Teaching

Columbia Math

• Spring 2024: TA for Differential Equations, taught by Ovidiu Savin.
• Fall 2023: TA for Calculus II, taught by Miodrag Iovanov.
• Spring 2023: TA for Algebraic Curves, taught by Akash Sengupta.
• Fall 2022: TA for Calculus II, taught by Ross Akhmechet.
• Summer 2022: Instructor for Linear Algebra Show more
  1. The syllabus for the course.
  2. My email: alm2297@columbia.edu
  3. Office hours: Mondays and Wednesdays 1.30pm-2.30pm, room 406 in the Math department.
  4. The TA for the course is Kevin Chang. His office hours are Tuesdays 2pm-5pm, Thursdays 10am-12pm and 4pm-5pm, room 406 in the Math department.
  5. Extra credit proposed projects list
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  6. Notes and recordings.
     Lecture 7/5. Notes. Recording.
     Lecture 7/6. Notes. Recording.
     Lecture 7/7. Notes. Recording.
     Lecture 7/8. Notes. Recording.
     Lecture 7/11. Notes. Recording.
     Lecture 7/12. Notes. No recording (technical issue).
     Lecture 7/13. Notes. Recording.
     Lecture 7/14 Notes. Recording.
     Lecture 7/18 Notes. Recording.
     Lecture 7/20 Notes. No recording (technical issue).
     Lecture 7/21 Notes. Recording.
     Lecture 7/25 Notes. Recording.
     Lecture 7/26 Notes. Recording.
     Lecture 7/27 Notes. Recording.
     Lecture 7/28 Notes. Recording.
     Lecture 8/1 Notes. Recording.
     Lecture 8/2 Notes. Recording.
     Lecture 8/3 Notes. Recording.
     Lecture 8/4 Notes. Recording.
  7. Homework Send me an email with your solutions if you don't have access to Gradescope yet.
     Homework 1. Due 7/8 at 23.59pm on Gradescope.
     Homework 2. Due 7/15 at 23.59pm on Gradescope.
     Homework 3. Due 7/22 at 23.59pm on Gradescope.
     Homework 4. Due 7/29 at 23.59pm on Gradescope.
     Homework 5. Due 8/5 at 23.59pm on Gradescope.
  8. Pre-class quizzes:
     Lecture 7/6: watch the following video.
     Lecture 7/7: read/watch the following blog post.
     Lecture 7/8: read/watch the following blog post.
     Lecture 7/11: read/watch the following blog post.
     Lecture 7/12: watch this video and this other video.
     Lecture 7/13: read/watch the following blog post.
     Lecture 7/14: no quiz.
     Lecture 7/20: watch the following blog post.
     Lecture 7/21: watch the following video.
     Lecture 7/25: watch the following video.
     Lecture 7/26: watch this video as well as this video.
     Lecture 7/27: watch the following video.
     Lecture 7/28: watch the following video.
     Lecture 8/1: watch this video and this video
     Lecture 8/2: watch the following video.
     Lecture 8/3: watch the following video.
  9. Mock midterm. Solutions.
  10. Midterm. Solutions.
  11. Mock final. Solutions.
• Spring 2022: TA for Modern Algebra I, taught by Jorge Pineiro.
• Fall 2021: TA for Calculus II, taught by Konstantin Aleshkin
• Summer 2021: TA for Modern Analysis II, taught by Dobrin Marchev
• Spring 2021: TA for Calculus III, taught by Nick Salter
• Fall 2020: TA for Linear Algebra, taught by Giulia Saccà

Columbia Science Honors Program

The SHP is a selective program for high school students who have a strong interest in the sciences and mathematics.

