1 of 3)

Speaker: 
Ram Murty
Date: 
Mon, May 30, 2011
Location: 
PIMS, University of Calgary
Conference: 
Analytic Aspects of L-functions and Applications to Number Theory
CRG: 
L-functions and Number Theory (2010-2013)
Abstract: 

After defining Artin L-series, we will discuss the Chebotarev density theorem and its applications.

This lecture is part of a series of 3.
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