UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 25, 2020

939 Evans Hall

3:45PM: Theta Surfaces

Bernd Sturmfels

A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincar\'e showed that theta surfaces are precisely the surfaces of double translation, i.e. obtained as the Minkowski sum of two space curves in two different ways. These curves are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

5:00PM: Tropical Support Vector Machines

Ruriko Yoshida

Most data in genome-wide phylogenetic analysis is essentially multidimensional, posing a major challenge to human comprehension and computational analysis. Also, we cannot directly apply statistical learning models in data science to a set of phylogenetic trees since tree space is not Euclidean. In fact, the space of phylogenetic trees is a tropical Grassmanian in the max-plus algebra. Therefore, to classify multi-locus data sets for phylogenetic analysis, we propose tropical Support Vector Machines (SVMs) over the space of phylogenetic trees. Like classical SVMs, a tropical SVM is a discriminative classifier defined by a tropical hyperplane which maximizes the minimum tropical distance from data points to itself to separate these data points into sectors. We show that, even though we have to solve exponentially many linear programming problems to find tropical SVMs, many instances are infeasible. Then we show the necessary and sufficient conditions for each instance to be feasible. In each case, there is an explicit formula for the optimal solution for the feasible linear programming problem. Based on our theorems, we develop novel methods to compute tropical SVMs, and computational experiments that show our methods work well.

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