personal_handwritten_collection

Here are some documents which are not written by LaTeX, and there are often some screenshots from the references. Those notes are only for the educational purpose.

I mean, the documents in the folder named "sketsch_of_lectures" are not useful for you. I keep them here only for the convenience of myself.

####Explanations about the missed date of "Eine Woche, ein Beispiel"(in Chinese, "日拱一卒")

November 28th-December 5th, 2021: focus on the "moduli".

December 19th, 2021: focus on the "moduli of vector bundles"

January 2nd, 2022: focus on the "moduli of elliptic curves"(haven't finished actually)

January 23rd, 2022: I saw the videos of the mini-course “$p$-adic functions on $\mathbb{Z}_p$” by Liang Xiao (Peking University), you can get it (videos, lecture notes and even exersices) here: https://ctnt-summer.math.uconn.edu/schedules-and-abstracts-2020-online/. Roughly speaking, we begin doing basic analysis on the $p$-adic world. Here are some highlights:

• Down-to-earth examples as well as beautiful picture of $\mathbb{Z}_p$.
• Understand the $p$-adic function space $\mathcal{C}(\mathbb{Z}_p;\mathbb{Q}_p)$: define Mahler basis(compare with Fourier analysis) and $p$-adic measure $\mathcal{D}(\mathbb{Z}_p;\mathbb{Q}_p)$.
• Define $p$-adic $L$-function and construct corresponding $p$-adic measures on Galois groups; As an application, prove Kummer's congruence whose statement is elementary.
• Philosophy: every $L$-function should have some $p$-adic version of $L$-function as well as corresponding measures.

February 6th, 2022: prepare for the exam.

March 6th, 2022: Read the survey "The Borel-Weil-Bott theorem in examples" by Liao Wang and saw the videos of the mini-course “Sieves” by Brandon Alberts. There are some typos, and the exercise of mini-course in the last 3 pages are rather difficult for me. Here are some highlights for the mini-course:

• Abel summation is a powerful tool in estimating sums.
• The Möbius function suits perfectly with the sieve of Eratosthenes.
• By applying Brun's sieve, one can get an upper bound of twin prime numbers.
• Brun's main theorem can give us both upper bound and lower bound.

March 20th, 2022: During these two weeks, I take part in the Klein AG seminar, travel to Hamburg, listen to two operas, play with the software LMMS, learn some knowledge of chords from Bilibili. So I want to have a rest. T_T

April 3th, 2022: I have to prepare two talks: the first one want to introduce the Auslander--Reiten theory in a personal way (with no proof), and the second one concludes the irreducible representations of $GL_2(\mathbb{Q}_p)$. (Finally I don't need to prepare anyone, but I don't want to fill in this blank ╮(╯▽╰)╭)

May 8th-May 22nd, 2022: I wanted to write irreducible representations of $GL_2(\mathbb{Q}_p)$, but I need to prepare the talk, so I have no time to do that... (After some discussion with Qirui Li, these blankness are now filled in)

July 31st-August 14th, 2022: Work on modular forms. Actually I should write something on perverse sheaf on $\mathbb{CP}^1$.

September 11th, 2022: Had one week of tourism in Paris and one week in Antwerp. Had no time to think out new examples, but you can definitely get lots of (highly non-trivial) examples in the videos of the conference: Noncommutative Shapes. I also learned the Alexander's theorem from my schoolmate, which can help us to parametrize knots and links, see Braids. (For example, up to ambient isotopy there are only countable many knots in $S^3$, one can compute the Alexander–Conway polynomial systematically, and we can let the computer know the knot we want.) Some explanations between configuration spaces and braid groups is shown here: Video, snapshots For the connection between knot and number theory, maybe I can see this: Arithmeticity of Knot Complements

Some formulas are collected here:

$$B_2=\left<\sigma_1,\sigma_2 \middle| \sigma_1\sigma_2\sigma_1=\sigma_2\sigma_1\sigma_2 \right>\cong \pi_1(\operatorname{UConf}_5(\mathbb{C}))$$ $$\mathcal{S}_2=\left<a=(\sigma_1\sigma_2)^3, b=\sigma_2^2, c= \sigma_2\sigma_1^2\sigma_2 \middle| aba^{-1}b^{-1}, aca^{-1}c^{-1} \right> \cong \pi_1(\operatorname{Conf}_5(\mathbb{C}))$$

$$1 \longrightarrow \mathcal{S}_k \longrightarrow B_k \longrightarrow S_k \longrightarrow 1 $$

$$1 \longrightarrow \left<\sigma_i^n \right>_i \longrightarrow B_k \longrightarrow B_k(n) \longrightarrow 1 $$

For $k,n \geqslant 3$, $# B_k(n) &lt; +\infty \Longleftrightarrow (k,n)= (3,3), (3,4), (4,3), (3,5), (5,3)$.

In those cases, we know $# B_k(n)$ in the snapshot.

October 9th-16th, 2022: No time to update. TOEFL exam is coming!

See Algebraic K-theory and Trace Methods if you want to understand some histories and introductions to algebraic K-theory.

December, 2022: Working on the master thesis.

From February to March, 2023: Sorry I was quite busy these two months, for PhD applications, 3 oral exams, 1 master thesis defense, 1 informal talk, 1 conference, 1 Klein AG and 2 exam invigilations.

April 2nd, 2023: I saw the videos of Solara570 about Ford circle: https://www.bilibili.com/video/BV16U4y1Y7aF . I really appreciate the part of Ford spheres! (don't like engineer applications, though) You can also see Ford Circles and Spheres by Sam Northshield.

