# Number Theory

The research of the number theory group encompasses classical and algebraic number theory, computational number theory, and especially the modern subject of arithmetic geometry.

Arithmetic geometry is the study of number-theoretic problems informed by the insights of geometry—among them algebraic geometry, topology, differential geometry, and discrete geometries related to graph theory and group theory. Part of the appeal of the subject comes from its combination of easily stated problems (such as: Which integers are the sum of two squares or two cubes? How can one explicitly find all the solutions of an explicit polynomial equation in two variables? Are there positive integers (a,b,c) satisfying a^n+b^n=c^n when n>2?) and breathtaking applications of the latest technology from the mathematical frontiers. Another attraction is the phenomenal progress that has been made in the late 20th and early 21st century, including proofs of such landmark results as the Weil conjectures, the Mordell conjecture, and of course Fermat's last theorem. The number of Fields Medals (the mathematical equivalent of the Nobel prize) awarded for work in the area is a testament to its centrality in modern fundamental mathematics. On the other hand, recent applications of arithmetic geometry to communications protocols (such as strong cryptography and error correcting information transmission schemes) have brought the tools of the field to bear on crucial problems of Internet technology.

Two of the Millenium Prize Problems in mathematics, offered by the Clay Mathematics Institute, are in the area of number theory and one more is closely related to number theory.

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## Members

## Bryden R Cais

## Kirti N Joshi

## Daniel Madden

## William G Mccallum

Professor, Mathematics

## William Yslas Velez

Professor, Mathematics