Course Announcement - Spring 2023

Math 270: Metric Algebraic Geometry

Instructor: Bernd Sturmfels

Office hours: e-mail me to set up an appointment.

Lecture times: Tuesdays, 8:10-9:30

First Day: Tuesday, January 17
Last Day: Tuesday, March 21

Prerequisites: Basics of commutative algebra and algebraic geometry. Experience with computer algebra and numerical computing.

Description: How can we compute the distance from a point in 3-space to a given curve or surface? This question arises for many geometric shapes and in many contexts, notably in optimization and data science. Metric algebraic geometry studies real algebraic varieties, with focus on metric properties -- distances, volumes, angles, and curvature. Our aim is to compute these quantities from defining polynomials. The term ``metric algebraic geometry'' was coined in 2021 by the PhD thesis of Maddie Weinstein.

Format: This 2-unit course offers an introduction to this field of research. Lectures by the instructor are supplemented by guest lectures and hands-on working sessions. Berkeley students will interact with participants from Germany and elsewhere.

Course materials: Notes are posted below. These are to be developed further for the Oberwolfach seminar
which will be led by Paul Breiding, Kathlén Kohn and myself during the week of May 28 - June 3, 2023.

Grading: Regular attendence is expected. Active participation and written work are encouraged.
This is a research-oriented course: You set your own goals; you decide how well you reached them.

Jan 17: Critical Equations
Jan 24: Svala Sverrisdóttir: HomotopyContinuation.jl
Jan 31: Wasserstein Distance
Feb 7: Voronoi Cells
Feb 14: Maximum Likelihood
Feb 21: Kathlén Kohn: Polar Classes
Feb 23: Tropical Geometry Methods (Thursday in 891 Evans!)
Feb 28: Maddie Weinstein: Curvature, Bottlenecks and Reach
Mar 2: Questions and Answers (Thursday in 891 Evans!)
Mar 7: Paul Breiding: Tensors
Mar 14: Volumes      (starts 16:10 in Europe)
Mar 21: Vahid Shahverdi: Machine Learning   (starts 16:10 in Europe)