Bea Araiza

Thesis Advisor: Dr. Dagan Karp (Harvey Mudd College - Department of Mathematics)

Second Reader: Dr. Edray Goins (Pomona College - Department of Mathematics)

Contact: baraiza@hmc.edu

Exploring Tropical Linear Series 

Abstract: In this thesis, I dive into the connections between tropical geometry and traditional algebraic geometry, focusing specifically on tropical linear series. I start by laying down the basics of the Riemann-Roch theorem and explore canonical divisors and how they relate to meromorphic forms, showing how the degree of a divisor plays a crucial role in determining the properties of rational functions.

 I introduce tropical numbers and define tropical polynomials and monomials, setting the stage for understanding complete linear series on tropical curves. By translating classical ideas into a tropical context, I hope to shed light on the geometry of curves and how rational functions behave when we use tropical methods. This work not only expands the ideas of algebraic geometry but also showcases the power of tropical approaches.