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“Understanding Number Fields Through the Distributions of their Arithmetic Invariants”


Come join us Friday January 19, 2018 at 12 pm in RM 507, Professor Ila Varma (Columbia University) will be giving a special lecture about “Understanding Number Fields Through the Distributions of their Arithmetic Invariants”


The most fundamental objects in number theory are number fields, field

extensions of the rational numbers that are finite dimensional as

vector spaces over Q. Their arithmetic is governed heavily by

certain invariants such as the discriminant, Artin conductors, and the

class group; for example, the ring of integers inside a number field

has unique prime factorization if and only if its class group is

trivial. The behavior of these invariants is truly mysterious: it is

not known how many number fields there are having a given discriminant

or conductor, and it is an open conjecture dating back to Gauss as to

how many quadratic fields have trivial class group.


Nonetheless, one may hope for statistical information regarding these

invariants of number fields, the most basic such question being “How

are such invariants distributed amongst number fields of degree d?”

To obtain more refined asymptotics, one may fix the Galois structure

of the number fields in question. There are many foundational

conjectures that predict the statistical behavior of these invariants

in such families; however, only a handful of unconditional results are

known. In this talk, I will describe a combination of algebraic,

analytic, and geometric methods to prove many new instances of these

conjectures, including some joint results with Altug, Bhargava, Ho,

Shankar, and Wilson.


Friday, January 19, 2018 at noon

Room 507, Mathematics Hall

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