updates

.gitignore

@@ -147,3 +147,6 @@ public/sitemap.xml
 
 # npm package lock file for different platforms
 package-lock.json
+
+# generated
+mcp-worker/generated

README.md

@@ -36,3 +36,18 @@ Update dependencies
 ```bash
 npx npm-check-updates -u
 ```
+
+### MCP access to notes
+
+This repository includes a minimal MCP server that exposes the existing `content/` notes as a knowledge base for AI tools over stdio.
+
+```bash
+npm install
+npm run mcp:notes
+```
+
+The server exposes:
+
+- `list_notes`
+- `read_note`
+- `search_notes`

content/CSE4303/CSE4303_L14.md

@@ -0,0 +1,156 @@
+# CSE4303 Introduction to Computer Security (Lecture 14)
+
+## Cryptography applications
+
+### Crypto summary
+
+- Cryptographic goals
+  - Confidentiality
+    - Symmetric-key ciphers
+      - Block ciphers
+      - Stream ciphers
+    - Public-key ciphers
+  - Data integrity
+    - Arbitrary length hash functions
+    - Message Authentication Codes (MACs)
+    - Digital signatures
+  - Authentication
+    - Entity authentication
+      - Authentication primitives
+    - Message authentication
+      - MACs
+      - Digital signatures
+  - Non-repudiation
+    - Digital signatures
+
+### Overview
+
+- Digital certificates
+  - PKI
+  - Intentional MITMs
+  - Other uses
+- Authentication (...of users, not data)
+  - SSH keys
+  - Time-based One-Time Password (TOTP)
+- Ransomware
+- Post-Quantum (PQ) crypto
+- Zero-Knowledge (ZK) proofs
+- Homomorphic encryption
+
+### Recall: key-exchange challenge
+
+- No matter the cipher, **algorithm** is known but **key** must remain secret
+  - (Security principle: open design)
+  - Symmetric-key system: entire key remains secret
+  - Public-key scheme: private keys remain secret
+- Q: before secure channel established, how to **reliably exchange** symmetric keys? How to reliably exchange public keys?
+  - Symmetric keys: via public/private keys
+  - Public keys: web of trust (e.g. GPG), what else?
+
+As presented, the protocol is insecure against active attacks
+
+MiTM can insert and create 2 separate secure sessions
+
+### Public-key distribution: digital certificates
+
+- Common public-key object
+  - Goal: bind identity to public key
+  - Contents:
+    - ID/key tuple
+    - Digital signature of some trusted signing authority: provides integrity/authenticity "guarantees" (under proper assumptions)
+  - Implementation:
+    - Can be chained: e.g. certificate for key of interest signed by Certificate Authority $CA_N$, cert for $CA_N$'s key signed by $CA_{N-1}$, ..., certificate for $CA_2$'s key signed by $CA_1$ (root CA), root CA's key pre-loaded in OS
+    - Allows for a hierarchy
+    - Requires entity to gain trust of signing authority and obtain certificate
+
+### Defeating MITM using Digital Certificates
+
+- Alice goes to a **trusted party** to get a certificate.
+- After verifying Alice's identity, the trusted party issues a certificate **signed by them** with Alice's name and her public key.
+- Alice sends the entire certificate to Bob.
+- Bob verifies the certificate using the trusted party's public key (i.e. he checks the **signature** on the cert).
+- Bob now knows the **true owner** of a public key.
+
+### Public key infrastructure
+
+- Certificate Authority (CA): a trusted party responsible for verifying the identity of users and binding the verified identity to public keys by issuing signed certificates to them
+- Digital Certificate: a document certifying that the public key included inside does belong to the identity described in the document (signed by CA)
+  - Standard governing certs: X.509
+
+### HTTPS layers of trust
+
+- Data
+  - Trust data b/c it's encrypted by session key
+- Session Key
+  - Trust session key b/c it was exchanged via public-key scheme
+- Public Key
+  - Trust public key b/c it's in a certificate
+- Certificate
+  - Trust certificate b/c it's signed by chain of CAs, ending with root CA
+- Certificate Authorities
+  - Trust root CA's sig b/c it's embedded in browser/OS
+- Browser/OS ...
+  - Trust browser/OS b/c
+    - we trust whoever installed the browser/OS
+    - and we trust the hardware that runs the OS
+    - (note: mechanisms for justifying these trusts exist too)
+- Hardware
+
+#### Application: PKI for public-key distribution
+
+- Q: How are encrypted communications usually sent across a network? (e.g. https traffic)
+- A: Multi-part strategy
+  - 1. Use symmetric-key cipher for encrypting traffic (speed advantage)
+  - 2. Use public-key cipher for exchanging symmetric key values (authenticity, consistency)
+  - 3. Use public-key infrastructure (PKI) to certify public keys (using certificates)
+- Full details and data traces: First Few Milliseconds of a TLS 1.2 Connection (2017)
+- [demo: ciphers currently used to encrypt session in Firefox, Chrome]
+- [demo: root CAs currently trusted]
+- How to mitigate attacks?
+  - Revocation cert
+- Real attacks
+  - DigiNotar attacked 2011: issued fraudulent certs for, e.g., google.com
+  - WoSign 2016: issued certs to domains rather than subdomains
+
+#### Legitimate (?) MITMs in PKI: private root certs
+
+- Recent trend: entity (company/ISP/govt) installs root cert on user's system (as requirement for access), and possibly accompanying app
+- Entity intercepts all secure sessions from user and MITMs them using root cert
+- Entity then has access to all user's traffic
+- Possibly legitimate uses:
+  - Enforce web usage policies: e.g. no gaming/social networking during work
+  - Enhance security: scan traffic for malware, phishing, etc. and respond
+- Possibly illegitimate uses:
+  - Record user's traffic, browsing habits, personal data
+  - Sell data to third party
+  - Censor content
+- More info:
+  - How to tell whether your https session is being MITM'ed
+  - Certificate hash lookup page
+  - Kazakhstan's ongoing MITM saga
+
+#### Related privacy question: visibility
+
+- Question: should govt agencies (including law enforcement) have access to encrypted communications?
+  - One "yes" argument: helps catch criminals, prevent terrorist attacks, etc.
+  - One "no" argument: invades privacy, gives too much eavesdropping power
+- Relevant history / case studies:
+  - Munitions restrictions on crypto circa late 1990's: DES
+  - Phil Zimmerman and PGP: e-mail encryption, govt attempts to suppress
+  - Edward Snowden revelations 2013: fears of privacy abuse are well-founded
+  - RSA Sec's use of NIST-recommended PRNG w/ECC: was apparently an NSA backdoor
+  - Syed Farook ("San Bernardino shooter") case 2015: FBI pressure on Apple to unlock user's iPhone
+    - Apple resisted
+    - Never resolved legally: FBI found 3rd party to grant access
+  - GCHQ "ghost protocol" proposal 2018
+    - Don't weaken encryption, but secretly add government to encrypted conversation at will
+      - Add extra private key to encrypted convos; suppress notifications about new user being added to convo
+    - Condemned by big tech companies June 2019
+- Question: should private companies have access to encrypted communications, or just metadata, or neither?
+- Relevant history / case studies:
+  - Zoom end-to-end encryption [Dec 2022 status]
+    - Note: without E2EE, Zoom holds keys but never decrypts conversations
+  - Facebook/WhatsApp terms of service update 2021 [Wired mag article]
+    - Note: WhatsApp still has E2EE for messages, but shares metadata
+
+

content/CSE4303/CSE4303_L15.md

@@ -0,0 +1,72 @@
+# CSE4303 Introduction to Computer Security (Lecture 15)
+
+## Cryptography applications
+
+### Authentication (...of users, not data)
+
+- Traditional authentication: password-based, single-factor
+  - Disadvantage: convenience often chosen over security
+    - Weak passwords
+    - Password re-use: one password compromised ==> many accounts compromised
+  - Attack: "credential stuffing" (testing single stolen password against many accounts)
+
+#### SSH keys
+
+- Idea: relieve burden of secure passwords
+  - Use public/private-key auth instead (can be automated!)
+  - Use secure password once to exchange public key
+- Protocol: challenge/response to verify possession of keys
+
+#### Case Study: SSH
+
+- SSH can use public-key based authentication to authenticate users
+- Generate a pair of public and private keys: `ssh-keygen -t rsa`
+- private key: `/home/seed/.ssh/id_rsa`
+- public key: `/home/seed/.ssh/id_rsa.pub`
+- Register public key with server:
+  - Send the public key file to the remote server using a secure channel
+  - Add public key to the authorization file `~/.ssh/authorized_keys`
+- Server can use key to authenticate clients
+
+#### Time-based One-Time Password (TOTP)
+
+- Goal: provide secure 2nd factor for authentication
+- Idea:
+  - Generate one-time (single-use) password for each login attempt
+  - Compute one-time password using secure HMAC with current time as a parameter
+- Key used for HMAC: exchanged once at setup
+- Protocol: open standard published by OATH, IETF
+  - HMAC-based One-Time Password (HOTP)
+  - e.g. $TOTP(k) = HOTP(k, C_t)$, where $C_t$ is absolute measure of current time interval
+  - Num digits taken from output: 6 to 10
+
+### Ransomware
+
+- Idea: attacker encrypts victim's data with symmetric cipher, requires ransom payment to decrypt (or provide key)
+  - System model: any data store (company database, municipal database, user's hard drive, etc.)
+  - Threat model: attacker who has already compromised victim's system
+  - Vuln: lack of backups (or prohibitive time to restore); whatever vuln allowed attacker into system
+  - Surface: exposed data, and surface of original compromise
+  - Vector: encrypt data store and erase or replace original data store
+  - Mitigation/defense: keep up-to-date backups (possibly "air-gapped") in separate location; practice restoring from backups
+- Enabler: anonymity of Bitcoin payments
+- Recent-ish examples:
+  - Baltimore 2019 (didn't pay, est. $18 million to fix)
+  - Atlanta 2018 (~$9 million to fix)
+  - Lake City & Riviera City, Florida 2019 (did pay, $500,000+ apiece)
+  - Many others since these
+  - One of the top projected trends in cybersecurity in 2021 (e.g. by CSO online)
+
+### Post-Quantum (PQ) crypto
+
+- Fundamentally different computation paradigm than "classical" von Neumann or dataflow models
+- Relies on properties of quantum physics to solve problems efficiently
+  - Superposition: state of quantum bit ("qubit") expressed by probability model over continuous range of values (vs. classic bit: 0 or 1 only)
+    - Like being able to operate on all possible bit combos of a register simultaneously, instead of operating on only one among all possibilities
+  - Entanglement: operating on one qubit affects others
+
+
+
+### Zero-Knowledge (ZK) proofs
+
+### Homomorphic encryption

content/CSE4303/CSE4303_L16.md

@@ -0,0 +1,160 @@
+# CSE4303 Introduction to Computer Security (Lecture 16)
+
+## System security
+
+- Why system security / platform security?
+  - All code runs on some physical machine!
+    - The cloud is not a cloud
+    - Web pages are just data and code copied from a server that also manages the transfer
+- Why Linux?
+  - Majority of web servers run Linux (esp. Cloud); popular in embedded, mobile devices
+
+### Operating system background
+
+Context: computing stack
+
+| Layer | Description |
+| --- | --- |
+| Application | Web browser, user apps, DNS |
+| OS:libs | Memory allocations, compiler/linker|
+| OS:kernel | Process control, networking, file system, access control|
+| OS:drivers | Manage hardware|
+| (Firmware) | Minimal hardware management (if no full OS)|
+|Hardware | Processor, cahce, RAM, disk, USB ports|
+
+#### Operating systems
+
+- Operating System:
+  - Provides easier to use and high level **abstractions** for resources such as address space for memory and files for disk blocks.
+  - Provides **controlled access** to hardware resources.
+  - Provides **isolation** between different processes and between the processes running untrusted/application code and the trusted operating system.
+
+- Need for trusting an operating system
+  - Why do we need to trust the operating system? (AKA a Trusted Computing Base or TCB)
+  - What requirements must it meet to be trusted?
+
+- TCB Requirements:
+  - 1. Tamper-proof
+  - 2. Complete mediation (reference monitor)
+  - 3. Correct
+
+Isolating OS from Untrusted User Code
+
+- How do we meet the first requirement of a TCB (e.g., isolation or tamper-proofness)?
+  - Hardware support for memory protection
+  - Processor execution modes (system AND user modes, execution rings)
+  - Privileged instructions which can only be executed in system mode
+  - System calls used to transfer control between user and system code
+
+
+System Calls: Going from User to OS Code
+
+- System calls used to transfer control between user and system code
+  - Such calls come through "call gates" and return back to user code.
+    - The processor execution mode or privilege ring changes when call and return happen.
+  - x86 `sysenter` / `sysexit` instructions
+
+Isolating User Processes from Each Other
+
+- How do we meet the user/user isolation and separation?
+  - OS uses hardware support for memory protection to ensure this.
+
+Virtualization
+
+- OS is large and complex, even different operating systems may be desired by different customers
+- Compromise of an OS impacts all applications
+
+Complete Mediation: The TCB
+
+- Make sure that no protected resource (e.g., memory page or file) could be accessed without going through the TCB
+- TCB acts as a reference monitor that cannot be bypassed
+- Privileged instructions
+
+Limiting the Damage oa a Hacked OS
+
+Use: Hypervisor, virtual machines, guest OS and applications
+
+Compromise of OS in VM1 only impacts applications running on VM1
+
+### Secure boot and Root of Trust (RoT)
+
+Goal: create chain of trust back to hardware-stored cryptographic keys
+
+#### Secure enclave: overview (Intel SGX)
+
+![Intel SGX](https://notenextra.com/CSE4303/Intel_SGX.png)
+
+Goal: keep sensitive data within hardware-isolated encrypted environment
+
+### Access control
+
+Controlling Accesses to Resources
+
+- TCB (reference monitor) sees a request for a resource, how does it decide whether it should be granted?
+  - Example: Should John's process making a request to read a certain file be allowed to do so?
+
+- Authentication establishes the source of a request (e.g., John's UID)
+- Authorization (or access control) answers the question if a certain source of a request (User ID) is allowed to read the file
+- Subject who owns a resource (creates it) should be able to control access to it (sometimes this is not true)
+
+- Access control
+  - Basically, it is about who is allowed to access what.
+  - Two parts
+    - Part I - Policy: decide who should have access to certain resources (access control policy)
+    - Part II - Enforcement: only accesses defined by the access control policy are granted.
+  - Complete mediation is essential for successful enforcement
+
+Discretionary Access Control
+
+- In discretionary access control (DAC), owner of a resource decides how it can be shared
+  - Owner can choose to give read or write access to other users
+- Two problems with DAC:
+  - You cannot control if someone you share a file with will not further share the data contained in it
+    - Cannot control "information flow"
+  - In many organizations, a user does not get to decide how certain type of data can be shared
+    - Typically the employer may mandate how to share various types of sensitive data
+  - Mandatory Access Control (MAC) helps address these problems
+
+Mandatory Access Control (MAC) Models
+
+- User works in a company and the company decides how data should be shared
+  - Hospital owns patient records and limits their sharing
+    - Regulatory requirements may limit sharing
+    - HIPAA for health information
+
+#### Example: Linux system controls
+
+Unix file access control list
+
+- Each file has owner and group
+- Permissions set by owner
+  - Read, write, execute
+  - Owner, group, other
+  - Represented by vector of four octal values
+- Only owner, root can change permissions
+  - This privilege cannot be delegated or shared
+- Setid bits -- Discuss in a few slides
+
+Process effective user id (EUID)
+
+- Each process has three IDs (+ more under Linux)
+  - Real user ID (RUID)
+    - Same as the user ID of parent (unless changed)
+    - Used to determine which user started the process
+  - Effective user ID (EUID)
+    - From set user ID bit on the file being executed, or sys call
+    - Determines the permissions for process
+      - File access and port binding
+  - Saved user ID (SUID)
+    - So previous EUID can be restored
+- Real group ID, effective group ID used similarly
+
+#### Weaknesses in Unix isolation, privileges
+
+- Shared resources
+  - Since any process can create files in `/tmp` directory, an untrusted process may create files that are used by arbitrary system processes
+- Time-of-Check-to-Time-of-Use (TOCTTOU), i.e. race conditions
+  - Typically, a root process uses system call to determine if initiating user has permission to a particular file, e.g. `/tmp/X`.
+  - After access is authorized and before the file open, user may change the file `/tmp/X` to a symbolic link to a target file `/etc/shadow`.
+
+### Hazard: race conditions

content/CSE4303/_meta.js

@@ -18,4 +18,7 @@ export default {
     CSE4303_L11: "Introduction to Computer Security (Lecture 11)",
     CSE4303_L12: "Introduction to Computer Security (Lecture 12)",
     CSE4303_L13: "Introduction to Computer Security (Lecture 13)",
+    CSE4303_L14: "Introduction to Computer Security (Lecture 14)",
+    CSE4303_L15: "Introduction to Computer Security (Lecture 15)",
+    CSE4303_L16: "Introduction to Computer Security (Lecture 16)"
 }

content/Math4202/Exam_reviews/Math4202_E1.md

@@ -163,9 +163,9 @@ Define the product $f*g$ of $f$ and $g$ to be the map $h:[0,1]\to X$.
 
