---
# System prepended metadata

title: semester review (2025 Spring)

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# semester review (2025 Spring)

大二下

[TOC]

## Analysis (Honor Program) (II)

分析二

>Nets Katz 是我共事過最聰明的人
>(Jean) Bourgain 的文章我個人覺得比 (Terence) Tao 的更有深度
> (沈俊嚴 week 14)

:::info
* number : MATH5229
* course type : elective / honor course 
 (cover the compulsory course "Introduction to Analysis (II)")
* lecturer : 沈俊嚴
* textbook & ingredient: (later)
* credits : 5 (4 main classes + 1 TA class)
:::

The course continues with [Analysis (Honor Program) (I)](https://hackmd.io/qSdnv_J_SX-8UmQuDY1gGA#Analysis-Honor-Program-I). Throughout the first three weeks, we covered the remaining material from the textbook we used last semester. Throughout the course, we followed a selection of topics from the following books:

1. Pugh. Real mathematical analysis. (chapter 6)
2. Torchinsky. Real-variable methods in harmonic analysis. (chapter 1~4)
3. Stein. Fourier analysis. (chapter 4, 5)

some supplementary topics:
* Hardy's conjecture
* For $A\subset \mathbb{Z}/p\mathbb{Z}$ with $|A|>p^\frac{3}{4}$, we have  $AA+AA+AA=\mathbb{Z}/p\mathbb{Z}$
* $f(x)\in L^1$ with Fourier series diverges almost everywhere

As in last semester, the exercises were taken from Pugh (chapter 6). However, a major difference is that only 5 homework sets were assigned, all coming before the midterm. The second half of the semester was non-examable, instead focusing on preparing for the final report and presentation. 

Topics of the final project:
1. BMO space and John-Nirenberg inequality
2. Ergotic theorems
3. Haar measure
4. Hilbert space, integral operator, and compact operator
5. Locally convex space
6. Quantitative Lebesgue differential theorem
7. Riesz interpolation theorem and its application
8. Theta function and zeta function

These topics are assigned by the lecturer and the TA, and we need to rank our prefered topics. Eventually my team was assigned with "BMO space and the John-Nirenberg inequality". We refered to Grafakos' *Modern Fourier Analysis* and explored a paper published in *Journal of Functional Analysis* to further study a dimension-free version of the John-Nirenberg inequality, which strengthens the "exponential integrability" of bounded mean ocsillation (BMO) functions.

:::spoiler my abstract for the final report
In this final report, we introduce the bounded mean oscillation functions on $\mathbb{R}^d$, discuss their fundamental properties, and present a proof that these functions are exponentially integrable via the John-Nirenberg inequality. While the upper bound given by the John-Nirenberg inequality depends on the dimension $d$, we prove that the dependence can be eliminated. To achieve this, we reduce the problem to the case of integer-valued BMO functions under additional assumptions, and estimate the measure of the sets
$$\{x:f(x)=k\}, \{x:f(x)<k\},\{x:f(x)>k\}$$
for some certain value $k$.

:::


Lastly, concerning the grading, the midterm exam average was exceptionally high -- around 90. Since students generally performed well(?) on the final presentation as well, the grading appears to be quite high. 

:::warning

grade : A

:::spoiler grading

* homework (45%) : $\approx$ 45/45
* midterm exam (25%) : 88
* final report & presentation (30%) : unrevealed 

theoretical score : ?

A+ rate: 11/29
A rate: 11/29

:::

## Algebra (Honor Program) (II)

代數二

>如果你們把Galois theory學好、會算Galois group，沒有人會說你代數沒學好的 
(林惠雯 week 8)

:::info
* number : MATH7008
* course type : elective / honor course
  (cover the compulsory course "Introduction to Algebra (II)")
* lecturer : 林惠雯
* textbook : No textbook
* ingredients : ring theory, commutative algebra, finite Galois theory (and an introduction to infinite Galois theory)
* credits : 5 (4 main classes + 1 TA class)
:::

The structure of of Algebra (Honor program) (II) closely follows that of [Algebra (Honor program) (I)](https://hackmd.io/qSdnv_J_SX-8UmQuDY1gGA#Algebra-Honor-Program-I). However, in this second semester, we spent relatively more time on classical algebra topics than Algebra (I). 

For students planning to take Algebra (Honor program) in the future, note that the division between semesters depends on the lecturer. In some cases, Galois theory may be covered in the first semester, while homological algebra and representation theory is the focus of the second.

:::spoiler syllabus :
![Screenshot 2025-06-07 120627](https://hackmd.io/_uploads/SJo4h4ZXxl.png)
:::

This semester I adjusted my study habits by reviewing the lesson notes more carefullyand consistently. Since the content also felt less abstract, I found it manageable to digest and more intuitive to follow. 

Despite having taken courses with the same lecturer for two years, I still couldn't predict what will appear on the exams. The lack of a clear central focus sometimes makes studying more time-consuming and inefficient, as it's challenging to prioritize what to review.


:::warning

grade : A-

:::spoiler grading

* homework (30%) : 85
* midterm exam (35%) : 58
* final exam (35%) : 70

theoretical score : 70.3

A+ & A rate: 9/25
A club rate: 18/25

:::



## Introduction to Probability Theory

機率導論

>As I mentioned last time, we'll be discussing the probabilistic method this week. Unfortunately, these materials won’t be covered on the exam, despite being more interesting.
(Shagnik Das, week 15)

:::info
* number : MATH2502
* course type : compulsory
* lecturer : Shagnik Das
* textbook : Sheldon M. Ross. A first course in probability
* range : chapter 1 ~ 8
* non-examable materials: Alon, Spencer. The probablistic method. Chapter 1.
* credits : 4
**This course is conducted in English**
:::

The lecturer is a combinatorist who frequently employs the probabilistic method. However, this course doesn't focus on them. Instead, it concentrates on the fundamental materials covered in the textbook. The professor provided numerous examples to illustrate each probability concept, which makes the lectures somewhat lengthy, while this approach should be effective-particularly for students unfamiliar with the probability theory.