• Spring 2023: Spaces and Symmetries. Show more
1. Week 7: Differential geometry: how do you curve space? Notes and exercises. Resources: CodeParade's video, NonEuclid, HyperRogue, Hypernom Exit feedback form
2. Week 6: Projective geometry: to infinity and beyond! Notes and exercises.
3. Week 5: Cellular homology: what is a hole? Notes and exercises.
4. Week 4: Classification of closed surfaces: the golden standard. Notes. Exercises. Resources: My surfaces tool.
5. Week 3: Equivalence relations: how to glue in math. Notes. Exercises
6. Week 2: Manifolds, knots and math software. Intro to manifolds. Planar codes and Sagemath. Resources: Sage
7. Week 1: Introduction to spaces and symmetries. Topology. Welcome form. Intro to topology.
• Spring 2022: Intro to Abstract Math through knots. Show more
1. Week 8: Algebraic topology and Brunnian links. Notes. Exercises. Resources: Sage
2. Week 7: Group theory and the knot group. Notes. Exercises. Resources: Sage
3. Week 6: Algebraic geometry and the knots of singularities of complex algebraic curves. Notes. Exercises. Resources: Desmos, Sage
4. Week 5: Hyperbolic geometry and hyperbolic knot theory. Notes. Exercises. Resources: CodeParade's video, NonEuclid, HyperRogue, Hypernom.
5. Week 4: Applications of surfaces to knots: Seifert surfaces. Notes. Exercises.
6. Week 3: Topology: surfaces and higher dimensions. Notes. Exercises.
7. Week 2: Combinatorics and our first invariant: n-colorability. Notes. Exercises. Resources: SnapPy
8. Week 1: The unknot, Reidemeister moves, and knot theory for computers. Notes. Exercises. Resources: Sage
• Fall 2021: Knot Theory. Map of what we did. Show more
1. 16. A look back. Notes.
2. 15. A party trick: Brunnian links. Notes. Exercises.
3. 14. The fundamental group of a space. Notes. Exercises.
4. 13. The fundamental group of a knot. Notes. Exercises.
5. 12. When two structures are secretly the same. Notes. Exercises.
6. 11. A lightning introduction to group theory. Notes. Exercises.
7. 10. Surfaces attached to knots. Notes. Exercises.
8. 9. The Euler characteristic. Notes. Exercises (8 and 9).
9. 8. One dimension up: surfaces. Notes.
10. 7. Higher dimensions and Lisa Piccirillo's result. Notes.
11. 6. The value of conjectures. Notes. Exercises.
12. 5. Knot Theory for computers. Notes. Exercises. Resources: Sage, SnapPy
13. 4. A link polynomial. Notes. Exercises.
14. 3. Multiplying knots, table of prime knots. Notes. Exercises.
15. 2. Invariants and tricolorability. Notes. Exercises.
16. 1. Reidemeister moves. Notes. Exercises (0 and 1).
17. 0. Intro and the unknot. Notes.
• Spring 2020: Diagrammatic Algebra, taught by Maithreya Sitaraman (brief substitution). Show more
• 2. Groups. The braid group.
• 1. String diagrams.
• Notes.
• Exercises.

Directed Reading Program

The DRP is a reading program in which undergraduate students at Columbia get paired with a graduate student volunteer for a semester. If you are a grad student at Columbia and are interested in the program, email Zoe, Robin or me and we will pair you with an undergrad student willing to read on some topic in your area of knowledge.

My previous mentees:

• Fall 2023: Gabriel Agustin Fernandez: Kauffman's classical paper on invariants of 3-manifolds. Final presentation: Hopf Algebras, Links, and a 3-Manifold Invariant.
• Spring 2023: Zhaocheng Dong: Lie algebras, their representations, and category O. Final presentation: The Weyl character formula.
• Fall 2022: Shiv Raj Yavnik: Knot invariants. Final presentation: An Extremely Compressed Overview of Khovanov’s Categorification of the Jones Polynomial.
• Spring 2022: Iris Rosenblum-Sellers, Lie theory: Lie groups, Lie algebras and their representations.
• Fall 2021: Tuan Dolmen, Algebraic Topology. Final presentation: Rubinzstein’s Topological Link Invariant