 #### Definition for equivalent classes of paths
 
-$\Pi_1(X,x)$ is the equivalent classes of paths starting and ending at $x$.
+$pi_1(X,x)$ is the equivalent classes of paths starting and ending at $x$.
 
-On $\Pi_1(X,x)$,, we define $\forall [f],[g],[f]*[g]=[f*g]$.
+On $pi_1(X,x)$,, we define $\forall [f],[g],[f]*[g]=[f*g]$.
 
 $$
 [f]\coloneqq \{f_i:[0,1]\to X|f_0(0)=f(0),f_i(1)=f(1)\}
@@ -191,6 +191,12 @@ $$
 
 ### Covering space
 
+#### Definition of partition into slice
+
+Let $p:E\to B$ be a continuous surjective map. The open set $U\subseteq B$ is said to be evenly covered by $p$ if it's inverse image $p^{-1}(U)$ can be written as the union of **disjoint open sets** $V_\alpha$ in $E$. Such that for each $\alpha$, the restriction of $p$ to $V_\alpha$ is a homeomorphism of $V_\alpha$ onto $U$.
+
+The collection of $\{V_\alpha\}$ is called a **partition** $p^{-1}(U)$ into slice.
+
 #### Definition of covering space
 
 Let $p:E\to B$ be a continuous surjective map.
@@ -225,3 +231,7 @@ Recall from previous lecture, we have unique lift for covering map.
 Let $p: E\to B$ be a covering map, and $e_0\in E$ and $p(e_0)=b_0$. Any path $f:I\to B$ beginning at $b_0$, has a unique lifting to a path starting at $e_0$.
 
 Back to the circle example, it means that there exists a unique correspondence between a loop starting at $(1,0)$ in $S^1$ and a path in $\mathbb{R}$ starting at $0$, ending in $\mathbb{Z}$.
+
+#### Theorem for induced homotopy for fundamental groups
+
+Suppose $f,g$ are two paths in $B$, and suppose $f$ and $g$ are path homotopy ($f(0)=g(0)=b_0$, and $f(1)=g(1)=b_1$, $b_0,b_1\in B$), then $\hat{f}:\pi_1(B,b_0)\to \pi_1(B,b_1)$ and $\hat{g}:\pi_1(B,b_0)\to \pi_1(B,b_1)$ are path homotopic.

content/Math4202/Math4202_L10.md

@@ -88,12 +88,12 @@ $\bar{f}_t=\bar{f}(1-ts)$ $s\in[\frac{1}{2},1]$.
 
 > [!CAUTION]
 >
-> Homeomorphism does not implies homotopy automatically.
+> Homeomorphism does not implies homotopy automatically. Homeomorphism doesn’t force a homotopy between that map and the identity (or between two given homeomorphisms).
 
 #### Definition for the fundamental group
 
 The fundamental group of $X$ at $x$ is defined to be
 
 $$
-(\Pi_1(X,x),*)
+(pi_1(X,x),*)
 $$

content/Math4202/Math4202_L11.md

@@ -8,9 +8,9 @@ The $*$ operation has the following properties:
 
 #### Properties for the path product operation
 
-Let $[f],[g]\in \Pi_1(X)$, for $[f]\in \Pi_1(X)$, let $s:\Pi_1(X)\to X, [f]\mapsto f(0)$ and $t:\Pi_1(X)\to X, [f]\mapsto f(1)$.
+Let $[f],[g]\in pi_1(X)$, for $[f]\in pi_1(X)$, let $s:pi_1(X)\to X, [f]\mapsto f(0)$ and $t:pi_1(X)\to X, [f]\mapsto f(1)$.
 
-Note that $t([f])=s([g])$, $[f]*[g]=[f*g]\in \Pi_1(X)$.
+Note that $t([f])=s([g])$, $[f]*[g]=[f*g]\in pi_1(X)$.
 
 This also satisfies the associativity. $([f]*[g])*[h]=[f]*([g]*[h])$.
 
@@ -51,33 +51,33 @@ Let $x_0\in X$. A path starting and ending at $x_0$ is called a loop based at $x
 The fundamental group of $X$ at $x$ is defined to be
 
 $$
-(\Pi_1(X,x),*)
+(pi_1(X,x),*)
 $$
 
-where $*$ is the product operation, and $\Pi_1(X,x)$ is the set o homotopy classes of loops in $X$ based at $x$.
+where $*$ is the product operation, and $pi_1(X,x)$ is the set o homotopy classes of loops in $X$ based at $x$.
 
 <details>
 <summary>Example of fundamental group</summary>
 
 Consider $X=[0,1]$, with subspace topology from standard topology in $\mathbb{R}$.
 
-$\Pi_1(X,0)=\{e\}$, (constant function at $0$) since we can build homotopy for all loops based at $0$ as follows $H(s,t)=(1-t)f(s)+t$.
+$pi_1(X,0)=\{e\}$, (constant function at $0$) since we can build homotopy for all loops based at $0$ as follows $H(s,t)=(1-t)f(s)+t$.
 
-And $\Pi_1(X,1)=\{e\}$, (constant function at $1$.)
+And $pi_1(X,1)=\{e\}$, (constant function at $1$.)
 
 ---
 
 Let $X=\{1,2\}$ with discrete topology.
 
-$\Pi_1(X,1)=\{e\}$, (constant function at $1$.)
+$pi_1(X,1)=\{e\}$, (constant function at $1$.)
 
-$\Pi_1(X,2)=\{e\}$, (constant function at $2$.)
+$pi_1(X,2)=\{e\}$, (constant function at $2$.)
 
 ---
 
 Let $X=S^1$ be the circle.
 
-$\Pi_1(X,1)=\mathbb{Z}$ (related to winding numbers, prove next week).
+$pi_1(X,1)=\mathbb{Z}$ (related to winding numbers, prove next week).
 
 </details>
 
@@ -85,7 +85,7 @@ A natural question is, will the fundamental group depends on the base point $x$?
 
 #### Definition for $\hat{\alpha}$
 
-Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$. $\alpha:[0,1]\to X$ such that $\alpha(0)=x_0$ and $\alpha(1)=x_1$. Define $\hat{\alpha}:\Pi_1(X,x_0)\to \Pi_1(X,x_1)$ as follows:
+Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$. $\alpha:[0,1]\to X$ such that $\alpha(0)=x_0$ and $\alpha(1)=x_1$. Define $\hat{\alpha}:pi_1(X,x_0)\to pi_1(X,x_1)$ as follows:
 
 $$
 \hat{\alpha}(\beta)=[\bar{\alpha}]*[f]*[\alpha]
@@ -93,12 +93,12 @@ $$
 
 #### $\hat{\alpha}$ is a group homomorphism
 
-$\hat{\alpha}$ is a group homomorphism between $(\Pi_1(X,x_0),*)$ and $(\Pi_1(X,x_1),*)$
+$\hat{\alpha}$ is a group homomorphism between $(pi_1(X,x_0),*)$ and $(pi_1(X,x_1),*)$
 
 <details>
 <summary>Proof</summary>
 
-Let $f,g\in \Pi_1(X,x_0)$, then $\hat{\alpha}(f*g)=\hat{\alpha}(f)\hat{\alpha}(g)$
+Let $f,g\in pi_1(X,x_0)$, then $\hat{\alpha}(f*g)=\hat{\alpha}(f)\hat{\alpha}(g)$
 
 $$
 \begin{aligned}
@@ -129,4 +129,4 @@ The other case is the same
 
 #### Corollary of fundamental group
 
-If $X$ is path-connected and $x_0,x_1\in X$, then $\Pi_1(X,x_0)$ is isomorphic to $\Pi_1(X,x_1)$.
+If $X$ is path-connected and $x_0,x_1\in X$, then $pi_1(X,x_0)$ is isomorphic to $pi_1(X,x_1)$.

content/Math4202/Math4202_L12.md

@@ -4,18 +4,18 @@
 
 ### Fundamental group
 
-Recall from last lecture, the $(\Pi_1(X,x_0),*)$ is a group, and for any two points $x_0,x_1\in X$, the group $(\Pi_1(X,x_0),*)$ is isomorphic to $(\Pi_1(X,x_1),*)$ if $x_0,x_1$ is path connected.
+Recall from last lecture, the $(pi_1(X,x_0),*)$ is a group, and for any two points $x_0,x_1\in X$, the group $(pi_1(X,x_0),*)$ is isomorphic to $(pi_1(X,x_1),*)$ if $x_0,x_1$ is path connected.
 
 > [!TIP]
 > 
-> How does the $\hat{\alpha}$ (isomorphism between $(\Pi_1(X,x_0),*)$ and $(\Pi_1(X,x_1),*)$) depend on the choice of $\alpha$ (path) we choose?
+> How does the $\hat{\alpha}$ (isomorphism between $(pi_1(X,x_0),*)$ and $(pi_1(X,x_1),*)$) depend on the choice of $\alpha$ (path) we choose?
 
 #### Definition of simply connected
 
 A space $X$ is simply connected if 
 
 - $X$ is [path-connected](https://notenextra.trance-0.com/Math4201/Math4201_L23/#definition-of-path-connected-space) ($\forall x_0,x_1\in X$, there exists a continuous function $\alpha:[0,1]\to X$ such that $\alpha(0)=x_0$ and $\alpha(1)=x_1$)
-- $\Pi_1(X,x_0)$ is the trivial group for some $x_0\in X$
+- $pi_1(X,x_0)$ is the trivial group for some $x_0\in X$
 
 <details>
 <summary>Example of simply connected space</summary>
@@ -59,7 +59,7 @@ $$
 
 #### Definition of group homomorphism induced by continuous map
 
-Let $h:(X,x_0)\to (Y,y_0)$ be a continuous map, define $h_*:\Pi_1(X,x_0)\to \Pi_1(Y,y_0)$ where $h(x_0)=y_0$. by $h_*([f])=[h\circ f]$.
+Let $h:(X,x_0)\to (Y,y_0)$ be a continuous map, define $h_*:pi_1(X,x_0)\to pi_1(Y,y_0)$ where $h(x_0)=y_0$. by $h_*([f])=[h\circ f]$.
 
 $h_*$ is called the group homomorphism induced by $h$ relative to $x_0$.
 
@@ -80,7 +80,7 @@ $$
 
 #### Theorem composite of group homomorphism
 
-If $h:(X,x_0)\to (Y,y_0)$ and $k:(Y,y_0)\to (Z,z_0)$ are continuous maps, then $k_* \circ h_*:\Pi_1(X,x_0)\to \Pi_1(Z,z_0)$ where $h_*:\Pi_1(X,x_0)\to \Pi_1(Y,y_0)$, $k_*:\Pi_1(Y,y_0)\to \Pi_1(Z,z_0)$,is a group homomorphism.
+If $h:(X,x_0)\to (Y,y_0)$ and $k:(Y,y_0)\to (Z,z_0)$ are continuous maps, then $k_* \circ h_*:pi_1(X,x_0)\to pi_1(Z,z_0)$ where $h_*:pi_1(X,x_0)\to pi_1(Y,y_0)$, $k_*:pi_1(Y,y_0)\to pi_1(Z,z_0)$,is a group homomorphism.
 
 <details>
 <summary>Proof</summary>
@@ -100,7 +100,7 @@ $$
 
 #### Corollary of composite of group homomorphism
 
-Let $\operatorname{id}:(X,x_0)\to (X,x_0)$ be the identity map. This induces $(\operatorname{id})_*:\Pi_1(X,x_0)\to \Pi_1(X,x_0)$.
+Let $\operatorname{id}:(X,x_0)\to (X,x_0)$ be the identity map. This induces $(\operatorname{id})_*:pi_1(X,x_0)\to pi_1(X,x_0)$.
 
 If $h$ is a homeomorphism with the inverse $k$, with
 
@@ -108,7 +108,7 @@ $$
 k_*\circ h_*=(k\circ h)_*=(\operatorname{id})_*=I=(\operatorname{id})_*=(h\circ k)_*
 $$
 
-This induced $h_*: \Pi_1(X,x_0)\to \Pi_1(Y,y_0)$ is an isomorphism.
+This induced $h_*: pi_1(X,x_0)\to pi_1(Y,y_0)$ is an isomorphism.
 
