Faculty Affairs and Academic Programs
Hannah Larson
Physical Sciences; Math
Mathematics
UC Berkeley
New Invariants in the Classification of Curves
The main goal of algebraic geometry is to classify shapes defined by polynomial equations. The first case of 1-dimensional shapes, known as algebraic curves, is central to progress not only in algebraic geometry but also in ground-breaking developments such as string theory and the most secure cryptography systems. The proposed research program tackles previously out-of-reach questions about algebraic curves. Because of their fundamental nature, such questions are likely to have far-reaching impact, both within mathematics and beyond. The starting point for the classification of algebraic curves is Riemann’s work in the 1850s. Riemann introduced the notion of an abstract curve, also called a Riemann surface, which may map into different spaces in different ways. This perspective naturally breaks the modern classification of curves into two parts: understand all Riemann surfaces, and understand all concrete realizations of a given Riemann surface in different spaces. The second part is called Brill--Noether theory, named after two researchers in the 1870s who made a stunning conjecture regarding all the possible ways to realize a Riemann surface in space. Over 100 years later, the conjecture was shown to hold for “most” Riemann surfaces. However, the conjecture fails for many important kinds of curves, including almost all curves lying in the plane, leaving large gaps in our knowledge. The proposed research project will study these special curves using a new invariant introduced by the PI, called the splitting type, as well as another proposed new invariant.
