Counting problems in number theory

Abstract: Many questions in number theory can be phrased as counting problems. How many primes are there? How many integral solutions to this system of equations are there? How many number fields are there? If the answer is “infinitely many,” we want to understand the order of growth for the “family" of objects we are counting. But in many settings we are also interested in finer-grained questions that zoom in to focus on just one part of the family. For example: how many number fields are there, with fixed degree and fixed discriminant? We know the answer is “finitely many,” but it would have important consequences if we could show the answer is always “very few indeed.” In this talk, we will describe several “counting problems” that remain mysterious, and explore how one way to prove finer-grained properties is by understanding the behavior of infinite families of mathematical objects. This talk will be radically accessible.

commutative algebraalgebraic geometrygeneral mathematicsnumber theoryrings and algebras

Audience: undergraduates

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PRiME: Pomona Research in Mathematics Experience

Series comments: PRiME is hosting a series of talks which will take place on Fridays in July. All are welcome to join us either in person at Pomona College or virtually over Zoom. There will be two types of series.

PROFESSIONAL DEVELOPMENT WORKSHOPS

We will have a series of 2-hour workshops geared for undergraduate students, graduate students, and faculty. The Morning Sessions will take place from 10:00 AM - 12:00 PM Pacific, while the Afternoon Sessions will take place from 2:00 PM - 4:00 PM Pacific. The Student Professional Development Series (geared for undergraduates) will meet at Pomona College in Estella 1051 (Argue Auditorium); while the Faculty Professional Development Series (geared for graduate students and junior faculty) will meet in Estella 1021 (Noether Auditorium).

COLLOQUIUM SERIES

We will have outside speakers to visit with us on Fridays from 4:30 PM - 6:00 PM Pacific. We will meet in person at Pomona College in Estella 1051 (Argue Auditorium).

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