 #### Corollary for homotopy and group homomorphism
 

content/Math4202/Math4202_L21.md

@@ -0,0 +1,67 @@
+# Math4202 Topology II (Lecture 21)
+
+## Algebraic Topology
+
+### Application of fundamental groups
+
+Recall from last Friday, $j:S^1\to \mathbb{R}^2-\{0\}$ is not null homotopic
+
+#### Hairy ball theorem
+
+Given a non-vanishing vector field on $B^2=\{(x,y)\in\mathbb{R}^2:x^2+y^2\leq 1\}$, ($v:B^2\to \mathbb{R}^2$ continuous and $v(x,y)\neq 0$ for all $(x,y)\in B^2$) there exists a point of $S^1$ where the vector field points directly outward, and a point of $S^1$ where the vector field points directly inward.
+
+<details>
+<summary>Proof</summary>
+
+By our assumption, then $v:B^2\to \mathbb{R}^2-\{0\}$ is a continuous vector field on $B^2$.
+
+$v|_{S^1}:S^1\to \mathbb{R}^2-\{0\}$ is null homotopic.
+
+We prove by contradiction.
+
+Suppose $v:B^2\to \mathbb{R}^2-\{0\}$ and $v|_{S^1}:S^1\to \mathbb{R}^2-\{0\}$ is everywhere outward. (for everywhere inward, consider $-v$ must be everywhere outward)
+
+Because $v|_{S^1}$ extends continuously to $B^2$, then $v|_{S^1}:B^2\to \mathbb{R}^2-\{0\}$ is null homotopic.
+
+We construct a homotopy for functions between $v|_{S^1}$ and $j$. (Recall $j:S^1\to \mathbb{R}^2-\{0\}$ is not null homotopic)
+
+Define $H:S^1\times I\to \mathbb{R}^2-\{0\}$ by affine combination
+
+$$
+H((x,y),t)=(1-t)v(x,y)+tj(x,y)
+$$
+
+we also need to show that $H$ is non zero.
+
+Since $v$ is everywhere outward, $v(x,y)\cdot j(x,y)$ is positive for all $(x,y)\in S^1$.
+
+$H((x,y),t)\cdot j(x,y)=(1-t)v(x,y)\cdot j(x,y)+tj(x,y)\cdot j(x,y)=(1-t)(v(x,y)\cdot j(x,y))+t$
+
+which is positive for all $t\in I$, therefore $H$ is non zero.
+
+So $H$ is a homotopy between $v|_{S^1}$ and $j$.
+</details>
+
+#### Corollary of the hairy ball theorem
+
+$\forall v:B^2\to \mathbb{R}^2$, if on $S^1$, $v$ is everywhere outward/inward, there is $(x,y)\in B^2$ such that $v(x,y)=0$.
+
+#### Brouwer's fixed point theorem
+
+If $f:B^2\to B^2$ is continuous, then there exists a point $x\in B^2$ such that $f(x)=x$.
+
+<details>
+<summary>Proof</summary>
+
+We proceed by contradiction again.
+
+Suppose $f$ has no fixed point, $f(x)-x\neq 0$ for all $x\in B^2$.
+
+Now we consider the map $v:B^2\to \mathbb{R}^2$ defined by $v(x,y)=f(x)-x$, this function is continuous since $f$ is continuous.
+
+$forall x\in S^1$, $v(x)\cdot x=f(x)\cdot x-x\cdot x=f(x)\cdot x-1$.
+
+Recall the cauchy schwartz theorem, $|f(x)\cdot x|\leq \|f(x)\|\cdot\|x\|\leq 1$, note that $f(x)\neq 0$ for all $x\in B^2$, $v(x)\cdot x<0$. This means that all $v(x)$ points inward.
+
+This is a contradiction to the hairy ball theorem, so $f$ has a fixed point.
+</details>

content/Math4202/Math4202_L22.md

@@ -0,0 +1,45 @@
+# Math4202 Topology II (Lecture 22)
+
+## Final reading, report, presentation
+
+- Mar 30: Reading topic send email or discuss in OH.
+- Apr 3: Finalize the plan.
+- Apr 22,24: Last two lectures: 10 minutes to present.
+- Final: type a short report, 2-5 pages.
+
+## Algebraic topology
+
+### Fundamental theorem of Algebra
+
+For arbitrary polynomial $f(z)=\sum_{i=0}^n a_i x^i$. Are there roots in $\mathbb{C}$?
+
+Consider $f(z)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0$ is a continuous map from $\mathbb{C}\to\mathbb{C}$.
+
+If $f(z_0)=0$, then $z_0$ is a root.
+
+By contradiction, Then $f:\mathbb{C}\to\mathbb{C}-\{0\}\cong \mathbb{R}^2-\{(0,0)\}$.
+
+#### Theorem for existence of n roots
+
+A polynomial equation $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0=0$ of degree $>0$ with complex coefficients has at least one complex root.
+
+There are $n$ roots by induction.
+
+#### Lemma
+
+If $g:S^1\to \mathbb{R}^2-\{(0,0)\}$ is the map $g(z)=z^n$, then $g$ is not nulhomotopic. $n\neq 0$, $n\in \mathbb{Z}$.
+
+> Recall that we proved that $g(z)=z$ is not nulhomotopic.
+
+Consider $k:S^1\to S^1$ by $k(z)=z^n$. $k$ is continuous, $k_*:\pi_1(S^1,1)\to \pi_1(S^1,1)$.
+
+Where $\pi_1(S^1,1)\cong \mathbb{Z}$.
+
+$k_*(n)=nk_*(1)$.
+
+Recall that the path in the loop $p:I\to S^1$ where $p:t\mapsto e^{2\pi it}$.
+
+$k_*(p)=[k(p(t))]$, where $n=\tilde{k\circ p}(1)$.
+
+$k_*$ is injective.
+

content/Math4202/Math4202_L23.md

@@ -0,0 +1,142 @@
+# Math4202 Topology II (Lecture 23)
+
+## Algebraic Topology
+
+### Fundamental Theorem of Algebra
+
+Recall the lemma $g:S^1\to \mathbb{R}-\{0\}$ is not nulhomotopic.
+
+$g=h\circ k$ where $k:S^1\to S^1$ by $z\mapsto z^n$, $k_*:\pi_1(S^1)\to \pi_1(S^1)$ is injective. (consider the multiplication of integer is injective)
+
+and $h:S^1\to \mathbb{R}-\{0\}$ where $z\mapsto z$. $h_*:\pi_1(S^1)\to \pi_1(\mathbb{R}-\{0\})$ is injective. (inclusion map is injective)
+
+Therefore $g_*:\pi_1(S^1)\to \pi_1(\mathbb{R}-\{0\})$ is injective, therefore $g$ cannot be nulhomotopic. (nulhomotopic cannot be injective)
+
+#### Theorem
+
+Consider $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0=0$ of degree $>0$.
+
+<details>
+<summary>Proof: part 1</summary>
+
+Step 1: if $|a_{n-1}|+|a_{n-2}|+\cdots+|a_0|<1$, then $x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0=0$ has a root in the unit disk $B^2$.
+
+We proceed by contradiction, suppose there is no root in $B^2$.
+
+Consider $f(x)=x^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0$.
+
+$f|_{B^2}$ is a continuous map from $B^2\to \mathbb{R}^2-\{0\}$.
+
+$f|_{S^1=\partial B^2}:S^1\to \mathbb{R}-\{0\}$ **is nulhomotopic**.
+
+> Recall that: Any map $g:S^1\to Y$ is nulhomotopic whenever it extends to a continuous map $G:B^2\to Y$.
+
+Construct a homotopy between $f|_{S^1}$ and $g$
+
+$$
+H(x,t):S^1\to \mathbb{R}-\{0\}\quad x^n+t(a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0)
+$$
+
+Observer on $S^1$, $\|x^n\|=1,\forall n\in \mathbb{N}$.
+
+$$
+\begin{aligned}
+\|t(a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0)\|&=t\|a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+\cdots+a_0\|\\
+&\leq 1(\|a_{n-1}x^{n-1}\|+\|a_{n-2}x^{n-2}\|+\cdots+\|a_0\|)\\
+&=\|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|\\
+&<1
+\end{aligned}
+$$
+
+Therefore $H(s,t)>0\forall 0<t<1$. is a well-defined homotopy between $f|_{S^1}$ and $g$.
+
+Therefore $f_*=g_*$ is injective, $f$ is not nulhomotopic. This contradicts our previous assumption that $f$ is nulhomotopic.
+
+Therefore $f$ must have a root in $B^2$.
+
+</details>
+
+<details>
+<summary>Proof: part 2</summary>
+
+If $\|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|< R$ has a root in the disk $B^2_R$. (and $R\geq 1$, otherwise follows part 1)
+
+Consider $\tilde{f}(x)=f(Rx)$.
+$$
+\begin{aligned}
+\tilde{f}(x)
+=f(Rx)&=(Rx)^n+a_{n-1}(Rx)^{n-1}+a_{n-2}(Rx)^{n-2}+\cdots+a_0\\
+&=R^n\left(x^n+\frac{a_{n-1}}{R}x^{n-1}+\frac{a_{n-2}}{R^2}x^{n-2}+\cdots+\frac{a_0}{R^n}\right)
+\end{aligned}
+$$
+
+$$
+\begin{aligned}
+\left\|\frac{a_{n-1}}{R}\right\|+\left\|\frac{a_{n-2}}{R^2}\right\|+\cdots+\left\|\frac{a_0}{R^n}\right\|&=\frac{1}{R}\|a_{n-1}\|+\frac{1}{R^2}\|a_{n-2}\|+\cdots+\frac{1}{R^n}\|a_0\|\\
+&<\frac{1}{R}\left(\|a_{n-1}\|+\|a_{n-2}\|+\cdots+\|a_0\|\right)\\
+&<\frac{1}{R}<1
+\end{aligned}
+$$
+
+By Step 1, $\tilde{f}$ must have a root $z_0$ inside the unit disk.
+
+$f(Rz_0)=\tilde{f}(z_0)=0$.
+
+So $f$ has a root $Rz_0$ in $B^2_R$.
+
+</details>
+
+### Deformation Retracts and Homotopy Type
+
+Recall previous section, $h:S^1\to \mathbb{R}-\{0\}$ gives $h_*:\pi_1(S^1,1)\to \pi_1(\mathbb{R}-\{0\},0)$ is injective.
+
+For this section, we will show that $h_*$ is an isomorphism.
+
+#### Lemma for equality of homomorphism
+
+Let $h,k: (X,x_0)\to (Y,y_0)$ be continuous maps. If $h$ and $k$ are homotopic, and if **the image of $x_0$ under the homotopy remains $y_0$**. The homomorphism $h_*$ and $k_*$ from $\pi_1(X,x_0)$ to $\pi_1(Y,y_0)$ are equal.
+
+<details>
+<summary>Proof</summary>
+
+Let $H:X\times I\to Y$ be a homotopy from $h$ to $k$ such that
+$$
+H(x,0)=h(x), \qquad H(x,1)=k(x), \qquad H(x_0,t)=y_0 \text{ for all } t\in I.
+$$
+
+To show $h_*=k_*$, let $[f]\in \pi_1(X,x_0)$ be arbitrary, where
+$f:I\to X$ is a loop based at $x_0$, so $f(0)=f(1)=x_0$.
+
+Define
+$$
+F:I\times I\to Y,\qquad F(s,t)=H(f(s),t).
+$$
+Since $H$ and $f$ are continuous, $F$ is continuous. For each fixed $t\in I$, the map
+$$
+s\mapsto F(s,t)=H(f(s),t)
+$$
+is a loop based at $y_0$, because
+$$
+F(0,t)=H(f(0),t)=H(x_0,t)=y_0
+\quad\text{and}\quad
+F(1,t)=H(f(1),t)=H(x_0,t)=y_0.
+$$
+Thus $F$ is a based homotopy between the loops $h\circ f$ and $k\circ f$, since
+$$
+F(s,0)=H(f(s),0)=h(f(s))=(h\circ f)(s),
+$$
+and
+$$
+F(s,1)=H(f(s),1)=k(f(s))=(k\circ f)(s).
+$$
+
+Therefore $h\circ f$ and $k\circ f$ represent the same element of $\pi_1(Y,y_0)$, so
+$$
+[h\circ f]=[k\circ f].
+$$
+Hence
+$$
+h_*([f])=[h\circ f]=[k\circ f]=k_*([f]).
+$$
+Since $[f]$ was arbitrary, it follows that $h_*=k_*$.
+</details>

content/Math4202/Math4202_L24.md

@@ -0,0 +1,84 @@
+# Math4202 Topology II (Lecture 24)
+
+## Algebraic Topology
+
+### Deformation Retracts and Homotopy Type
+
+Recall from last lecture, let $h,k:(X,x_0)\to (Y,y_0)$ be continuous maps. If there exists a homotopy of $h,y$ such that $H:X\times I\to Y$ that $H(x_0,t)=y_0$.
+
+Then $h_*=k_*:\pi_1(X,x_0)\to \pi_1(Y,y_0)$.
+
+We can prove this by showing that all the loop $f:I\to X$ based at $x_0$, $h_*([f])=k_*([f])$.
+
+That is $[h\circ f]=[k\circ f]$.
+
+This is a function $I\times I \to Y$ by $(s,t)\mapsto H(f(s),t)$.
+
+We need to show that this is a homotopy between $h\circ f$ and $k\circ f$.
+
+#### Theorem
+
+The Inclusion map $j:S^n\to \mathbb{R}^n-\{0\}$ induces on isomorphism of fundamental groups
+
+$$
+j_*:\pi_1(S^n)\to \pi_1(\mathbb{R}^n-\{0\})
+$$
+
+The function is injective.
+
+> Recall we showed that $S^1\to \mathbb{R}-\{0\}$ is injective by $x\mapsto \frac{x}{|x|}$.
+
+We want to show that $j_*\circ r_*=id_{\pi_1(S^n)}\quad r_*\circ j_*=id_{\pi_1(\mathbb{R}^n-\{0\})}$, then $r_*$, $j_*$ are isomorphism.
+
+<details>
+<summary>Proof</summary>
+
+**Homotopy is well defined**.
+
+Consider $H:(\mathbb{R}^n-\{0\})\times I\to \mathbb{R}^n-\{0\}$.
+
+Given $(x,t)\mapsto tx+(1-t)\frac{x}{\|x\|}$.
+
+Note that $(t-\frac{1-t}{\|x\|})x=0\implies t=0\land t=1$.
+
+So this map is well defined.
+
+**Base point is fixed**.
+
+On point $(1,0)$ (or anything on the sphere), $H(x,0)=x$.
+
+</details>
+
+#### Definition of deformation retract
+
+Let $A$ be a subspace of $X$, we say that $A$ is a deformation retract of $X$ if the identity map of $X$ is homotopic to a map that carries all $X$ to $A$ such that each point of $A$ remains fixed during the homotopy.
+
+Equivalently, there exists a homotopy $H:X\times I\to X$ such that:
+
+- $H(x,0)=x$ forall $x\in X$
+- $H(a,t)=a$ for all $a\in A$, $t\in I$
+- $H(x,1)\in A$ for all $x\in X$
+
+Equivalently,
+
+$r:H(x,1):X\to A$ is a retract.
+
+If we let $j:A\to X$ be the inclusion map, then $r\circ j=id_A$, and $j\circ r\sim id_X$ (with $A$ fixed.)
+
+<details>
+<summary>Example of deformation retract</summary>
+
+$S^1$ is a deformation retract of $\mathbb{R}^2-\{0\}$ 
+
+---
+
+Consider $\mathbb{R}^2-p=q$, the doubly punctured plane. "The figure 8" space is the deformation retract.
+
+![Retraction of doubly punctured plane](https://notenextra.trance-0.com/Math4202/Retraction_of_doubly_punctured_plane.jpg)
+
+</details>
+
+#### Theorem for Deformation Retract
+
+If $A$ is a deformation retract of $X$, then $A$ and $X$ have the same fundamental group.
+

