## Student Number Theory Seminar 2019-20

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9/9/19: Andy Hardt -- * What is Number Theory?*

**Abstract:** Number Theory is infamous for being hard to pin down. The subject is very broad, and different areas can seem totally unrelated. Perhaps a simple slogan could be: we find integer solutions to algebraic equations by relating geometry to analysis. In this talk, we'll start to get a handle on what that means.

This talk will discuss the broad contours of number theory, and will double as an introduction for those who are new to the area. We’ll mention, very roughly and certainly without proofs: Diophantine equations, elliptic curves, L-functions, automorphic forms, automorphic/p-adic representations, and the Langlands program. The idea is not to gain any deep understanding, but to say a bunch of words, put them into context vis-a-vis other words, and set the stage for later talks.

**Helpful links**

Elliptic Curves and Congruent Numbers

Elliptic Curves

Congruent Numbers

Elliptic Curves, and a lot else

Modular/Automorphic Forms

Counting Primes

Modular/Automorphic Forms

Why are automorphic forms important?

Significance of Tate's Thesis

Tate's Thesis

Langlands Program