# Number Theory

## Undecidability in Number Theory

Hilbert’s Tenth Problem asked for an algorithm that, given a multivariable polynomial equation with integer coefficients, would decide whether there exists a solution in integers. Around 1970, Matiyasevich, building on earlier work of Davis, Putnam, and Robinson, showed that no such algorithm exists. However, the answer to the analogous question with integers replaced by rational numbers is still unknown, and there is not even agreement among experts as to what the answer should be.

## The rank of elliptic curves

After quadratic equations in two variables come cubic equations, or elliptic curves. The set of rational points on an elliptic curve has the structure of a finitely generated abelian group. I will recall the basic theory of elliptic curves, then discuss the conjecture of Birch and Swinnerton-Dyer, which attempts to predict the rank of the group of rational points from the number of solutions (mod p) for all primes p. I will also discuss some recent results on the average rank, due to Manjul Bhargava and his collaborators. (PIMS-UBC Distinguished Colloquium)

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## Native American Mathematics

One sometimes hears that the indigenous peoples of the Americas are for some reason not predisposed to be able to do mathematics. This belief is surprising, since the mathematical traditions of the Western Hemisphere prior to European contact were already rich and extensive. This talk will focus on some of those traditions, primarily Central American but with some information about mathematical traditions in Algonkian cultures such as the Blackfoot. Almost all of this talk will be accessible to any interested listener, with perhaps five minutes in the middle using a small amount of very elementary number theory. Along the way any listener who has ever eaten an 18 Rabbits granola bar will learn why doing so celebrates indigenous mathematics.

**ABOUT THE RICHARD AND LOUISE GUY LECTURE SERIES:**

The Richard & Louise Guy lecture series celebrates the joy of discovery and wonder in mathematics for everyone. Indeed, the lecture series was a 90th birthday present from Louise Guy to Richard in recognition of his love of mathematics and his desire to share his passion with the world. Richard Guy is the author of over 100 publications including works in combinatorial game theory, number theory and graph theory. He strives to make mathematics accessible to all. The other contributions to the lecture series have been made by Elwyn Berlekamp (2006), John Conway (2007), Richard Nowakowski (2008), William Pulleyblank (2009), Erik Demaine (2010), Noam Elkies (2011), Ravi Vakil (2012) and Carl Pomerance (2013).

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

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

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

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.

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## Ranks of elliptic curves

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.

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

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

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

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

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

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