Curves, Surfaces, and Applied Algebraic Geometry

Abstract: Applied Algebraic Geometry (AG) is an inclusive community involving industry, scientists, and mathematicians. It combines centuries worth of AG ideas with modern computing power to drive research into applications from new angles. The success and growth in this area comes from the fact that many problems can be restated in terms of solving systems of polynomial equations. The solution sets to these equations are algebraic sets, also known as algebraic varieties. For instance, algebraic curves like the parabola are one dimensional varieties, while finite sets of points are zero dimensional.

In this talk I will (1) present motivating examples of polynomial systems appearing in applications and (2) illustrate the core ideas of methods to describe curves. As an added bonus, and if time permits, I will present a case study on finding the nearest point to an algebraic curve. This will involve the Euclidean distance degree, Euler characteristics, and a few results that are found in this recent survey arxiv.org/abs/2305.19842 .

No prior background knowledge in algebraic geometry will be assumed.

commutative algebraalgebraic geometryalgebraic topologygeneral 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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