Course Announcement - Spring 2020

Math 191: Invitation to Nonlinear Algebra

Instructor: Bernd Sturmfels

Office hours: Tuesdays 8:00-10:00 and by appointment
Contact: bernd at math, 925 Evans

Time and Place: Mondays, Wednesdays and Fridays, 9:00-10:00, 9 Evans Hall

Prerequisites: Very strong foundation in undergraduate mathematics. Complete mastery of linear algebra (Math 110)
and abstract algebra (Math 113). Familiarity with other upper division courses, such as Math 114, 143, 170 or 172.

Consultants: Madeleine Weinstein and Leon Zhang will assist. Questions can be directed to either them or me.

Text Book: We will cover the text book I am currently writing with Mateusz Michalek:
Invitation to Nonlinear Algebra, American Mathematical Society, 2021.

Syllabus: The book has 13 chapters, one for each of the weeks in the semester:
1. Polynomials
2. Varieties
3. Solving and Decomposing
4. Mapping and Projecting
5. Linear Spaces and Grassmannians
6. Nullstellensätze
7. Tropical Algebra
8. Toric Varieties
9. Tensors
10. Representation Theory
11. Invariant Theory
12. Semidefinite Programming
13. Combinatorics
Each chapter has three sections, one for each of the three lectures per week.

Homework: The homework is due on Wednesdays, starting on January 29 and ending on April 29.
Each assignment is to work out six exercises from the chapter covered in the previous week.
No late homework! You can make up points by submitting more than six exercises in later weeks.

Collecting Solutions: Solutions to exercises will be collected on Overleaf. Click to see the requested Latex format.
This URL is hosted by Kyle Huang, Yuhan Jiang and Elliot Stahnke. Please e-mail invited solutions to one of them.

Software Illustrations: Many thanks to Aidan Abdulali, Kevin An, Max Smolin, and Tyler Zhu for posting
numerous helpful examples at this website. Please e-mail your comments and questions to one of them.

Grading: The course grade is based on the homework (78%) and on supplementary work (22%).
Please e-mail a brief report about your 22% effort to Leon, Maddie or Bernd by April 30.
Yes, bug reports count!

Further Reading: It is recommended to follow pointers to other sources that are given in the textbook.

prepare by reading 1.1. Ideals
January 22: 1.2. Gröbner Bases
January 24: 1.3. Dimension and Degree
January 27: 2.1. Affine Varieties
January 29: 2.2. Projective Varieties
January 31: 2.3. Geometry in Low Dimensions
February 3: 3.1. Zero-Dimensional Ideals
February 5: 3.2. Primary Decomposition
February 7: 3.3. Linear PDE with Constant Coefficients
February 10: 4.1. Elimination
February 12: 4.2. Implicitization
February 14: 4.3. The Image of a Polynomial Map
self-study: 5.1. Coordinates for Linear Spaces
February 19: 5.2. Plücker Relations
February 21: 5.3. Schubert Calculus
February 24: 6.1. Certificates for Infeasibility
February 26: 6.2. Hilbert's Nullstellensatz
February 28: 6.3. Let's Get Real    (Isabelle Shankar)
March 2: 7.1. Tropical Algebra
March 4: 7.2. Linear Algebra
March 6: 7.3. Tropical Varieties
March 9: 8.1. The Affine Story
March 11: 8.2. Varieties from Polytopes
March 13: 8.3. The World is Toric
March 16: 9.1. Eigenvectors
March 18: 9.2. Tensor Rank
March 20: 9.3. Matrix Multiplication

Read the book chapters below and click titles to see lectures:

April 6-10: Chapter 10: Representation Theory
April 8: Homework for Chapter 9 is due.
April 10, Friday: Class meeting via Zoom (9-10am)

April 13-17: Chapter 11: Invariant Theory
April 15: Homework for Chapter 10 is due.
April 17, Friday: Class meeting via Zoom (9-10am)

April 20-24: Chapter 12: Semidefinite Programming
April 22: Homework for Chapter 11 is due.
April 24, Friday: Class meeting via Zoom (9-10am)

April 29: Homework for Chapter 12 is due.
April 30: Term papers and 22% reports are due.