May 7th, 2023: Read 2/3 of my coursemate's master thesis (really nice with pictures and examples!), read 1/2 of Prof. Wang's course note "Functional Analysis" (learned a lot about the facts for the dual space!), and prepare for the Langlands correspondence summer school in Bonn.

June 18th, 2023: upload unramified L-parameters in Archimedean case: GL_n case.

July 16th, 2023: prepare for the exams of global analysis and topology II.

August 13th, 2023: found many materials in Joshua Ruiter's Homepage. There are many interesting notes which can be a potential topic in this program. Especially, I read his notes Group cohomology, Brauer groups, and algebraic K-theory. I learned a lot!

August 20th-September 10th, 2023: I am sorry that I'm doing bad in this program these weeks. In fact, I have thought carefully about the topics of these four weeks, and should definitely clean them up (despite not before the KleinAG). To remind myself, the 4(+4) topics are:

• the classification of multiplicative group (with an eye toward the Dieudonné-Manin classification theorem). It can help me to understand the rigid inner twist finally.

• the ramified covering.

• the relationships between the ramified covering and the functional field extension.

• from RS to algebraic curve over real/finite field, and finally the number field (mixed characteristic in two meanings: char 0+char finite, and different characters). In these cases, try to explain

  • the Frob,
  • the difference between the "residue field automorphism group" and "Deck transformation group"
  • the Chebotarev density theorem
  • In the number field, I should explain why the ramification points can be computed by the discriminant, and recompute the ramification points of RS in the same way.

• How the ramification theory interacts with the Brauer group as well as quartenion algebras. There are no direct relationships, but I think it to be fun.

• (local field extension)

  • of $\mathbb{C}((t))$, and discuss Newton polygons
  • of NA local fields
  • local-global compatability
    • If the field extension is determined by the ramification information
    • If the global Galois group is generated by local Galois group

In the same time I should also prepare the KleinAG talks.

September 17th-October 8th, 2023: It took me so much time to move from Bonn to Berlin. The lecture notes of KleinAG talk is typed, and the computer has been cleaned. Will have my own office!

October 15th, 2023: Try to fill the hole I made before. I made too many strands not completed, which is bad. Anyhow, at least part of the stuff are left, and I don't need to struggle from the beginning.

October 22nd, 2023: Read Conway's book The sensual (quadratic) form. It is an amazing book, with lots of figures and thousands of combinatorical results. Unfortunately, most results are not conclude in a independent theorem, so it becomes harder to cite the results. In ptc, he classifies

• quadratic forms up to rational equivalence in Lec 4, p94-97;

• integral binary quadratic forms up to integrally equivalence in p16-26;

• definite integral 3-dim quadratic forms in Lec 3;

• indefinite integral quadratic forms in Lec 4, p125;

• even unimodular lattices of dim 8, 16, 24 in p38, p57;

• cubic and isocubic lattices, tetralattices, root lattices, Niemeier lattices... in Lec 2+afterthoughts;

Reference for ultrametric space:

https://en.wikipedia.org/wiki/Ultrametric_space

November 5th, 2023: Read Laurenţiu G. Maxim's book "Intersection Homology & Perverse Sheaves. This book establishes a paradigm for comprehending perverse sheaves and vanishing cycles.

• Singular spaces frequently arise in research, with famous examples like Schubert cells, moduli spaces, and some equation-cut varieties. In section 2.1, the author introduces the concepts of topologically stratified spaces and topological pseudomanifolds as fundamental elements of this book.
• To gain a deeper understanding of their geometric properties, we need to develop the theory of (Borel-Moore) intersection homology, which lies in between the cohomology and the BM homology. They provide us the Kähler package, and they reflect some properties of the singularities. As a compromise, they are no longer homotopy-equivalent invariants. In practice, the computation of intersection homology often reduces to the computation of usual homology.
• To view intersection homology as a form of sheaf cohomology, the intersection cohomology complex $IC^{\cdot}{\bar{p}}$ is introduced. This sheaf is uniquely determined by certain axioms and can be explicitly constructed in two ways: as singular chains or as the Deligne complex (via induction). On a smooth oriented manifold, $IC^{\cdot}{\bar{p}}$ is equivalent to the constant sheaf $\underline{\mathbb{Z}}$.
• $IC^{\cdot}{\bar{p}}$ lies naturally in the category of (derived category of) constructable sheaves, and it even lies in the heart. Therefore, $IC^{\cdot}{\bar{p}}$ becomes a buiding object of the perverse sheaf.
• Cohomology provides valuable information. Kähler package, Hodge structure, Characteristic class, ... These can all be generalized in the study of singular spaces.

December 24th-31st, 2023: Christmas break! Got ill and recovered.

January 7th, 2024: try to compute equations on computers, using singular. With the help of Computational Aspects of Singularities, we are able to compute:

• varieties defined by equations, intersection and union of varieties
• Jacobian matrix and minors
• ideal of singular locus (and its radical)
• Milnor and Tjurina Numbers
• Type of hypersurface singularities
• intersection matrix of exceptional divisors

February 11th-25th, 2024: These days I'm having a really long holiday with people I know. I got a little bored with the current program: after all, not many people will see my drafts, and it become harder to identify the knowledge and the corresponding reference.

I'm thinking to reschedule my references in a more efficient way. My schoolmate recommends me to use the software called "zotero". I'm learning about it these days.

March 3rd-10th: stayed in USTC. Don't have computer to write notes. :-< Focusing on discussions and communications instead.

March 24th-April 14th: Tourism. Have a great rest. Try to work on Zotero, to make the learning flow smoother. Try to learn the name of mathematical journals.