Host
Hugh Kuhn
Podcast Content
Analytic number theory uses tools of analysis to study the integers, and is usually concerned with questions of the asymptotics and distributions of arithmetic data, such as prime numbers, class groups of number fields, discriminants of number fields, and so forth. A third branch of number theory is arithmetic geometry, which is concerned with finding solutions of polynomial equations on fields which are not algebraically closed, like rational numbers. Arithmetic geometry is closely related to algebraic geometry, which is the study of geometric objects defined by polynomial equations. There is an emphasis on arithmetic geometry, including the dynamics of arithmetic, elliptic curves, and higher-genus curves.
They encompass arithmetic as well as classical algebraic geometry; theory of automorphic, geometric, and p-adic representations; the Shimura varieties and Galois representations; the Hodge theory of p-adic representations; harmonic analysis and analytic number theory; stability of representations and commutative algebra; and algorithmic and computational number theory. Commutative algebra, taught in MAT 447, is the fundamental tool for learning algebraic number theory and algebraic geometry, and it is good practice to learn commutative algebra before doing too much algebraic geometry . There are many algebraic structures relevant to studying algebraic integers, and especially to the number fields; these problems are part and parcel of algebraic number theory, including its central piece, the field theory of classes. Number theory is the branch of mathematics dedicated mainly to the study of the integers, their additive and multiplicative structures, and the properties which distinguish them from other rings .
Problems in number theory are usually better understood by studying analytical objects which in some way encode properties of integers, primes, or other number-theoretic objects . The Riemann hypothesis, a problem from the Clay millennium, is a piece of analytic number theory, that uses analytic methods to make sense of integers. The Riemann hypothesis requires properties about the Riemann zeta function, a function that plays a critical role in the distribution of the prime numbers. Analytic number theory has given further refinement to Greek mathematician Euclid, by determining a function which measures how tightly distributed the primes are across all integers.
Many mathematicians, including Mersenne and Euler, tried to come up with a formula that would determine all prime numbers. Some of these formulas are quite complex, but the best known ones are quite simple, such as Fermats theorem below, which shows whether or not a number is prime. Using one of the many Fermat theorems, a computer can rapidly calculate whether a number -- even a big one -- is prime.
The hard analysis is particularly significant, and indeed most of the latter half of MAT 335 is concerned with the proof of the Prime Number Theorem, which was a seminal work in analytical number theory. Typical problems are distributions of prime numbers, and studies of integer solutions to algebraic equations with integer coefficients, also called Diophantine equations.
Number theory is currently a coalescing field; it encapsulates additive number theory and, one might argue, part of the geometry of numbers, along with some fast-developing new materials. This theory has been developed for higher-rank reductive groups, and has a variety of powerful applications to the understanding of the connections between l-functions and Galois representations, which are central to contemporary studies of algebraic number theory and arithmetic geometry.
Wiles proof of his moduli-lifting theorem is an ideal illustration of the techniques of p-adics in number theory, with deformations of Galois representations, consistency among moduli forms, and their profound connections to the special values of L-functions. In essence, his congruence theory allows humans to decompose an infinite set of integers into smaller, manageable pieces and to do calculus on small ones. The importance of Gausss theory of congruence is that it created a formula which allows for a huge number of arithmetic operations based on various sets of numbers.
Fermat thought that if one were to take the sum of two squared, then take the sum of two squared up to some higher power, which he called n , then the result would be merely the primes. Despite such isolated results, there was no general theory of numbers. Very early in civilisation, humans grasped the idea of multiplicity by, and thus took their first steps towards a study of numbers.
For instance, as explained later, the topic of algorithms in number theory is quite old, older, in a way, than the notion of a proof; meanwhile, the modern study of computability dates from the 1930s and the 1940s, and that of computational complexity only from the 1970s. The multi-Dichlet series theory introduced in 1980s is now emerging as a major tool in getting estimates for the abrupt growths in Z-functions and L-functions, a major classical number-theoretical problem, which has applications in algebraic geometry. While Babylonian number theory--or what remains of Babylonian mathematics to call it so--consists of this one, glaring fragment, Babylonian algebra is extraordinarily well developed. The Langlands Program, one of mathematics major ongoing programs on a grand scale, is sometimes described as attempting to generalize the field theory of classes over the non-Abelian extensions of the field of numbers.