SPECIAL SEMINAR
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”
ABSTRACT
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