content/Math4202/Math4202_L25.md

@@ -0,0 +1,89 @@
+# Math4202 Topology II (Lecture 25)
+
+## Algebraic Topology
+
+### Deformation Retracts and Homotopy Type
+
+Recall from last lecture, Let $A\subseteq X$, if there exists a continuous map (deformation retraction) $H:X\times I\to X$ such that
+
+- $H(x,0)=x$ for all $x\in X$
+- $H(x,1)\in A$ for all $x\in X$
+- $H(a,t)=a$ for all $a\in A$, $t\in I$
+
+then the inclusion map$\pi_1(A,a)\to \pi_1(X,a)$ is an isomorphism.
+
+<details>
+<summary>Example for more deformation retract</summary>
+
+Let $X=\mathbb{R}^3-\{0,(0,0,1)\}$.
+
+Then the two sphere with one point intersect is a deformation retract of $X$.
+
+---
+
+Let $X$ be $\mathbb{R}^3-\{(t,0,0)\mid t\in \mathbb{R}\}$, then the cyclinder is a deformation retract of $X$.
+
+</details>
+
+#### Definition of homotopy equivalence
+
+Let $f:X\to Y$ and $g:Y\to X$ be a continuous maps.
+
+Suppose 
+
+- the map $g\circ f:X\to X$ is homotopic to the identity map $\operatorname{id}_X$.
+- the map $f\circ g:Y\to Y$ is homotopic to the identity map $\operatorname{id}_Y$.
+
+Then $f$ and $g$ are **homotopy equivalences**, and each is said to be the **homotopy inverse** of the other.
+
+$X$ and $Y$ are said to be **homotopy equivalent**.
+
+<details>
+<summary>Example</summary>
+
+Consider the punctured torus $X=S^1\times S^1-\{(0,0)\}$.
+
+Then we can do deformation retract of the glued square space to boundary of the square.
+
+After glueing, we left with the figure 8 space.
+
+Then $X$ is homotopy equivalent to the figure 8 space.
+
+</details>
+
+Recall the lemma, [Lemma for equality of homomorphism](https://notenextra.trance-0.com/Math4202/Math4202_L23/#lemma-for-equality-of-homomorphism)
+
+Let $f:X\to Y$ and $g:X\to Y$, with homotopy $H:X\times I\to Y$, such that
+
+- $H(x,0)=f(x)$ for all $x\in X$
+- $H(x,1)=g(x)$ for all $x\in X$
+- $H(x,t)=y_0$ for all $t\in I$, and $y_0\in Y$ is fixed.
+
+Then $f_*=g_*:\pi_1(X,x_0)\to \pi_1(Y,y_0)$ is an isomorphism.
+
+We wan to know if it is safe to remove the assumption that $y_0$ is fixed.
+
+<details>
+<summary>Idea of Proof</summary>
+
+Let $k$ be any loop in $\pi_1(X,x_0)$.
+
+We can correlate the two fundamental group $f\cric k$ by the function $\alpha:I\to Y$, and $\hat{\alpha}:\pi_1(Y,y_0)\to \pi_1(Y,y_1)$. (suppose $f(x_0)=y_0, g(x_0)=y_1$), it is sufficient to show that
+
+$$
+f\circ k\simeq \alpha *(g\circ k)*\bar{\alpha}
+$$
+
+</details>
+
+#### Lemma of homotopy equivalence
+
+Let $f,g:X\to Y$ be continuous maps. let $f(x_0)=y_0$ and $g(x_0)=y_1$. If $f$ and $g$ are homotopic, then there is a path $\alpha:I\to Y$ such that $\alpha(0)=y_0$ and $\alpha(1)=y_1$.
+
+Defined as the restriction of the homotopy to $\{x_0\}\times I$, satisfying $\hat{\alpha}\circ f_*=g_*$.
+
+Imagine a triangle here:
+
+- $\pi_1(X,x_0)\to \pi_1(Y,y_0)$ by $f_*$
+- $\pi_1(Y,y_0)\to \pi_1(Y,y_1)$ by $\hat{\alpha}$
+- $\pi_1(Y,y_1)\to \pi_1(X,x_0)$ by $g_*$

content/Math4202/Math4202_L26.md

@@ -0,0 +1,90 @@
+# Math4202 Topology II (Lecture 26)
+
+## Algebraic Topology
+
+### Deformation Retracts and Homotopy Type
+
+#### Lemma of homotopy equivalence
+
+Let $f,g:X\to Y$ be continuous maps. let
+
+$$
+f_*=\pi_1(X,f(x_0))\quad\text{and}\quad g_*=\pi_1(Y,g(x_0))
+$$
+
+And $H:X\times I\to Y$ is a homotopy from $f$ to $g$ with a path $H(x_0,t)=\alpha(t)$ for all $t\in I$.
+
+Then $\hat{\alpha}\circ f_*=[\bar{\alpha}*(f\circ \gamma)*\alpha]=[g\circ \gamma]=g_*$. where $\gamma$ is a loop in $X$ based at $x_0$.
+
+<details>
+<summary>Proof</summary>
+$I\times I\xrightarrow{\gamma_{id}} X\times I\xrightarrow{H} Y$
+
+- $I\times \{0\}\mapsto f\circ\gamma$
+- $I\times \{1\}\mapsto g\circ\gamma$
+- $\{0\}\times I\mapsto \alpha$
+- $\{1\}\times I\mapsto \alpha$
+
+As $I\times I$ is convex, $I\times \{0\}\simeq (\{0\}\times I)*(I\times \{1\})*(\{1\}\times I)$.
+
+</details>
+
+#### Corollary for homotopic continuous maps
+
+Let $h,k$ be homotopic continuous maps. And let $h(x_0)=y_0,k(x_0)=y_1$. If $h_*:\pi_1(X,x_0)\to \pi_1(Y,y_0)$ is injective, then $k_*:\pi_1(X,x_0)\to \pi_1(Y,y_1)$ is injective.
+
+<details>
+<summary>Proof</summary>
+
+$\hat{\alpha}$ is an isomorphism of $\pi_1(Y,y_0)$ to $\pi_1(Y,y_1)$.
+
+</details>
+
+#### Corollary for nulhomotopic maps
+
+Let $h:X\to Y$ be nulhomotopic. Then $h_*:\pi_1(X,x_0)\to \pi_1(Y,h(x_0))$ is a trivial group homomorphism (mapping to the constant map on $h(x_0)$).
+
+#### Theorem for fundamental group isomorphism by homotopy equivalence
+
+Let $f:X\to Y$ be a continuous map. Let $f(x_0)=y_0$. If $f$ is a [homotopy equivalence](https://notenextra.trance-0.com/Math4202/Math4202_L25/#definition-of-homotopy-equivalence) ($\exists g:Y\to X$ such that $fg\simeq id_X$, $gf\simeq id_Y$), then
+
+$$
+f_*:\pi_1(X,x_0)\to \pi_1(Y,y_0)
+$$
+is an isomorphism.
+
+<details>
+<summary>Proof</summary>
+
+Let $g:Y\to X$ be the homotopy inverse of $f$. 
+
+Then,
+
+$f_*\circ g_*=\alpha \circ id_{\pi_1(Y,y_0)}=\alpha$
+
+And $g_*\circ f_*=\bar{\alpha}\circ id_{\pi_1(X,x_0)}=\bar{\alpha}$
+
+So $f_*\circ (g_*\circ \hat{\alpha}^-1)=id_{\pi_1(X,x_0)}$
+
+And $g_*\circ (f_*\circ \hat{\alpha}^-1)=id_{\pi_1(Y,y_0)}$
+
+So $f_*$ is an isomorphism (have left and right inverse).
+</details>
+
+### Fundamental group of higher dimensional sphere
+
+$\pi_1(S^n,x_0)=\{e\}$ for $n\geq 2$.
+
+We can decompose the sphere to the union of two hemisphere and compute $\pi_1(S^n_+,x_0)=\pi_1(S^n_-,x_0)=\{e\}$
+
+But for $n\geq 2$, $S^n_+\cap S^n_-=S^{n-1}$, where $S^1_+\cap S^1_-$ is two disjoint points.
+
+#### Theorem for "gluing" fundamental group
+
+Suppose $X=U\cup V$, where $U$ and $V$ are open subsets of $X$. Suppose that $U\cap V$ is path connected, and $x\in U\cap V$. Let $i,j$ be the inclusion maps of $U$ and $V$ into $X$, the images of the induced homomorphisms
+
+$$
+i_*:\pi_1(U,x_0)\to \pi_1(X,x_0)\quad j_*:\pi_1(V,x_0)\to \pi_1(X,x_0)
+$$
+
+The image of the two map generate $\pi_1(X,x_0)$.

content/Math4202/Math4202_L27.md

@@ -0,0 +1,69 @@
+# Math4202 Topology II (Lecture 27)
+
+## Algebraic Topology
+
+### Fundamental Groups for Higher Dimensional Sphere
+
+#### Theorem for "gluing" fundamental group
+
+Suppose $X=U\cup V$, where $U$ and $V$ are open subsets of $X$. Suppose that $U\cap V$ is path connected, and $x\in U\cap V$. Let $i,j$ be the inclusion maps of $U$ and $V$ into $X$, the images of the induced homomorphisms
+
+$$
+i_*:\pi_1(U,x_0)\to \pi_1(X,x_0)\quad j_*:\pi_1(V,x_0)\to \pi_1(X,x_0)
+$$
+
+The image of the two map generate $\pi_1(X,x_0)$.
+
+$G$ is a group, and let $S\subseteq G$, where $G$ is generated by $S$, if $\forall g\in G$, $\exists s_1,s_2,\ldots,s_n\in S$ such that $g=s_1s_2\ldots s_n\in G$. (We can write $G$ as a word of elements in $S$.)
+
+<details>
+<summary>Proof</summary>
+
+Let $f$ be a loop in $X$, $f\simeq g_1*g_2*\ldots*g_n$, where $g_i$ is a loop in $U$ or $V$.
+
+For example, consider the function, $f=f_1*f_2*f_3*f_4$, where $f_1\in S_+$, $f_2\in S_-$, $f_3\in S_+$, $f_4\in S_-$.
+
+Take the functions $\bar{\alpha_1}*\alpha_1\simeq e_{x_1}$ where $x_1$ is the intersecting point on $f_1$ and $f_2$.
+
+Therefore,
+
+$$
+\begin{aligned}
+f&=f_1*f_2*f_3*f_4\\
+&(f_1*\bar{\alpha})*(\alpha_1*f_2*\bar{\alpha_2})*(\alpha_2*f_3*\bar{\alpha_3})*(\alpha_4*f_4)
+\end{aligned}
+$$
+
+This decompose $f$ into a word of elements in either $S_+$ or $S_-$.
+
+---
+
+Note that $f$ is a continuous function $I\to X$, for $t\in I$, $\exists I_t$ being a small neighborhood of $t$ such that $f(I_t)\subseteq U$ or $f(I_t)\subseteq V$.
+
+Since $U_{t\in I}I_t=I$, then $\{I_t\}_{t\in I}$ is an open cover of $I$.
+
+By compactness of $I$, there is a finite subcover $\{I_{t_1},\ldots,I_{t_n}\}$.
+
+Therefore, we can create a partition of $I$ into $[s_i,s_{i+1}]\subseteq I_{t_k}$ for some $k$.
+
+Then with the definition of $I_{t_k}$, $f([s_i,s_{i+1}])\subseteq U$ or $V$.
+
+Then we can connect $x_0$ to $f(s_i)$ with a path $\alpha_i\subseteq U\cap V$.
+
+$$
+\begin{aligned}
+f&=f|_{[s_0,s_1]}*f|_{[s_1,s_2]}*\ldots**f|_{[s_{n-1},s_n]}\\
+&\simeq f|_{[s_0,s_1]}*(\bar{\alpha_1}*\alpha_1)*f|_{[s_1,s_2]}*(\bar{\alpha_2}*\alpha_2)*\ldots*f|_{[s_{n-1},s_n]}*(\bar{\alpha_n}*\alpha_n
+)\\
+&=(f|_{[s_0,s_1]}*\bar{\alpha_1})*(\alpha_1*f|_{[s_1,s_2]}*\bar{\alpha_2})*\ldots*(\alpha_{n-1}*f|_{[s_{n-1},s_n]}*\bar{\alpha_n})\\
+&=g_1*g_2*\ldots*g_n
+\end{aligned}
+$$
+
+</details>
+
+#### Corollary in higher dimensional sphere
+
+Since $S^n_+$ and $S^n_-$ are homeomorphic to open balls $B^n$, then $\pi_1(S^n_+,x_0)=\pi_1(S^n_-,x_0)=\pi_1(B^n,x_0)=\{e\}$ for $n\geq 2$.
+
+> Preview: Van Kampen Theorem

content/Math4202/Math4202_L4.md

@@ -17,6 +17,7 @@ An $m$-dimensional **manifold** is a topological space  $X$ that is
 > Try to find some example that satisfies some of the properties above but not a manifold.
 
 1. Non-Hausdorff
+   - Real line with two origin, as discussed in homework problem
 2. Non-countable basis
    - Consider $\mathbb{R}^\delta$ where the set is $\mathbb{R}$ with discrete topology. The basis must include all singleton sets in $\mathbb{R}$ therefore $\mathbb{R}^\delta$ is not second countable.
 3. Non-local euclidean

content/Math4202/Math4202_L9.md

@@ -7,7 +7,7 @@
 Consider the space of paths up to homotopy equivalence.
 
 $$
-\operatorname{Path}/\simeq_p(X) =\Pi_1(X)
+\operatorname{Path}/\simeq_p(X) =pi_1(X)
 $$
 
 We want to impose some group structure on $\operatorname{Path}/\simeq_p(X)$.
@@ -33,9 +33,9 @@ Define the product $f*g$ of $f$ and $g$ to be the map $h:[0,1]\to X$.
 
 #### Definition for equivalent classes of paths
 
-$\Pi_1(X,x)$ is the equivalent classes of paths starting and ending at $x$.
+$pi_1(X,x)$ is the equivalent classes of paths starting and ending at $x$.
 
