>. Notes [notes]

Some of my mathematical notes, use with caution!

Reinforcement learning for finding counterexamples in graph theory [rl4fc_practice_2023]

July 9, 2023
Cheni Yuki Yang, Karl Schamel, Hugo Fernandez
Blog post

This is a report for HEGL Praktikum in SoSe 2023. This is a multi-author article, I wrote it with my groupmates and post it here now as a reference.

Leanduction from scratch? [formal_seminar_2023]

July 26, 2023
Universität Heidelberg
Cheni Yuki Yang, Nicolas Arnold
Slide

This is the final project presentation for the HEGL seminar on formalizing mathematics in summer term, 2023. We inspected the induction principle in Lean 4.

Bachelor’s Thesis. Cell complexes in homotopy type theory [bachelor]

In this thesis we study the theory of CW-complexes in a synthetic way. We first introduced some important concepts of homotopy type theory and used them to define cell complexes as a homotopical pushout. After that, we have proven some topological properties of CW-complexes and calculated the homotopy groups of some examples with methods in type theory.

Exercise session of differential geometry 1 [diffgeo_teaching_2025]

My notes for doing TAs of elementary differential geometry course.

Seminar on elliptic curves [ec_seminar_2025]

This is a talk about Mordell-Weil theorem, following Silverman’s book.

Seminar on intersection homology [ih_seminar_2025]

This is a talk about how to build sheaves of intersection complexes, following Banagl’s book.

Build a reliable math library [lean_talk_2025]

September 13, 2025
Online
Yuki
Slide

I gave a talk on the 6th plenary meeting of NJUPT SAST MathSIG, about Lean 4 and Mathlib.

Seminar on modular forms and sphere packings [mf_seminar_2025]

I explained the motivation and method of Viazovska for finding magic functions of sphere packings as in her fantastic annal paper.

References

Reference. Topological invariants of stratified spaces [banagl2007topo]

2007
Markus Banagl

The homology of manifolds enjoys a remarkable symmetry: Poincaré duality. If the manifold is triangulated, then this duality can be established by associating to a s- plex its dual block in the barycentric subdivision. In a manifold, the dual block is a cell, so the chain complex based on the dual blocks computes the homology of the manifold. Poincaré duality then serves as a cornerstone of manifold classi cation theory. One reason is that it enables the de nition of a fundamental bordism inva- ant, the signature. Classifying manifolds via the surgery program relies on modifying a manifold by executing geometric surgeries. The trace of the surgery is a bordism between the original manifold and the result of surgery. Since the signature is a b- dism invariant, it does not change under surgery and is thus a basic obstruction to performing surgery. Inspired by Hirzebruch’s signature theorem, a method of Thom constructs characteristic homology classes using the bordism invariance of the s- nature. These classes are not in general homotopy invariants and consequently are ne enough to distinguish manifolds within the same homotopy type. Singular spaces do not enjoy Poincaré duality in ordinary homology. After all, the dual blocks are not cells anymore, but cones on spaces that may not be spheres. This book discusses when, and how, the invariants for manifolds described above can be established for singular spaces.

@book{banaglTopologicalInvariantsStratified2007,
  title = {Topological Invariants of Stratified Spaces},
  author = {Banagl, Markus},
  year = 2007,
  series = {Springer Monographs in Mathematics},
  publisher = {Springer},
  address = {Berlin Heidelberg New York},
  isbn = {978-3-540-38585-1 978-3-540-38587-5},
}

Reference. The arithmetic of elliptic curves [silverman2009ec]

2009
Joseph H. Silverman

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel’s theorem on finiteness of integral points.

@book{silvermanArithmeticEllipticCurves2009,
  title = {The Arithmetic of Elliptic Curves},
  author = {Silverman, Joseph H.},
  year = 2009,
  series = {Graduate Texts in Mathematics},
  volume = {106},
  publisher = {Springer New York},
  address = {New York, NY},
  doi = {10.1007/978-0-387-09494-6},
  isbn = {978-0-387-09493-9 978-0-387-09494-6},
}

Reference. The sphere packing problem in dimension 8 [viazovska2017sp8]

2017
Maryna Viazovska
10.4007/annals.2017.185.3.7

In this paper we prove that no packing of unit balls in Euclidean space R8 has density greater than that of the E8-lattice packing.

@article{viazovskaSpherePackingProblem2017,
  title = {The Sphere Packing Problem in Dimension \$8\$},
  author = {Viazovska, Maryna},
  year = 2017,
  month = may,
  journal = {Annals of Mathematics},
  volume = {185},
  number = {3},
  issn = {0003-486X},
  doi = {10.4007/annals.2017.185.3.7},
}