If tasked with the problem of coming up with a zk-SNARK protocol, many people would make their way to this point and then get stuck and give up. How can a verifier possibly check every single piece of the computation, without looking at each piece of the computation individually? But it turns out that there is a clever solution.

Polynomials

Polynomials are a special class of algebraic expressions of the form:

• x+5
• x^4
• x^3+3x^2+3x+1
• 628x^{271}+318x^{270}+530x^{269}+…+69x+381

i.e. they are a sum of any (finite!) number of terms of the form cx^k

There are many things that are fascinating about polynomials. But here we are going to zoom in on a particular one: polynomials are a single mathematical object that can contain an unbounded amount of information (think of them as a list of integers and this is obvious). The fourth example above contained 816 digits of tau, and one can easily imagine a polynomial that contains far more.

Furthermore, a single equation between polynomials can represent an unbounded number of equations between numbers. For example, consider the equation A(x)+ B(x) = C(x). If this equation is true, then it's also true that:

• A(0)+B(0)=C(0)
• A(1)+B(1)=C(1)
• A(2)+B(2)=C(2)
• A(3)+B(3)=C(3)

And so on for every possible coordinate. You can even construct polynomials to deliberately represent sets of numbers so you can check many equations all at once. For example, suppose that you wanted to check:

• 12+1=13
• 10+8=18
• 15+8=23
• 15+13=28

You can use a procedure called Lagrange interpolation to construct polynomials A(x) that give (12,10,15,15) as outputs at some specific set of coordinates (eg. (0,1,2,3)), B(x) the outputs (1,8,8,13) on thos same coordinates, and so forth. In fact, here are the polynomials:

• A(x)=-2x^3+\frac{19}{2}x^2-\frac{19}{2}x+12
• B(x)=2x^3-\frac{19}{2}x^2+\frac{29}{2}x+1
• C(x)=5x+13

Checking the equation A(x)+B(x)=C(x) with these polynomials checks all four above equations at the same time.

Comparing a polynomial to itself

You can even check relationships between a large number of adjacent evaluations of the same polynomial using a simple polynomial equation. This is slightly more advanced. Suppose that you want to check that, for a given polynomial F, F(x+2)=F(x)+F(x+1) with the integer range {0,1…89} (so if you also check F(0)=F(1)=1, then F(100) would be the 100th Fibonacci number)

As polynomials, F(x+2)-F(x+1)-F(x) would not be exactly zero, as it could give arbitrary answers outside the range x={0,1…98}. But we can do something clever. In general, there is a rule that if a polynomial P is zero across some set S=\{x_1,x_2…x_n\} then it can be expressed as P(x)=Z(x)*H(x), where Z(x)=(x-x_1)*(x-x_2)*…*(x-x_n) and H(x) is also a polynomial. In other words, any polynomial that equals zero across some set is a (polynomial) multiple of the simplest (lowest-degree) polynomial that equals zero across that same set.

Why is this the case? It is a nice corollary of polynomial long division: the factor theorem. We know that, when dividing P(x) by Z(x), we will get a quotient Q(x) and a remainder R(x) is strictly less than that of Z(x). Since we know that P is zero on all of S, it means that R has to be zero on all of S as well. So we can simply compute R(x) via polynomial interpolation, since it's a polynomial of degree at most n-1 and we know n values (the zeros at S). Interpolating a polynomial with all zeroes gives the zero polynomial, thus R(x)=0 and H(x)=Q(x).

Going back to our example, if we have a polynomial F that encodes Fibonacci numbers (so F(x+2)=F(x)+F(x+1) across x=\{0,1…98\}), then I can convince you that F actually satisfies this condition by proving that the polynomial P(x)=F(x+2)-F(x+1)-F(x) is zero over that range, by giving you the quotient:
H(x)=\frac{F(x+2)-F(x+1)-F(x)}{Z(x)}
Where Z(x) = (x-0)*(x-1)*…*(x-98).
You can calculate Z(x) yourself (ideally you would have it precomputed), check the equation, and if the check passes then F(x) satisfies the condition!

Now, step back and notice what we did here. We converted a 100-step-long computation into a single equation with polynomials. Of course, proving the N'th Fibonacci number is not an especially useful task, especially since Fibonacci numbers have a closed form. But you can use exactly the same basic technique, just with some extra polynomials and some more complicated equations, to encode arbitrary computations with an arbitrarily large number of steps.

Tools for managing your PhD projects, reading lists, notes, and inspiration.

As a researcher, you have to know everything. But knowledge is useless if it cannot be accessed quickly. An easy-to-use method of archiving information makes taking notes effortless and enjoyable.
As a PhD student in Artificial Intelligence, I use Obsidian (https://obsidian.md) to manage my knowledge.

The article has three parts:

1. What is a note, how to organize notes, tags, folders, and links? This section is tool-agnostic, so you can use most of these ideas with any note-taking app.
2. Instructions for using Obsidian, managing notes, reading lists, and useful plugins. This section demonstrates how I use Obsidian, my preferred knowledge management tool.
3. Workflows: How to use Zotero to take notes from papers, manage multiple projects' notes, create MOCs with Dataview, and more. This section explains how to use Obsidian to solve common scientific problems and manage/maintain your knowledge effectively.

This list is not perfect or complete, but it is my current solution to problems I've encountered during my PhD. Please leave additional comments or contact me if you have any feedback. I'll try to update this article.
Throughout the article, I'll refer to your digital library as your "Obsidian Vault" or "Zettelkasten".
Other useful resources are listed at the end of the article.