-On $\Pi_1(X,x)$,, we define $\forall [f],[g],[f]*[g]=[f*g]$.
+On $pi_1(X,x)$,, we define $\forall [f],[g],[f]*[g]=[f*g]$.
 
 $$
 [f]\coloneqq \{f_i:[0,1]\to X|f_0(0)=f(0),f_i(1)=f(1)\}
@@ -141,5 +141,5 @@ Continue next time.
 The fundamental group of $X$ at $x$ is defined to be
 
 $$
-(\Pi_1(X,x),*)
+(pi_1(X,x),*)
 $$

content/Math4202/_meta.js

@@ -1,3 +1,5 @@
+import { MathJax } from "nextra/components";
+
 export default {
     index: "Course Description",
     "---":{
@@ -24,4 +26,11 @@ export default {
     Math4202_L18: "Topology II (Lecture 18)",
     Math4202_L19: "Topology II (Lecture 19)",
     Math4202_L20: "Topology II (Lecture 20)",
+    Math4202_L21: "Topology II (Lecture 21)",
+    Math4202_L22: "Topology II (Lecture 22)",
+    Math4202_L23: "Topology II (Lecture 23)",
+    Math4202_L24: "Topology II (Lecture 24)",
+    Math4202_L25: "Topology II (Lecture 25)",
+    Math4202_L26: "Topology II (Lecture 26)",
+    Math4202_L27: "Topology II (Lecture 27)",
 }

content/Math4302/Math4302_L22.md

@@ -0,0 +1,109 @@
+# Math4302 Modern Algebra (Lecture 22)
+
+## Groups
+
+### Group acting on a set
+
+Let $X$ be a $G$-set, recall that the orbit of $x\in X$ is $\{g\cdot x|g\in G\}$.
+
+#### The orbit-stabilizer theorem
+
+For any $x\in X$, ,$G_x=\{g\in G|g\cdot x=x\}\leq G$.
+
+Let $(G:G_x)$ denote the index of $G_x$ in $G$, then $(G:G_x)=\frac{|G|}{|G_x|}$, which equals to the number of left cosets of $G_x$ in $G$.
+
+<details>
+<summary>Proof</summary>
+
+Define $\alpha:gG_x\mapsto g\cdot x$.
+
+$\alpha$ is well-defined and injective.
+
+$$
+gG_x=g'G_x\iff g^{-1}g'\in G_x\iff (g^{-1}g')\cdot x=x\iff g^{-1}\cdot(g'\cdot x)=x\iff g'\cdot x=g\cdot x
+$$
+
+$\alpha$ is surjective, therefore $\alpha$ is a bijection.
+
+</details>
+
+<details>
+<summary>Example</summary>
+
+Number of elements in the orbit of $x$ is $1$ if and only if $g\cdot x=x$ for all $g\in G$.
+
+if and only if $G_x=G$.
+
+</details>
+
+#### Theorem for orbit with prime power groups
+
+Suppose $X$ is a $G$-set, and $|G|=p^n$ for some prime $p$. Let $X_G$ be the set of all elements in $X$ whose orbit has size $1$. (Recall the orbit divides $X$ into disjoint partitions.) Then $|X|\equiv |X_G|\mod p$.
+
+<details>
+<summary>Examples</summary>
+
+Let $G=D_4$ acting on $\{1,2,3,4\}=X$.
+
+$X_G=\emptyset$ since there is no element whose orbit has size $1$.
+
+---
+
+Let $G=\mathbb{Z}_{11}$ acting on a set with $|X|=20$ if the action is not trivial, then what is $|X_G|$?
+
+Using the theorem we have $|X_G|\equiv 20\mod 11=9$. Therefore $|X_G|=9$ or $20$, but the action is not trivial, $|X_G|=9$.
+
+An instance for such $X=\mathbb{Z}_{11}\sqcup\{x_1,x_2,\ldots,x_9\}$, where $\mathbb{Z}_{11}$ acts on $\{x_1,x_2,\ldots,x_9\}$ trivially. and $\mathbb{Z}_{11}$ acts on $x_1$ with addition.
+
+</details>
+
+<details>
+<summary>Proof</summary>
+
+If $x\in X$ such that $|Gx|\geq 2$, then $\frac{|G|}{|G_x|}=|Gx|\geq 2$.
+
+So $|G|=|G_x||Gx|\implies |Gx|$ divides $|G|$.
+
+So $|Gx|=p^m$ for some $m\geq 1$.
+
+Note that $X$ is the union of subset of elements with orbit of size $1$, and distinct orbits of sizes $\geq 2$. (each of them has size positive power of $p$)
+
+So $p|(|X|-|X_G|)$.
+
+this implies that $|X_G|\equiv |X_G|\mod p$.
+
+</details>
+
+#### Corollary: Cauchy's theorem
+
+If $p$ is prime and $p|(|G|)$, then $G$ has a subgroup of order $p$.
+
+> This does not hold when $p$ is not prime.
+>
+> Consider $A_4$ with order $12$, and $A_4$ has no subgroup of order $6$.
+
+<details>
+<summary>Proof</summary>
+
+It is enough to show, there is $a\in G$ which has order $p$: $\{e,a,a^2,\ldots,a^{p-1}\}\leq G$.
+
+Let $X=\{(g_1,g_2,\ldots,g_p)|g_i\in G,g_1g_2g_3\ldots g_p=e\}$.
+
+Then $|X|=|G|^{p-1}$ since $g_p$ is determined uniquely by $g_p=(g_1,g_2,\ldots,g_{p-1})^{-1}$.
+
+Therefore we can define $\mathbb{Z}_p$ acts on $X$ by shifting.
+
+$i\in \mathbb{Z}_p$ $i\cdot (g_1,g_2,\ldots,g_p)=(g_{i+1},g_2,\ldots,g_p,g_1,\ldots,g_i)$.
+
+$X$ is a $\mathbb{Z}_p$-set.
+
+- $0\cdot (g_1,g_2,\ldots,g_p)=(g_1,g_2,\ldots,g_p)$.
+- $j\cdot (i\cdot (g_1,g_2,\ldots,g_p))=(i+j)\cdot (g_1,g_2,\ldots,g_p)$.
+
+By the previous theorem, $|X|\equiv |X_G|\mod p$.
+
+Since $p$ divides $|G|^{p-1}$, $p$ also divides $|X_G|$. Therefore $(e,e,e,\ldots,e)\in X_G$. Therefore $|X_G|\geq 1$.
+
+So $|X_G|\geq 2$, we have $(a,a,\ldots,a)\in X_G$, $a\neq e$, but $a^p=e$, so $ord(a)=p$.
+
+</details>

content/Math4302/Math4302_L23.md

@@ -0,0 +1,133 @@
+# Math4302 Modern Algebra (Lecture 23)
+
+## Group
+
+### Group acting on a set
+
+#### Theorem for the orbit of a set with prime power group
+
+Suppose $X$ is a $G$-set, and $|G|=p^n$ where $p$ is prime, then $|X_G|\equiv |X|\mod p$.
+
+Where $X_G=\{x\in X|g\cdot x=x\text{ for all }g\in G\}=\{x\in X|\text{orbit of }x\text{ is trivial}\}$
+
+#### Corollary: Cauchy's theorem
+
+If $p$, where $p$ is a prime, divides $|G|$, then $G$ has a subgroup of order $p$. (equivalently, $g$ has an element of order $p$)
+
+> This does not hold when $p$ is not prime.
+>
+> Consider $A_4$ with order $12$, and $A_4$ has no subgroup of order $6$.
+
+#### Corollary: Center of prime power group is non-trivial
+
+If $|G|=p^m$, then $Z(G)$ is non-trivial. ($Z(G)\neq \{e\}$)
+
+<details>
+<summary>Proof</summary>
+
+Let $G$ act on $G$ via conjugation, then $g\cdot h=ghg^{-1}$. This makes $G$ to a $G$-set.
+
+Apply the theorem, the set of elements with trivial orbit is; Let $X=G$, then $X_G=\{h\in G|g\cdot h=h\text{ for all }g\in G\}=\{h\in G|ghg^{-1}=h\text{ for all }g\in G\}=Z(G)$.
+
+Therefore $|Z(G)|\equiv |G|\mod p$.
+
+So $p$ divides $|Z(G)|$, so $|Z(G)|\neq 1$, therefore $Z(G)$ is non-trivial.
+</details>
+
+#### Proposition: Prime square group is abelian
+
+If $|G|=p^2$, where $p$ is a prime, then $G$ is abelian.
+
+<details>
+<summary>Proof</summary>
+
+Since $Z(G)$ is a subgroup of $G$, $|Z(G)|$ divides $p^2$ so $|Z(G)|=1, p$ or $p^2$.
+
+By corollary center of prime power group is non-trivial, $Z(G)\neq 1$.
+
+If $|Z(G)|=p$. If $|Z(G)|=p$, then consider the group $G/Z(G)$ (Note that $Z(G)\trianglelefteq G$). We have $|G/Z(G)|=p$ so $G/Z(G)$ is cyclic (by problem 13.39), therefore $G$ is abelian.
+
+If $|Z(G)|=p^2$, then $G$ is abelian.
+
+</details>
+
+### Classification of small order
+
+Let $G$ be a group
+
+- $|G|=1$
+  - $G=\{e\}$
+- $|G|=2$
+  - $G\simeq\mathbb{Z}_2$ (prime order)
+- $|G|=3$
+  - $G\simeq\mathbb{Z}_3$ (prime order)
+- $|G|=4$
+  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2$
+  - $G\simeq\mathbb{Z}_4$
+- $|G|=5$
+  - $G\simeq\mathbb{Z}_5$ (prime order)
+- $|G|=6$
+  - $G\simeq S_3$
+  - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_2\simeq \mathbb{Z}_6$
+<details>
+<summary>Proof</summary>
+
+$|G|$ has an element of order $2$, namely $b$, and an element of order $3$, namely $a$.
+
+So $e,a,a^2,b,ba,ba^2$ are distinct.
+
+Therefore, there are only two possibilities for value of $ab$. ($a,a^2$ are inverse of each other, $b$ is inverse of itself.)
+
+If $ab=ba$, then $G$ is abelian, then $G\simeq \mathbb{Z}_2\times \mathbb{Z}_3$.
+
+If $ab=ba^2$, then $G\simeq S_3$.
+</details>
+
+- $|G|=7$
+  - $G\simeq\mathbb{Z}_7$ (prime order)
+- $|G|=8$
+  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2$
+  - $G\simeq\mathbb{Z}_4\times \mathbb{Z}_2$
+  - $G\simeq\mathbb{Z}_8$
+  - $G\simeq D_4$
+  - $G\simeq$ quaternion group $\{e,i,j,k,-1,-i,-j,-k\}$ where $i^2=j^2=k^2=-1$, $(-1)^2=1$. $ij=l$, $jk=i$, $ki=j$, $ji=-k$, $kj=-i$, $ik=-j$.
+- $|G|=9$
+  - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_3$
+  - $G\simeq\mathbb{Z}_9$ (apply the corollary, $9=3^2$, these are all the possible cases)
+- $|G|=10$
+  - $G\simeq\mathbb{Z}_5\times \mathbb{Z}_2\simeq \mathbb{Z}_{10}$
+  - $G\simeq D_5$
+- $|G|=11$
+  - $G\simeq\mathbb{Z}_11$ (prime order)
+- $|G|=12$
+  - $G\simeq\mathbb{Z}_3\times \mathbb{Z}_4$
+  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_3$
+  - $A_4$
+  - $D_6\simeq S_3\times \mathbb{Z}_2$
+  - ??? One more
+- $|G|=13$
+  - $G\simeq\mathbb{Z}_{13}$ (prime order)
+- $|G|=14$
+  - $G\simeq\mathbb{Z}_2\times \mathbb{Z}_7$
+  - $G\simeq D_7$
+
+#### Lemma for group of order $2p$ where $p$ is prime
+
+If $p$ is prime, $p\neq 2$, and $|G|=2p$, then $G$ is either abelian $\simeq \mathbb{Z}_2\times \mathbb{Z}_p$ or $G\simeq D_p$
+
+<details>
+<summary>Proof</summary>
+
+We know $G$ has an element of order 2, namely $b$, and an element of order $p$, namely $a$.
+
+So $e,a,a^2,\dots ,a^{p-1},ba,ba^2,\dots,ba^{p-1}$ are distinct elements of $G$.
+
+Consider $ab$, if $ab=ba$, then $G$ is abelian, then $G\simeq \mathbb{Z}_2\times \mathbb{Z}_p$.
+
+If $ab=ba^{p-1}$, then $G\simeq D_p$.
+
+$ab$ cannot be inverse of other elements, if $ab=ba^t$, where $2\leq t\leq p-2$, then $bab=a^t$, then $(bab)^t=a^{t^2}$, then $ba^tb=a^{t^2}$, therefore $a=a^{t^2}$, then $a^{t^2-1}=e$, so $p|(t^2-1)$, therefore $p|t-1$ or $p|t+1$.
+
+This is not possible since $2\leq t\leq p-2$.
+
+</details>

content/Math4302/Math4302_L24.md

@@ -0,0 +1,147 @@
+# Math4302 Modern Algebra (Lecture 24)
+
+## Rings
+
+### Definition of ring
+
+A ring is a set $R$ with binary operation $+$ and $\cdot$ such that:
+
+- $(R,+)$ is an abelian group.
+- Multiplication is associative: $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
+- Distribution property: $a\cdot (b+c)=a\cdot b+a\cdot c$, $(b+c)\cdot a=b\cdot a+c\cdot a$. (Note that $\cdot$ may not be abelian, may not even be a group, therefore we need to distribute on both sides.)
+
+> [!NOTE]
+>
+> $a\cdot b=ab$ will be used for the rest of the sections.
+
+<details>
+<summary>Examples of rings</summary>
+
+$(\mathbb{Z},+,*)$, $(\mathbb{R},+,*)$ are rings.
+
+---
+
+$(2\mathbb{Z},+,\cdot)$ is a ring.
+
+---
+
+$(M_n(\mathbb{R}),+,\cdot)$ is a ring.
+
+---
+
+$(\mathbb{Z}_n,+,\cdot)$ is a ring, where $a\cdot b=a*b\mod n$.
+
+e.g. in $\mathbb{Z}_{12}, 4\cdot 8=8$.
+
+</details>
+
+> [!TIP]
+>
+> If $(R+,\cdot)$ is a ring, then $(R,\cdot)$ may not be necessarily a group.
+
+#### Properties of rings
+
+Let $0$ denote the identity of addition of $R$. $-a$ denote the additive inverse of $a$.
+
+- $0\cdot a=a\cdot 0=0$
+- $(-a)b=a(-b)=-(ab)$, $\forall a,b\in R$
+- $(-a)(-b)=ab$, $\forall a,b\in R$
+
+<details>
+<summary>Proof</summary>
+
+1) $0\cdot a=(0+0)\cdot a=0\cdot a+0\cdot a$, by cancellation, $0\cdot a=0$.  
+Similarly, $a\cdot 0=0\cdot a=0$.
+
+2) $(a+(-a))\cdot b=0\cdot b=0$ by (1), So $a\cdot b +(-a)\cdot b=0$, $(-a)\cdot b=-(ab)$. Similarly, $a\cdot (-b)=-(ab)$.
+
+3) $(-a)(-b)=(a(-b))$ by (2), apply (2) again, $-(-(ab))=ab$.
+</details>
+
+#### Definition of commutative ring
+
+A ring $(R,+,\cdot)$ is commutative if $a\cdot b=b\cdot a$, $\forall a,b\in R$.
+
+<details>
+<summary>Example of non commutative ring</summary>
+
+$(M_n(\mathbb{R}),+,\cdot)$ is not commutative.
+
+</details>
+
+#### Definition of unity element
+
+A ring $R$ has unity element if there is an element $1\in R$ such that $a\cdot 1=1\cdot a=a$, $\forall a\in R$.
+
+> [!NOTE]
+>
+> Unity element is unique.
+>
+> Suppose $1,1'$ are unity elements, then $1\cdot 1'=1'\cdot 1=1$, $1=1'$.
+
+<details>
+<summary>Example of field have no unity element</summary>
+
+$(2\mathbb{Z},+,\cdot)$ does not have unity element.
+</details>
+
+#### Definition of unit
+
+Suppose $R$ is a ring with unity element. An element $a\in R$ is called a unit if there is $b\in R$ such that $a\cdot b=b\cdot a=1$.
+
+In this case $b$ is called the inverse of $a$.
+
+> [!TIP]
+>
+> If $a$ is a unit, then its inverse is unique. If $b,b'$ are inverses of $a$, then $b'=1b'=bab'=b1=b$.
+
+We use $a^{-1}$ or $\frac{1}{a}$ to represent the inverse of $a$.
+
+Let $R$ be a ring with unity, then $0$ is not a unit. (identity of addition has no multiplicative inverse)
+
+If $0b=b0=1$, then $\forall a\in R$, $a=1a=0a=0$.
+
+#### Definition of division ring
+
+If every $a\neq 0$ in $R$ has a multiplicative inverse (is a unit), then $R$ is called a division ring.
+
+#### Definition of field
+
+A commutative division ring is called a field.
+
+<details>
+<summary>Example of field</summary>
+
+$(\mathbb{R},+,\cdot)$ is a field.
+
+---
+
+$(\mathbb{Z}_p,+,\cdot)$ is a field, where $p$ is a prime number.
+</details>
+
+#### Lemma $\mathbb{Z}_p$ is a field
+
+$\mathbb{Z}_p$ is a field if and only if $p$ is prime.
+
+<details>
+<summary>Proof</summary>
+
+If $\mathbb{Z}_n$ is a field, then $n$ is prime.
+
+<!-- This is equivalent to the statement that: If $\mathbb{Z}_p$ is a field and $1\leq m\leq n-1$, then $\operatorname{gcd}(m,n)=1$.
+
+We $\operatorname{gcd}(m,n)=d>1$,  -->
+
+We proceed by contradiction. Suppose $n$ is not a prime, then $d|n$ for some $2\leq d\leq n-1$, then $[d]$ does not have inverse.
+
+If $[d][x]=[1]$, then $dx\equiv 1\mod n$, so $dx-1=ny$ for some $y\in \mathbb{Z}$, but $d|dx$, and $d|ny$, so $d|1$ which is impossible.
+
+Therefore, $n$ is prime.
+
+---
+
+If $p$ is prime, then $\mathbb{Z}_p$ is a field.
+
+Since $p$ is a prime, then $\operatorname{gcd}(m,n)=1$ for $1\leq m\leq n-1$. So $1=mx+ny$ for some $x,y\in \mathbb{Z}_p$. Then $[x]$ (the remainder of $x$ when divided by $p$) is the multiplicative inverse of $[m]$. $[m][x]=[mx]=[1-ny]=[1]$.
+
+</details>