Over a semester, a total of 12 homeworks were assigned, each spanning 1 week. The one before midterm could be submitted with a one-week extension, and the ones before the final were optional bonus tasks. Most of the homework were closely aligned with the lectures and were generally manageable, although some exercises involved lengthy or intricate calculations.

In week 15, the course briefly introduced some classic problems concerning the probablistic method. The central principle discussed was that if $\mathbb{P}(E)>0$ (the probability of $E$ is positive), then the event $E$ must happen in at least one case. Using this idea, we explored partial results on topics such as sum-free sets and the Ramsey theory. Our last homework was based on this and offered more glimpses into how probability is used in combinatorics.


:::warning

grade : A+

:::spoiler grading

* homework (30%) : 30/30 (12 homework / 3% each / not exceeding 30%)
* midterm exam (30%) : 90
* final exam (40%) : 96

theoretical score : 95.4

A+ rate: 36.27%

:::

## Principle of Macroeconomics (with Recitation)

總體經濟學原理與實習

>大家跟我唸一遍：「Inflation is always and everywhere a monetary phenomenon.」
>(林明仁 week 12)

:::info
* number : ECON1021-01
* course type : elective (elective-compulsory for math major)
* lecturer : 林明仁
* textbook : Acemoglu, Laibson, List. Ecomomics. (Third Edition)
* range : part 4 ~ 9 : chapter 15, 16, 19, 20, 21, 22, 23, 24, 25, 28
* credits : 4 (3 main classes + online TA classes)
:::

The formation and the features of the course is almost the same as [Principle of Microeconomics (with recitation)](https://hackmd.io/qSdnv_J_SX-8UmQuDY1gGA#Principle-of-Microeconomics-with-Recitation)



:::warning

grade : A

:::spoiler grading

* homework (30%) : 26.3/30 ?
* midterm exam (30%) : 82
* final exam (40%) : 76

theoretical score : 81.3 ?

:::spoiler grade distribution
![IMG_3716](https://hackmd.io/_uploads/SyY9UAQEgl.jpg)

:::



## French (II)(2)

法文二下

:::info
* number : FL3012-05
* course type : elective
* lecturer : Maximilien Dersch du Mesgnild (達利安 Max)
* textbook : Amical 2 & Campagnon A2
* range : lesson 13 ~ 24
* credits : 3 (4 classes a week)
**This course is conducted in English**
:::

I didn't continue with Louisa's [French II(1)](https://hackmd.io/qSdnv_J_SX-8UmQuDY1gGA#French-II1) due to a scheduling conflict with Introduction to Probability Theory. So I switched to this class this semester.

Compared to other French courses I've taken, this course involves relatively challenging in-class exercises, homework, and exams. As a transfer student, the focus shifted from memorizing grammar and vocabulary to applying French more actively. I recommend this course to anyone planning to visit French-speaking countries, whether for exchange or further studies.

Texts in *Amical 2* still serves as the primary material we follow. To provide more comprehensive practice, the lecturer designed additional material *Campagnon*, on which we also spent a fair amount of time on. However, we didn't have many opportunities for spoken interaction in French during class. In terms of skill emphasis and time we spent, I personally felt the course workload could be ranked as: *writing > listening > reading > speaking*.

The exams deviated from traditional paper-based tests. Each exam consisted of 4 or 5 tasks that tested reading, listening, writing and speaking. While the content related to the topics we discussed in our lessons, it didn't directly follow any text in *Amical* or *Campagnon*. Writing seemed to be especially emphasized: the reading/listening section could require writing short responses or articles. Electronic devices were permissible only for watching the video in the listening section and recording our speaking responses.

All exams were completed in pairs. While this approach aimed to encourage collaboration, it sometimes caused frustration. With limited time and differing ideas on how to approach tasks and what to answer, disagreements could make it difficult to finish everything on time. (Nonetheless, I appreciate my partner's effort.)


:::warning

grade : A+

:::spoiler grading

* workshops (20%) : no grade is revealed ( $\approx$ 5 homework regarding writing short articles / working in pairs)
* journal (15%) : :white_check_mark: (4 diaries per semester / each in A5 paper)
* language practice (5%) : :white_check_mark: (attend once in the [bla bla night](https://www.instagram.com/bla.bla.night/))
* midterm exam (30%) : no grade is revealed
* final exam (30%) : no grade is revealed
* bonus(+5%) : :white_check_mark: (duolingo streak for 30 days without freeze)

A+ rate: 41.94%

:::

## general 

![image](https://hackmd.io/_uploads/SJ2wSAAVee.png)

total credits: 21
GPA: 4.03 (before using the "explore credits")

---

# summer vacation

## plan

I plan to participate in student seminars on the Green–Tao theorem, the probabilistic method, and generating functions, with a stronger focus on the first two. I will be leading the one on the Green–Tao theorem, aiming to work through the proof of not only the Green–Tao theorem but also the Gauss prime constellation theorem from the ground up.

In addition, I intend to complete a number of exercises in the probabilistic method to deepen my understanding. I also hope to spend some time browsing recent problems and papers in extremal combinatorics to explore current developments and identify areas of interest for future research.
