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.

** Schedule:**

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.