content/Math4302/Math4302_L25.md

@@ -0,0 +1,141 @@
+# Math4302 Modern Algebra (Lecture 25)
+
+Midterm next, next Wednesday
+
+## Rings
+
+### Definitions
+
+- commutative ring: elements $a\cdot b=b\cdot a$, $\forall a,b\in R$
+- ring with unity: elements $a\cdot 1=1\cdot a=a$, $\forall a\in R$
+- units: elements such that there is $a\cdot b=1$ for some $b\in R$.
+- division ring: every element $a\neq 0$ has a multiplicative inverse $a^{-1}$ such that $a\cdot a^{-1}=1$.
+- field: division ring that is commutative
+
+<details>
+<summary>Examples of division ring that is not a field</summary>
+
+Quaternions
+
+Let $i^2=-1$, $j^2=-1$, $k^2=-1$, with $ij=k$, $jk=i$, $ki=j$.
+
+$R=\{a+bi+ci+dj\mid a,b,c,d\in \mathbb{R}\}$
+
+$R$ is not commutative since $ij\neq ji$, but $R$ is a division ring.
+
+Let $x=a+bi+cj+dk$ be none zero, then $\bar{x}=a-bi-cj-dk$, $x^{-1}=\frac{\bar{x}}{a^2+b^2+c^2+d^2}$ is also non zero and $xx^{-1}=1$.
+
+</details>
+
+Recall from last time $\mathbb{Z}_n$ is a field if and only if $n$ is prime.
+
+#### Units in $\mathbb{Z}_n$ is coprime to $n$
+
+More generally, $[m]\in \mathbb{Z}_n$ is a unit if and only if $\operatorname{gcd}(m,n)=1$.
+
+<details>
+<summary>Proof</summary>
+
+Let $d=\operatorname{gcd}(m,n)$ and $[m]$ is a unit, then $\exists [x]\in \mathbb{Z}_n$ with $[m][z]=[1]$, so $mz\equiv 1\mod n$. so $mz-1=nt$ for some $t\in \mathbb{Z}$, but $d|m$, $d|t$, so $d|1$ implies $d=1$.
+
+If $d=1$, so $1=mr+ns$ for some $r,s\in \mathbb{Z}_n$. If $x=r\mod n$, then $[x]$ is the inverse of $[m]$. $mr\equiv 1\mod n\implies [m][x]=[1]$.
+
+</details>
+
+### Integral Domains
+
+#### Definition of zero divisors
+
+If $a,b\in R$ with $a,b\neq 0$ and $ab=0$, then $a,b$ are called zero divisors.
+
+<details>
+<summary>Example of zero divisors</summary>
+
+Consider $\mathbb{Z}_6$, then $2\cdot 3=0$, so $2$ and $3$ are zero divisors.
+
+And $4\cdot 3=0$, so $4$ and $3$ are zero divisors.
+
+> If $a$ is a unit, then $a$ is not a zero divisor.
+
+$ab=0\implies a^{-1}ab=0\implies 1b=0\implies b=0$.
+
+</details>
+
+> [!NOTE]
+>
+> If an element is not unit, it may not be a zero divisor.
+>
+> Consider $R=\mathbb{Z}$ and $2$ is not a unit, but $2$ is not a zero divisor.
+
+#### Zero divisors in $\mathbb{Z}_n$
+
+$[m]\in \mathbb{Z}_n$ is a zero divisor if and only if $\operatorname{gcd}(m,n)>1$ ($m$ is not a unit).
+
+<details>
+<summary>Proof</summary>
+
+If $d=\operatorname{gcd}(m,n)=1$, then $[m]$ is a unit, so $[m]$ is not a zero divisor.
+
+Therefore $[m]$ is a zero divisor if $\operatorname{gcd}(m,n)>1$.
+
+---
+
+If $d=\operatorname{gcd}(m,n)>1$, then $n=n_1d,m=m_1d, 1\leq n_1<n$.
+
+Then $mn_1=m_1dn_1=m_1n$, $n|mn_1$ $[m][n_1]=[0]$, $n_1\neq 0$, $[m]$ is a zero divisor.
+
+</details>
+
+#### Definition of integral domain
+
+A commutative ring with unity is called a integral domain (or just a domain) if it has no zero divisors.
+
+<details>
+<summary>Example of integral domain</summary>
+
+$\mathbb{Z}$ is a integral domain.
+
+---
+
+Any field is a integral domain.
+</details>
+
+#### Corollaries of integral domain
+
+If $R$ is a integral domain, then we have cancellation property $ab=ac,a\neq 0\implies b=c$.
+
+#### Units with multiplication forms a group
+
+If $R$ is a ring with unity, then the units in $R$ forms a group under multiplication.
+
+<details>
+<summary>Proof</summary>
+
+if $a,b$ are units, then $ab$ is a unit $(ab)^{-1}=b^{-1}a^{-1}$.
+
+</details>
+
+In particular, non-zero elements of any field form an abelian group under multiplication.
+
+<details>
+<summary>Example</summary>
+
+Consider $\mathbb{Z}_p$ field, then $(\{1,2,\cdots,p-1\},\cdot)$ forms an abelian group of size $p-1$.
+
+---
+
+Consider $\mathbb{Z}_5$, then we have a group of size $4$ under multiplication.
+
+- $1$ has order 1
+- $2$ has order 4 $2,4,3,1$.
+- $3$ has order 4 $3,4,2,1$.
+- $4$ has order 2 $4,1$.
+
+Therefore $\mathbb{Z}_5\simeq \mathbb{Z}_4$.
+
+---
+
+Therefore in $R=\mathbb{Z}_p$, $\mathbb{Z}_p^*=\{[1],[2],\cdots,[p-1]\}$ is a group of order $p-1$.
+
+Therefore, for every $a\in \mathbb{Z}_p$, $[a]^{p-1}=[1]$, then $a^{p-1}\equiv 1\mod p$ (Fermat's little theorem).
+</details>

content/Math4302/Math4302_L26.md

@@ -0,0 +1,111 @@
+# Math4302 Modern Algebra (Lecture 26)
+
+## Rings
+
+### Fermat’s and Euler’s Theorems
+
+Recall from last lecture, we consider $\mathbb{Z}_p$ and $\mathbb{Z}_p^*$ denote the group of units in $\mathbb{Z}_p$ with multiplication.
+
+$$
+\mathbb{Z}_p^* = \{1,2,\cdots,p-1\}, \quad |\mathbb{Z}_p^*| = p-1
+$$
+
+Let $[a]\in \mathbb{Z}_p^*$, then $[a]^{p-1}=[1]$, this implies that $a^{p-1}\mod p=1$. 
+
+Now if $m\in \mathbb{Z}$ and $a=$ remainder of $m$ by $p$, $[a]\in \mathbb{Z}_p$, implies $m\equiv a\mod p$.
+
+Then $m^{p-1}\equiv a^{p-1}\mod p$.
+
+So
+
+#### Fermat’s little theorem
+
+If $p$ is not a divisor of $m$, then $m^{p-1}\equiv 1\mod p$.
+
+#### Corollary of Fermat’s little theorem
+
+If $m\in \mathbb{Z}$, then $m^p\equiv m\mod p$.
+
+<details>
+<summary>Proof</summary>
+
+If $p|m$, then $m^{p-1}\equiv 0\equiv m\mod p$.
+
+If $p\not|m$, then by Fermat’s little theorem, $m^{p-1}\equiv 1\equiv m\mod p$, so $m^p\equiv m\mod p$.
+
+</details>
+
+<details>
+<summary>Example</summary>
+
+Find the remainder of $40^{100}$ by $19$.
+
+$40^{100}\equiv 2^{100}\mod 19$
+
+$2^{100}\equiv 2^{10}\mod 19$ (Fermat’s little theorem $2^18\equiv 1\mod 19, 2^{90}\equiv 1\mod 19$)
+
+$2^10\equiv (-6)^2\mod 19\equiv 36\mod 19\equiv 17\mod 19$
+
+---
+
+For every integer $n$, $15|(n^{33}-n)$.
+
+$15=3\cdot 5$, therefore enough to show that $3|(n^{33}-n)$ and $5|(n^{33}-n)$.
+
+Apply the corollary of Fermat’s little theorem to $p=3$: $n^3\equiv n\mod 3$, $(n^3)^11\equiv n^{11}\equiv (n^3)^3 n^2=n^3 n^2\equiv n^3\mod 3\equiv n\mod 3$.
+
+Therefore $3|(n^{33}-n)$.
+
+Apply the corollary of Fermat’s little theorem to $p=5$: $n^5\equiv n\mod 5$, $n^30 n^3\equiv (n^5)^6 n^3\equiv n^6 n^3\equiv n^5\mod 5\equiv n\mod 5$.
+
+Therefore $5|(n^{33}-n)$.
+
+</details>
+
+#### Euler’s totient function
+
+Consider $\mathbb{Z}_6$, by definition for the group of units, $\mathbb{Z}_6^*=\{1,5\}$.
+
+$$
+\phi(n)=|\mathbb{Z}_n^*|=|\{1\leq x\leq n:gcd(x,n)=1\}|
+$$
+
+<details>
+<summary>Example</summary>
+
+$\phi(8)=|\{1,3,5,7\}|=4$
+
+</details>
+
+If $[a]\in \mathbb{Z}_n^*$, then $[a]^{\phi(n)}=[1]$. So $a^{\phi(n)}\equiv 1\mod n$.
+
+#### Theorem
+
+If $m\in \mathbb{Z}$, and $gcd(m,n)=1$, then $m^{\phi(n)}\equiv 1\mod n$.
+
+<details>
+<summary>Proof</summary>
+
+If $a$ is the remainder of $m$ by $n$, then $m\equiv a\mod n$, and $\operatorname{gcd}(a,n)=1$, so $m^{\phi(n)}\equiv a^{\phi(n)}\equiv 1\mod n$.
+
+</details>
+
+#### Applications on solving modular equations
+
+Solving equations of the form $ax\equiv b\mod n$.
+
+Not always have solution, $2x\equiv 1\mod 4$ has no solution since $1$ is odd.
+
+Solution for $2x\equiv 1\mod 3$
+
+- $x\equiv 0\implies 2x\equiv 0\mod 3$
+- $x\equiv 1\implies 2x\equiv 2\mod 3$
+- $x\equiv 2\implies 2x\equiv 1\mod 3$
+
+So solution for $2x\equiv 1\mod 3$ is $\{3k+2|k\in \mathbb{Z}\}$.
+
+#### Theorem for exsistence of solution of modular equations
+
+$ax\equiv b\mod n$ has a solution if and only if $\operatorname{gcd}(a,n)|b$ and in that case the equation has $d$ solutions in $\mathbb{Z}_n$.
+
+Proof on next lecture.

content/Math4302/Math4302_L27.md

@@ -0,0 +1,126 @@
+# Math4302 Modern Algebra (Lecture 27)
+
+## Rings
+
+### Fermat’s and Euler’s Theorems
+
+Recall from last lecture, $ax\equiv b \mod n$, if $x\equiv y\mod n$, then $x$ is a solution if and only if $y$ is a solution.
+
+#### Theorem for existence of solution of modular equations
+
+$ax\equiv b\mod n$ has a solution if and only if $d=\operatorname{gcd}(a,n)|b$ And if there is a solution, then there are exactly $d$ solutions in $\mathbb{Z}_n$.
+
+<details>
+<summary>Proof</summary>
+
+For the forward direction, we proved if $ax\equiv b\mod n$ then $ax-b=ny$, $y\in\mathbb{Z}$.
+
+then $b=ax-ny$, $d|(ax-ny)$ implies that $d|b$.
+
+---
+
+For the backward direction, assume $d=\operatorname{gcd}(a,n)=1$. Then we need to show, there is exactly $1$ solution between $0$ and $n-1$.
+
+If $ax\equiv b\mod n$, then in $\mathbb{Z}_n$, $[a][x]=[b]$. (where $[a]$ denotes the remainder of $a$ by $n$ and $[b]$ denotes the remainder of $b$ by $n$)
+
+Since $\operatorname{gcd}(a,n)=1$, then $[a]$ is a unit in $\mathbb{Z}_n$, so we can multiply the above equation by the inverse of $[a]$. and get $[x]=[a]^{-1}[b]$.
+
+Now assume $d=\operatorname{gcd}(a,n)$ where $n$ is arbitrary. Then $a=a'd$, then $n=n'd$, with $\operatorname{gcd}(a',n')=1$.
+
+Also $d|b$ so $b=b'd$. So
+
+$$
+\begin{aligned}
+ax\equiv b \mod n&\iff n|(ax-b)\\
+&\iff n'd|(a'dx-b'd)\\
+&\iff n'|(a'x-b')\\
+&\iff a'x\equiv b'\mod n'
+\end{aligned}
+$$.
+
+Since $\operatorname{gcd}(a',n')=1$, there is a unique solution $x_0\in \mathbb{Z}_{n'}$. $0\leq x_0\leq n'+1$. Other solution in $\mathbb{Z}$ are of the form $x_0+kn'$ for $k\in \mathbb{Z}$.
+
+And there will be $d$ solutions in $\mathbb{Z}_n$,
+
+</details>
+
+<details>
+<summary>Examples</summary>
+
+Solve $12x\equiv 25\mod 7$.
+
+$12\equiv 5\mod 7$, $25\equiv 4\mod 7$. So the equation becomes $5x\equiv 4\mod 7$.
+
+$[5]^{-1}=3\in \mathbb{Z}_7$, so $[5][x]\equiv [4]$ implies $[x]\equiv [3][4]\equiv [5]\mod 7$.
+
+So solution in $\mathbb{Z}$ is $\{5+7k:k\in \mathbb{Z}\}$.
+
+---
+
+Solve $6x\equiv 32\mod 20$.
+
+$\operatorname{gcd}(6,20)=2$, so $6x\equiv 12\mod 20$ if and only if $3x\equiv 6\mod 10$.
+
+$[3]^{-1}=[7]\in \mathbb{Z}_{10}$, so $[3][x]\equiv [6]$ implies $[x]\equiv [7][6]\equiv [2]\mod 10$.
+
+So solution in $\mathbb{Z}_{20}$ is $[2]$ and $[12]$
+
+So solution in $\mathbb{Z}$ is $\{2+10k:k\in \mathbb{Z}\}$
+
+</details>
+
+### Ring homomorphisms
+
+#### Definition of ring homomorphism
+
+Let $R,S$ be two rings, $f:R\to S$ is a ring homomorphism if $\forall a,b\in R$,
+
+- $f(a+b)=f(a)+f(b)\implies f(0)=0, f(-a)=-f(a)$
+- $f(ab)=f(a)f(b)$
+
+#### Definition of ring isomorphism
+
+If $f$ is a ring homomorphism and a bijection, then $f$ is called a ring isomorphism.
+
+<details>
+<summary>Example</summary>
+Let $f:(\mathbb{Z},+,\times)\to(2\mathbb{Z},+,\times)$ by $f(a)=2a$.
+
+Is not a ring homomorphism since $f(ab)\neq f(a)f(b)$ in general.
+
+---
+
+Let $f:(\mathbb{Z},+,\times)\to(\mathbb{Z}_n,+,\times)$ by $f(a)=a\mod n$
+
+Is a ring homomorphism.
+
+</details>
+
+### Integral domains and their file fo fractions.
+
+Let $R$ be an integral domain: (i.e. $R$ is commutative with unity and no zero divisors).
+
+#### Definition of field of fractions
+
+If $R$ is an integral domain, we can construct a field containing $R$ called the field of fractions (or called field of quotients) of $R$.
+
+$$
+S=\{(a,b)|a,b\in R, b\neq 0\}
+$$
+
+a relation on $S$ is defined as follows:
+
+$(a,b)\sim (c,d)$ if and only if $ad=bc$.
+
+<details>
+<summary>This equivalence relation is well defined</summary>
+
+- Reflectivity: $(a,b)\sim (a,b)$ $ab=ab$
+- Symmetry: $(a,b)\sim (c,d)\Rightarrow (c,d)\sim (a,b)$
+- Transitivity: $(a,b)\sim (c,d)$ and $(c,d)\sim (e,f)\Rightarrow (a,b)\sim (e,f)$
+  - $ad=bc$, and $cf=ed$, we want to conclude that $af=be$. since $ad=bc$, then $adf=bcf$, since $cf=ed$, then $cfb=edb$, therefore $adf=edb$.
+  - Then $d(af-be)=0$ since $d\neq 0$ then $af=be$.
+
+</details>
+
+Then $S/\sim$ is a field.

