Courses

This is a collection of lecture notes, assignments, and some other related materials of the courses I have taken.

1. FL23-SP24 Math5031-5032 Algebra I-II, Prof. Roya Beheshti Zavareh

Thanks to Albert Peng for his permission to edit his tex file of Math5031 and also J.S. Milne’s tex file. The lecture note also adds many thoughts and exercises from other sources (see the bibliography).

Math5031-32 Lecture Note, with homework incorporated as solved exercises

2. FL24 Math539 Topics in Algebraic Geometry, Prof. Roya Beheshti Zavareh

We use Algebraic Geometry by Robin Hartshorne as textbook but not follow it closely for some parts of the course. There will be homework on most weeks, and some of them will be chosen from Hartshorne. Some resources:

The red book of Varieties and Schemes, David Mumford. Springer Lecture Notes, 1999.

Lecture note by Ravi Vakil.

Lecture note by Andreas Gathmann.

3. FL24 Math547 Lie Algbera and Representation Theory, Prof. Martha Precup

See syllabus here.

4. FL24 Math560 Compact Lie Group, Prof. Quo-Shin Chi

The course based on Lectures on Lie Groups by Frank Adams is structured in three stages. First, it proves that the universal cover of a compact connected Lie group is the direct product of Euclidean space and compact, simply connected simple Lie groups, which aids in classifying all compact Lie groups. Next, it explores the maximal torus, root systems, Weyl groups, and their role in classifying these groups. Finally, the course examines the relationship between the weights of a maximal torus and irreducible complex representations, particularly focusing on exceptional Lie groups and Spin representations.

5. FL24 Math5047 Riemannian Geometry, Prof. Renato Feres

See syllabus here.

6. FL23-SP24 Math5021-5022 Complex Analysis I-II, Prof. Henri Martinkainen

I transcribe Prof. Martinkainen’s written manuscripts into latex file, with several tikz illustrations.

I also add some useful remarks made by Prof. Steven Krantz in undergraduate complex analysis course Math416.

Math5021-22 Lecture Note

Above is the image of a cat by Cayley map $\frac{z-i}{z+i}$ visualized through online conformal mapping viewer. Cute cat becomes a bear.

7. FL22 and SP25 (plan to take) Math5051-5052 Measure Theory and Functional Analysis I-II, Prof. Henri Martikainen

dyadic cubes

Math5051-52 Lecture Note

8. FL23 Math586 Network Statistics, Prof. Robert Lunde

Lecture Note for the course (I scribed lecture 9 and 10)

Math586 Lecture Note

Final project on cross-validation on network with Aaron Luo.

Beamer presentaton

pianist Emaneul Ax's signature on Lee's Intro to SM

9. SP23 Math5046 Differential Topology, Prof. Rachel Roberts

We used John M. Lee’s Introduction to Smooth Manifold (e2). I am AI and grader for this course in SP24 with Prof. Roberts again, this time using Tu’s Introduction to Manifold (e2). Some other texts are also very interesting, including a classical little book Milnor’s Topology from the Differentiable Viewpoint and Smooth Manifolds and Observables by a group named themselves Jet Nestruev.

Lecture Note and Homework soln

10. SP24 Math547 Topics in Geometry: Theory of Polytopes, Prof. Laura Escobar

permutohedron conv(S3)

This is a topic course for polytopes. Topics include various types of polytopes, simple polytopes, integral polytopes, permutohedrons, zonotopes, etc.; their representations, by vertices and inequalities; and their characterizations by graphs, discrete volumes, etc.

The following lecture note also includes my final project, Fourier analysis on polytopes.

Lecture Note

11. SP22 Math350 Dynamical System, Prof. John McCarthy

Dynamical systems can be defined as broadly as a semigroup $G$ action on a manifold $M$. Primary examples are $G=\mathbb{Z}$ (iterated maps) and $G=\mathbb{R}$ (flows and differential equations). In this Lecture note, I begin with a summary of classical ode theory partly from the summer course taught by Soumya Sinha Babu in 2021 (and also from various references (see bibliography in the pdf.)) and then use Prof. McCarthy’s Math350 (SP22) where we used Strogatz’s textbook as a source for supplementary examples to a more systematic approach to dynamics, chaos, and fractals, etc. See also my reading in Measures, Dimensions, and Analytic Capacities for geometry of fractals. The last part of the note is on analysis and dynamics on Riemann surfaces.

Lecture Note

12. SP24 CSE546T Computational Geometry, Prof. Tao Ju

A course in computational geometry. Topics include convex hull algorithms, sweeping line algorithm, line arrangement algorithm, trapezoidal maps, Art gallery problem, Voronoi diagrams, Delaunay triangulations, KD trees, range trees, and interval trees, motion planning, DCEL, and Hamsandwich cuts …

Course slides, Homework solns, and final project

Demo

David M. Mount’s lecture note