Apollonian Circle Packings, Integers, and Higher-Dimensional Sphere Packings

Abstract: Apollonian circle packings are certain circle configurations in which the circles do not intersect but may be tangent to each other. What do Apollonian circle packings have to do with integers? Under certain conditions, each circle in an Apollonian circle packing has a bend (1/radius) that is an integer! When all the bends are integral, which integers appear as bends? It turns out that this is a hard question to answer. However, it is easier to answer a similar question for analogous higher-dimensional sphere packings.

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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