content/Math4302/Math4302_L28.md

@@ -0,0 +1,153 @@
+# Math4302 Modern Algebra (Lecture 28)
+
+## Rings
+
+### Field of quotients
+
+Let $R$ be an integral domain ($R$ has unity and commutative with no zero divisors).
+
+Consider the pair $S=\{(a,b)|a,b\in R, b\neq 0\}$.
+
+And define the equivalence relation on $S$ as follows:
+
+$(a,b)\sim (c,d)$ if and only if $ad=bc$.
+
+We denote $[(a,b)]$ as set of all elements in $S$ equivalent to $(a,b)$.
+
+Let $F$ be the set of all equivalent classes. We define addition and multiplication on $F$ as follows:
+
+$$
+[(a,b)]+[(c,d)]=[(ad+bc,bd)]
+$$
+
+$$
+[(a,b)]\cdot[(c,d)]=[(ac,bd)]
+$$
+
+<details>
+<summary>The multiplication and addition is well defined </summary>
+
+Addition:
+
+If $(a,b)\sim (a',b')$, and $(c,d)\sim (c',d')$, then we want to show that $(ad+bc,bd)\sim (a'd+c'd,b'd)$.
+
+Since $(a,b)\sim (a',b')$, then $ab'=a'b$; $(c,d)\sim (c',d')$, then $cd'=dc'$,
+
+So $ab'dd'=a'bdd'$, and $cd'bb'=dc'bb'$.
+
+ $adb'd'+bcb'd'=a'd'bd+b'c'bd$, therefore $(ad+bc,bd)\sim (a'd+c'd,b'd)$.
+
+---
+
+Multiplication:
+
+If $(a,b)\sim (a',b')$, and $(c,d)\sim (c',d')$, then we want to show that $(ac,bd)\sim (a'c',b'd')$.
+
+Since $(a,b)\sim (a',b')$, then $ab'=a'b$; $(c,d)\sim (c',d')$, then $cd'=dc'$, so $(ac,bd)\sim (a'c',b'd')$
+
+</details>
+
+#### Claim (F,+,*) is a field
+
+- additive identity: $(0,1)\in F$
+- additive inverse: $(a,b)\in F$, then $(-a,b)\in F$ and $(-a,b)+(a,b)=(0,1)\in F$
+- additive associativity: bit long.
+
+- multiplicative identity: $(1,1)\in F$
+- multiplicative inverse: $[(a,b)]$ is non zero if and only if $a\neq 0$, then $a^{-1}=[(b,a)]\in F$.
+- multiplicative associativity: bit long
+
+- distributivity: skip, too long.
+
+Such field is called a quotient field of $R$.
+
+And $F$ contains $R$ by $\phi:R\to F$, $\phi(a)=[(a,1)]$.
+
+This is a ring homomorphism.
+
+- $\phi(a+b)=[(a+b,1)]=[(a,1)][(b,1)]\phi(a)+\phi(b)$
+- $\phi(ab)=[(ab,1)]=[(a,1)][(b,1)]\phi(a)\phi(b)$
+
+and $\phi$ is injective.
+
+If $\phi(a)=\phi(b)$, then $a=b$.
+
+<details>
+<summary>Example</summary>
+
+Let $D\subset \mathbb R$ and
+
+$$
+\mathbb Z \subset D\coloneqq \{a+b\sqrt{2}:a,b\in \mathbb Z\}
+$$
+
+Then $D$ is a subring of $\mathbb R$, and integral domain, with usual addition and multiplication.
+
+$$
+(a+b\sqrt{2})(c+d\sqrt{2})=(ac+2bd)+(ad+bc)\sqrt{2}
+$$
+
+$$
+-(a+b\sqrt{2})=(-a)+(-b)\sqrt{2})
+$$
+
+...
+
+$D$ is a integral domain since $\mathbb R$ has no zero divisors, therefore $D$ has no zero divisors.
+
+Consider the field of quotients of $D$. $[(a+b\sqrt{2},c+d\sqrt{2})]$. This is isomorphic to $\mathbb Q(\sqrt2)=\{r+s\sqrt{2}:r,s\in \mathbb Q\}$
+
+$$
+m+n\sqrt{2}=\frac{m}{n}+\frac{m'}{n'}\sqrt{2}\mapsto [(mn'+nm'\sqrt{2},nn')]
+$$
+
+And use rationalization on the forward direction.
+
+</details>
+
+#### Polynomial rings
+
+Let $R$ be a ring, a polynomial with coefficients in $R$ is a sum
+
+$$
+a_0+a_1x+\cdots+a_nx^n
+$$
+
+where $a_i\in R$. $x$ is indeterminate, $a_0,a_1,\cdots,a_n$ are called coefficients. $a_0$ is the constant term.
+
+If $f$ is a non-zero polynomial, then the degree of $f$ is defined as the largest $n$ such that $a_n\neq 0$.
+
+<details>
+<summary>Example</summary>
+
+Let $f=1+2x+0x^2-1x^3+0x^4$, then $deg f=3$
+
+</details>
+
+If $R$ has a unity $1$, then we write $x^m$ instead of $1x^m$.
+
+Let $R[x]$ denote the set of all polynomials with coefficients in $R$.
+
+We define multiplication and addition on $R[x]$.
+
+$f:a_0+a_1x+\cdots+a_nx^n$
+
+$g:b_0+b_1x+\cdots+b_mx^m$
+
+Define,
+
+$$
+f+g=a_0+b_0+a_1x+b_1x+\cdots+a_nx^n+b_mx^m
+$$
+
+$$
+fg=(a_0b_0)+(a_1b_0)x+\cdots+(a_nb_m)x^m
+$$
+
+In general, the coefficient of $x^m=\sum_{i=0}^{m}a_ix^{m-i}$.
+
+> [!CAUTION]
+>
+> The field $R$ may not be commutative, follow the order of computation matters.
+
+We will show that this is a ring and explore additional properties.

content/Math4302/_meta.js

@@ -24,4 +24,11 @@ export default {
     Math4302_L19: "Modern Algebra (Lecture 19)",
     Math4302_L20: "Modern Algebra (Lecture 20)",
     Math4302_L21: "Modern Algebra (Lecture 21)",
+    Math4302_L22: "Modern Algebra (Lecture 22)",
+    Math4302_L23: "Modern Algebra (Lecture 23)",
+    Math4302_L24: "Modern Algebra (Lecture 24)",
+    Math4302_L25: "Modern Algebra (Lecture 25)",
+    Math4302_L26: "Modern Algebra (Lecture 26)",
+    Math4302_L27: "Modern Algebra (Lecture 27)",
+    Math4302_L28: "Modern Algebra (Lecture 28)",
 }

mcp-server.mjs

@@ -0,0 +1,207 @@
+import fs from 'node:fs/promises'
+import path from 'node:path'
+import process from 'node:process'
+
+import { Server } from '@modelcontextprotocol/sdk/server/index.js'
+import { StdioServerTransport } from '@modelcontextprotocol/sdk/server/stdio.js'
+import {
+  CallToolRequestSchema,
+  ListToolsRequestSchema
+} from '@modelcontextprotocol/sdk/types.js'
+
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+const SNIPPET_RADIUS = 220
+
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+    }
+
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+  }
+
+  const notes = await walkNotes()
+  return notes.find((note) =>
+    note.slug === normalized ||
+    note.relativePath === normalized ||
+    note.relativePath.replace(/\.(md|mdx)$/i, '') === normalized
+  ) ?? null
+}
+
+function buildSnippet(content, index, query) {
+  const start = Math.max(0, index - SNIPPET_RADIUS)
+  const end = Math.min(content.length, index + query.length + SNIPPET_RADIUS)
+  return content
+    .slice(start, end)
+    .replace(/\s+/g, ' ')
+    .trim()
+}
+
+function textResponse(text) {
+  return {
+    content: [{ type: 'text', text }]
+  }
+}
+
+const server = new Server(
+  {
+    name: 'notenextra-notes',
+    version: '1.0.0'
+  },
+  {
+    capabilities: {
+      tools: {}
+    }
+  }
+)
+
+server.setRequestHandler(ListToolsRequestSchema, async () => ({
+  tools: [
+    {
+      name: 'list_notes',
+      description: 'List available notes from the Next.js content directory.',
+      inputSchema: {
+        type: 'object',
+        properties: {
+          course: {
+            type: 'string',
+            description: 'Optional course or directory prefix, for example CSE442T or Math4201.'
+          }
+        }
+      }
+    },
+    {
+      name: 'read_note',
+      description: 'Read a note by slug or relative path, for example CSE442T/CSE442T_L1.',
+      inputSchema: {
+        type: 'object',
+        properties: {
+          noteId: {
+            type: 'string',
+            description: 'Note slug or relative path inside content/.'
+          }
+        },
+        required: ['noteId']
+      }
+    },
+    {
+      name: 'search_notes',
+      description: 'Search the notes knowledge base using a simple text match over all markdown content.',
+      inputSchema: {
+        type: 'object',
+        properties: {
+          query: {
+            type: 'string',
+            description: 'Search term or phrase.'
+          },
+          limit: {
+            type: 'number',
+            description: `Maximum results to return, capped at ${MAX_SEARCH_RESULTS}.`
+          }
+        },
+        required: ['query']
+      }
+    }
+  ]
+}))
+
+server.setRequestHandler(CallToolRequestSchema, async (request) => {
+  const { name, arguments: args = {} } = request.params
+
+  if (name === 'list_notes') {
+    const notes = await walkNotes()
+    const course = typeof args.course === 'string'
+      ? args.course.trim().toLowerCase()
+      : ''
+    const filtered = course
+      ? notes.filter((note) => note.relativePath.toLowerCase().startsWith(`${course}/`))
+      : notes
+
+    return textResponse(filtered.map((note) => note.slug).join('\n') || 'No notes found.')
+  }
+
+  if (name === 'read_note') {
+    const note = await resolveNote(args.noteId)
+    if (!note) {
+      return textResponse('Note not found.')
+    }
+
+    const content = await fs.readFile(note.fullPath, 'utf8')
+    return textResponse(`# ${note.slug}\n\n${content}`)
+  }
+
+  if (name === 'search_notes') {
+    const query = typeof args.query === 'string' ? args.query.trim() : ''
+    if (!query) {
+      return textResponse('Query must be a non-empty string.')
+    }
+
+    const limit = Math.max(1, Math.min(Number(args.limit) || 5, MAX_SEARCH_RESULTS))
+    const queryLower = query.toLowerCase()
+    const notes = await walkNotes()
+    const matches = []
+
+    for (const note of notes) {
+      const content = await fs.readFile(note.fullPath, 'utf8')
+      const haystack = `${note.slug}\n${content}`
+      const index = haystack.toLowerCase().indexOf(queryLower)
+      if (index === -1) {
+        continue
+      }
+
+      matches.push({
+        note,
+        index,
+        snippet: buildSnippet(haystack, index, query)
+      })
+    }
+
+    matches.sort((a, b) => a.index - b.index || a.note.slug.localeCompare(b.note.slug))
+
+    return textResponse(
+      matches
+        .slice(0, limit)
+        .map(({ note, snippet }) => `- ${note.slug}\n${snippet}`)
+        .join('\n\n') || 'No matches found.'
+    )
+  }
+
+  throw new Error(`Unknown tool: ${name}`)
+})
+
+const transport = new StdioServerTransport()
+await server.connect(transport)

mcp-worker/src/index.mjs

@@ -0,0 +1,227 @@
+import { Server } from '@modelcontextprotocol/sdk/server/index.js'
+import { WebStandardStreamableHTTPServerTransport } from '@modelcontextprotocol/sdk/server/webStandardStreamableHttp.js'
+import {
+  CallToolRequestSchema,
+  ListToolsRequestSchema
+} from '@modelcontextprotocol/sdk/types.js'
+
+import { notesData } from '../generated/notes-data.mjs'
+
+const MAX_SEARCH_RESULTS = 10
+const SNIPPET_RADIUS = 220
+const transports = new Map()
+
+function normalizeNoteId(noteId = '') {
+  const normalized = String(noteId).trim().replaceAll('\\', '/').replace(/^\/+|\/+$/g, '')
+  if (!normalized || normalized.includes('..')) {
+    return null
+  }
+
+  return normalized
+}
+
+function resolveNote(noteId) {
+  const normalized = normalizeNoteId(noteId)
+  if (!normalized) {
+    return null
+  }
+
+  return notesData.find((note) =>
+    note.slug === normalized ||
+    note.relativePath === normalized ||
+    note.relativePath.replace(/\.(md|mdx)$/i, '') === normalized
+  ) ?? null
+}
+
+function buildSnippet(content, index, query) {
+  const start = Math.max(0, index - SNIPPET_RADIUS)
+  const end = Math.min(content.length, index + query.length + SNIPPET_RADIUS)
+  return content
+    .slice(start, end)
+    .replace(/\s+/g, ' ')
+    .trim()
+}
+
+function textResponse(text) {
+  return {
+    content: [{ type: 'text', text }]
+  }
+}
+
+function createServer() {
+  const server = new Server(
+    {
+      name: 'notenextra-notes-worker',
+      version: '1.0.0'
+    },
+    {
+      capabilities: {
+        tools: {}
+      }
+    }
+  )
+
+  server.setRequestHandler(ListToolsRequestSchema, async () => ({
+    tools: [
+      {
+        name: 'list_notes',
+        description: 'List available notes from the generated notes knowledge base.',
+        inputSchema: {
+          type: 'object',
+          properties: {
+            course: {
+              type: 'string',
+              description: 'Optional course or directory prefix, for example CSE442T or Math4201.'
+            }
+          }
+        }
+      },
+      {
+        name: 'read_note',
+        description: 'Read a note by slug or relative path, for example CSE442T/CSE442T_L1.',
+        inputSchema: {
+          type: 'object',
+          properties: {
+            noteId: {
+              type: 'string',
+              description: 'Note slug or relative path inside content/.'
+            }
+          },
+          required: ['noteId']
+        }
+      },
+      {
+        name: 'search_notes',
+        description: 'Search the notes knowledge base using a simple text match over all markdown content.',
+        inputSchema: {
+          type: 'object',
+          properties: {
+            query: {
+              type: 'string',
+              description: 'Search term or phrase.'
+            },
+            limit: {
+              type: 'number',
+              description: `Maximum results to return, capped at ${MAX_SEARCH_RESULTS}.`
+            }
+          },
+          required: ['query']
+        }
+      }
+    ]
+  }))
+
+  server.setRequestHandler(CallToolRequestSchema, async (request) => {
+    const { name, arguments: args = {} } = request.params
+
+    if (name === 'list_notes') {
+      const course = typeof args.course === 'string'
+        ? args.course.trim().toLowerCase()
+        : ''
+      const filtered = course
+        ? notesData.filter((note) => note.relativePath.toLowerCase().startsWith(`${course}/`))
+        : notesData
+
+      return textResponse(filtered.map((note) => note.slug).join('\n') || 'No notes found.')
+    }
+
+    if (name === 'read_note') {
+      const note = resolveNote(args.noteId)
+      if (!note) {
+        return textResponse('Note not found.')
+      }
+
+      return textResponse(`# ${note.slug}\n\n${note.content}`)
+    }
+
+    if (name === 'search_notes') {
+      const query = typeof args.query === 'string' ? args.query.trim() : ''
+      if (!query) {
+        return textResponse('Query must be a non-empty string.')
+      }
+
+      const limit = Math.max(1, Math.min(Number(args.limit) || 5, MAX_SEARCH_RESULTS))
+      const queryLower = query.toLowerCase()
+      const matches = []
+
+      for (const note of notesData) {
+        const haystack = `${note.slug}\n${note.content}`
+        const index = haystack.toLowerCase().indexOf(queryLower)
+        if (index === -1) {
+          continue
+        }
+
+        matches.push({
+          note,
+          index,
+          snippet: buildSnippet(haystack, index, query)
+        })
+      }
+
+      matches.sort((a, b) => a.index - b.index || a.note.slug.localeCompare(b.note.slug))
+
+      return textResponse(
+        matches
+          .slice(0, limit)
+          .map(({ note, snippet }) => `- ${note.slug}\n${snippet}`)
+          .join('\n\n') || 'No matches found.'
+      )
+    }
+
+    throw new Error(`Unknown tool: ${name}`)
+  })
+
+  return server
+}
+
+async function handleMcpRequest(request) {
+  const sessionId = request.headers.get('mcp-session-id')
+  let transport = sessionId ? transports.get(sessionId) : undefined
+
+  if (!transport && request.method === 'POST') {
+    transport = new WebStandardStreamableHTTPServerTransport({
+      sessionIdGenerator: () => crypto.randomUUID(),
+      enableJsonResponse: true,
+      onsessioninitialized: (newSessionId) => {
+        transports.set(newSessionId, transport)
+      },
+      onsessionclosed: (closedSessionId) => {
+        transports.delete(closedSessionId)
+      }
+    })
+
+    transport.onclose = () => {
+      if (transport.sessionId) {
+        transports.delete(transport.sessionId)
+      }
+    }
+
+    const server = createServer()
+    await server.connect(transport)
+  }
+
+  if (!transport) {
+    return new Response('Invalid or missing MCP session.', { status: 400 })
+  }
+
+  return transport.handleRequest(request)
+}
+
+export default {
+  async fetch(request) {
+    const url = new URL(request.url)
+
+    if (url.pathname === '/health') {
+      return Response.json({
+        status: 'ok',
+        notes: notesData.length
+      })
+    }
+
+    if (url.pathname === '/mcp') {
+      return handleMcpRequest(request)
+    }
+
+    return new Response('Not found.', { status: 404 })
+  }
+}

mcp-worker/wrangler.toml

@@ -0,0 +1,4 @@
+name = "notenextra-mcp"
+main = "src/index.mjs"
+compatibility_date = "2025-02-13"
+compatibility_flags = ["nodejs_compat"]

package.json

@@ -7,9 +7,16 @@
     "build:test": "cross-env ANALYZE=true NODE_OPTIONS='--inspect --max-old-space-size=4096' next build",
     "build:analyze": "cross-env ANALYZE=true NODE_OPTIONS='--max-old-space-size=16384' next build",
     "postbuild": "next-sitemap && pagefind --site .next/server/app --output-path out/_pagefind",
-    "start": "next start"
+    "start": "next start",
+    "mcp:notes": "node ./mcp-server.mjs",
+    "mcp:worker:build-data": "node ./scripts/generate-mcp-worker-data.mjs",
+    "mcp:worker:deploy": "npm run mcp:worker:build-data && npx wrangler deploy --config mcp-worker/wrangler.toml",
+    "mcp:worker:deploy:dry-run": "npm run mcp:worker:build-data && npx wrangler deploy --dry-run --config mcp-worker/wrangler.toml",
+    "test:mcp": "node ./test/test-mcp-server.mjs",
+    "test:mcp:worker": "node ./test/test-mcp-worker.mjs"
   },
   "dependencies": {
+    "@modelcontextprotocol/sdk": "^1.18.1",
     "@docsearch/css": "^4.3.1",
     "@docsearch/react": "^4.3.1",
     "@napi-rs/simple-git": "^0.1.22",

scripts/generate-mcp-worker-data.mjs

@@ -0,0 +1,47 @@
+import fs from 'node:fs/promises'
+import path from 'node:path'
+import process from 'node:process'
+
+const CONTENT_ROOT = path.join(process.cwd(), 'content')
+const OUTPUT_DIR = path.join(process.cwd(), 'mcp-worker', 'generated')
+const OUTPUT_FILE = path.join(OUTPUT_DIR, 'notes-data.mjs')
+const NOTE_EXTENSIONS = new Set(['.md', '.mdx'])
+
+async function walkNotes(dir = CONTENT_ROOT) {
+  const entries = await fs.readdir(dir, { withFileTypes: true })
+  const notes = await Promise.all(entries.map(async (entry) => {
+    const fullPath = path.join(dir, entry.name)
+
+    if (entry.isDirectory()) {
+      return walkNotes(fullPath)
+    }
+
+    if (!entry.isFile() || !NOTE_EXTENSIONS.has(path.extname(entry.name))) {
+      return []
+    }
+
+    const relativePath = path.relative(CONTENT_ROOT, fullPath).replaceAll('\\', '/')
+    const slug = relativePath.replace(/\.(md|mdx)$/i, '')
+    const content = await fs.readFile(fullPath, 'utf8')
+
+    return [{
+      slug,
+      relativePath,
+      title: path.basename(slug),
+      content
+    }]
+  }))
+
+  return notes.flat().sort((a, b) => a.relativePath.localeCompare(b.relativePath))
+}
+
+const notes = await walkNotes()
+
+await fs.mkdir(OUTPUT_DIR, { recursive: true })
+await fs.writeFile(
+  OUTPUT_FILE,
+  `export const notesData = ${JSON.stringify(notes, null, 2)};\n`,
+  'utf8'
+)
+
+process.stdout.write(`Generated ${notes.length} notes for MCP worker.\n`)

test/test-mcp-server.mjs

@@ -0,0 +1,74 @@
+import assert from 'node:assert/strict'
+import path from 'node:path'
+import process from 'node:process'
+
+import { Client } from '@modelcontextprotocol/sdk/client/index.js'
+import { StdioClientTransport } from '@modelcontextprotocol/sdk/client/stdio.js'
+
+const transport = new StdioClientTransport({
+  command: process.execPath,
+  args: [path.join(process.cwd(), 'mcp-server.mjs')],
+  cwd: process.cwd(),
+  stderr: 'pipe'
+})
+
+let stderrOutput = ''
+transport.stderr?.setEncoding('utf8')
+transport.stderr?.on('data', (chunk) => {
+  stderrOutput += chunk
+})
+
+const client = new Client({
+  name: 'notenextra-mcp-test',
+  version: '1.0.0'
+})
+
+async function main() {
+  await client.connect(transport)
+
+  const toolListResponse = await client.listTools()
+  const toolNames = toolListResponse.tools.map((tool) => tool.name).sort()
+  assert.deepEqual(toolNames, ['list_notes', 'read_note', 'search_notes'])
+
+  const listNotesResponse = await client.callTool({
+    name: 'list_notes',
+    arguments: {
+      course: 'CSE442T'
+    }
+  })
+  const listedNotes = listNotesResponse.content[0].text
+  assert.match(listedNotes, /CSE442T\/CSE442T_L1/, 'list_notes should include CSE442T lecture notes')
+
+  const readNoteResponse = await client.callTool({
+    name: 'read_note',
+    arguments: {
+      noteId: 'about'
+    }
+  })
+  const aboutText = readNoteResponse.content[0].text
+  assert.match(aboutText, /# about/i)
+  assert.match(aboutText, /This is a static server for me to share my notes/i)
+
+  const searchResponse = await client.callTool({
+    name: 'search_notes',
+    arguments: {
+      query: "Kerckhoffs' principle",
+      limit: 3
+    }
+  })
+  const searchText = searchResponse.content[0].text
+  assert.match(searchText, /CSE442T\/CSE442T_L1/, 'search_notes should find the cryptography lecture')
+  assert.match(searchText, /Kerckhoffs/i)
+}
+
+try {
+  await main()
+  process.stdout.write('MCP server test passed.\n')
+} catch (error) {
+  const suffix = stderrOutput ? `\nServer stderr:\n${stderrOutput}` : ''
+  process.stderr.write(`${error.stack || error}${suffix}\n`)
+  process.exitCode = 1
+} finally {
+  await client.close().catch(() => {})
+  await transport.close().catch(() => {})
+}

test/test-mcp-worker.mjs

@@ -0,0 +1,121 @@
+import assert from 'node:assert/strict'
+import process from 'node:process'
+
+import worker from '../mcp-worker/src/index.mjs'
+import { LATEST_PROTOCOL_VERSION } from '@modelcontextprotocol/sdk/types.js'
+
+function makeJsonRequest(url, body, headers = {}) {
+  return new Request(url, {
+    method: 'POST',
+    headers: {
+      accept: 'application/json, text/event-stream',
+      'content-type': 'application/json',
+      ...headers
+    },
+    body: JSON.stringify(body)
+  })
+}
+
+const baseUrl = 'https://example.com'
+
+const healthResponse = await worker.fetch(new Request(`${baseUrl}/health`))
+assert.equal(healthResponse.status, 200)
+const healthJson = await healthResponse.json()
+assert.equal(healthJson.status, 'ok')
+assert.ok(healthJson.notes > 0)
+
+const initializeResponse = await worker.fetch(makeJsonRequest(`${baseUrl}/mcp`, {
+  jsonrpc: '2.0',
+  id: 1,
+  method: 'initialize',
+  params: {
+    protocolVersion: LATEST_PROTOCOL_VERSION,
+    capabilities: {},
+    clientInfo: {
+      name: 'notenextra-worker-test',
+      version: '1.0.0'
+    }
+  }
+}))
+
+assert.equal(initializeResponse.status, 200)
+const sessionId = initializeResponse.headers.get('mcp-session-id')
+assert.ok(sessionId, 'initialize should return an MCP session ID')
+const protocolVersion = initializeResponse.headers.get('mcp-protocol-version') || LATEST_PROTOCOL_VERSION
+const initializeJson = await initializeResponse.json()
+assert.ok(initializeJson.result, 'initialize should return a result payload')
+
+const toolListResponse = await worker.fetch(makeJsonRequest(`${baseUrl}/mcp`, {
+  jsonrpc: '2.0',
+  id: 2,
+  method: 'tools/list',
+  params: {}
+}, {
+  'mcp-protocol-version': protocolVersion,
+  'mcp-session-id': sessionId
+}))
+
+assert.equal(toolListResponse.status, 200)
+const toolListJson = await toolListResponse.json()
+const toolNames = toolListJson.result.tools.map((tool) => tool.name).sort()
+assert.deepEqual(toolNames, ['list_notes', 'read_note', 'search_notes'])
+
+const listNotesResponse = await worker.fetch(makeJsonRequest(`${baseUrl}/mcp`, {
+  jsonrpc: '2.0',
+  id: 3,
+  method: 'tools/call',
+  params: {
+    name: 'list_notes',
+    arguments: {
+      course: 'CSE442T'
+    }
+  }
+}, {
+  'mcp-protocol-version': protocolVersion,
+  'mcp-session-id': sessionId
+}))
+
+assert.equal(listNotesResponse.status, 200)
+const listNotesJson = await listNotesResponse.json()
+assert.match(listNotesJson.result.content[0].text, /CSE442T\/CSE442T_L1/)
+
+const readNoteResponse = await worker.fetch(makeJsonRequest(`${baseUrl}/mcp`, {
+  jsonrpc: '2.0',
+  id: 4,
+  method: 'tools/call',
+  params: {
+    name: 'read_note',
+    arguments: {
+      noteId: 'about'
+    }
+  }
+}, {
+  'mcp-protocol-version': protocolVersion,
+  'mcp-session-id': sessionId
+}))
+
+assert.equal(readNoteResponse.status, 200)
+const readNoteJson = await readNoteResponse.json()
+assert.match(readNoteJson.result.content[0].text, /This is a static server for me to share my notes/i)
+
+const searchResponse = await worker.fetch(makeJsonRequest(`${baseUrl}/mcp`, {
+  jsonrpc: '2.0',
+  id: 5,
+  method: 'tools/call',
+  params: {
+    name: 'search_notes',
+    arguments: {
+      query: "Kerckhoffs' principle",
+      limit: 3
+    }
+  }
+}, {
+  'mcp-protocol-version': protocolVersion,
+  'mcp-session-id': sessionId
+}))
+
+assert.equal(searchResponse.status, 200)
+const searchJson = await searchResponse.json()
+assert.match(searchJson.result.content[0].text, /CSE442T\/CSE442T_L1/)
+
+process.stdout.write('MCP worker test passed.\n')