1. Philosophy: Taking and organizing notes

Before diving into Obsidian, let's establish a Personal Knowledge Management System and a Zettelkasten. You can skip to Section 2 if you already know these terms.
Niklas Luhmann, a prolific sociologist who wrote 400 papers and 70 books, inspired this section and much of Zettelkasten. Zettelkasten means “slip box” (or library in this article). His Zettlekasten had around 90000 physical notes, which can be found here.
There are now many tools available to help with this process. Obsidian's website has a good introduction section: https://publish.obsidian.md/hub/

Notes

We'll start with "What is a note?" Although it may seem trivial, the answer depends on the topic or your note-taking style. The idea is that a note is as “atomic” (i.e. You should read the note and get the idea right away.
The resolution of your notes depends on their detail. Deep Learning, for example, could be a general description of Neural Networks, with a few notes on the various architectures (eg. Recurrent Neural Networks, Convolutional Neural Networks etc..).
Limiting length and detail is a good rule of thumb. If you need more detail in a specific section of this note, break it up into smaller notes. Deep Learning now has three notes:

• Deep Learning
• Recurrent Neural Networks
  - Convolutional Neural Networks

Repeat this step as needed until you achieve the desired granularity. You might want to put these notes in a “Neural Networks” folder because they are all about the same thing. But there's a better way:

The main issue with folders is that they are not flexible and assume that all notes in the folder belong to a single category. This makes it difficult to make connections between topics.
Deep Learning has been used to predict protein structure (AlphaFold) and classify images (ImageNet). Imagine a folder structure like this:

- /Proteins/ 
     - Protein Folding
- /Deep Learning/
     - /Proteins/ 

Your notes about Protein Folding and Convolutional Neural Networks will be separate, and you won't be able to find them in the same folder.
This can be solved in several ways. The most common one is to use tags rather than folders. A note can be grouped with multiple topics this way. Obsidian tags can also be nested (have subtags).

You can also link two notes together. You can build your “Knowledge Graph” in Obsidian and other note-taking apps like Obsidian.

My Knowledge Graph. Green: Biology, Red: Machine Learning, Yellow: Autoencoders, Blue: Graphs, Brown: Tags.

My Knowledge Graph and the note “Backrpropagation” and its links.

Backpropagation note and all its links

Why use Folders?

Folders help organize your vault as it grows. The main suggestion is to have few folders that "weakly" collect groups of notes or better yet, notes from different sources.
Among my Zettelkasten folders are:

My Zettelkasten's 5 folders

They usually gather data from various sources:

MOC: Map of Contents for the Zettelkasten.
Projects: Contains one note for each side-project of my PhD where I log my progress and ideas. Notes are linked to these.
Bio and ML: These two are the main content of my Zettelkasten and could theoretically be combined.
Papers: All my scientific paper notes go here. A bibliography links the notes. Zotero .bib file
Books: I make a note for each book I read, which I then split into multiple notes.

Keeping images separate from other files can help keep your main folders clean.

I will elaborate on these in the Workflow Section.

My general recommendation is to use tags and links instead of folders.

Maps of Content (MOC) 

Making Tables of Contents is a good solution (MOCs).
These are notes that "signposts" your Zettelkasten library, directing you to the right type of notes. It can link to other notes based on common tags. This is usually done with a title, then your notes related to that title. As an example:

An example of a Machine Learning MOC generated with Dataview.

As shown above, my Machine Learning MOC begins with the basics. Then it's on to Variational Auto-Encoders. Not only does this save time, but it also saves scrolling through the tag search section.
So I keep MOCs at the top of my library so I can quickly find information and see my library. These MOCs are generated automatically using an Obsidian Plugin called Dataview (https://github.com/blacksmithgu/obsidian-dataview).
Ideally, MOCs could be expanded to include more information about the notes, their status, and what's left to do. In the absence of this, Dataview does a fantastic job at creating a good structure for your notes.
In the absence of this, Dataview does a fantastic job at creating a good structure for your notes.

2. Tools: Knowing Obsidian

Obsidian is my preferred tool because it is free, all notes are stored in Markdown format, and each panel can be dragged and dropped. You can get it here: https://obsidian.md/

Obsidian interface. 

Obsidian is highly customizable, so here is my preferred interface:

The theme is customized from https://github.com/colineckert/obsidian-things

Alternatively, each panel can be collapsed, moved, or removed as desired. To open a panel later, click on the vertical "..." (bottom left of the note panel).

My interface is organized as follows:

How my Obsidian Interface is organized.

Folders/Search:
This is where I keep all relevant folders. I usually use the MOC note to navigate, but sometimes I use the search button to find a note.

Tags:
I use nested tags and look into each one to find specific notes to link.

cMenu:
Easy-to-use menu plugin cMenu (https://github.com/chetachiezikeuzor/cMenu-Plugin)

Global Graph:
The global graph shows all your notes (linked and unlinked). Linked notes will appear closer together. Zoom in to read each note's title. It's a bit overwhelming at first, but as your library grows, you get used to the positions and start thinking of new connections between notes.

Local Graph:
Your current note will be shown in relation to other linked notes in your library. When needed, you can quickly jump to another link and back to the current note.

Links:
Finally, an outline panel and the plugin Obsidian Power Search (https://github.com/aviral-batra/obsidian-power-search) allow me to search my vault by highlighting text.

Start using the tool and worry about panel positioning later. I encourage you to find the best use-case for your library.

Plugins

An additional benefit of using Obsidian is the large plugin library. I use several (Calendar, Citations, Dataview, Templater, Admonition):
Obsidian Calendar Plugin: https://github.com/liamcain
It organizes your notes on a calendar. This is ideal for meeting notes or keeping a journal.

Calendar addon from hans/obsidian-citation-plugin
Obsidian Citation Plugin: https://github.com/hans/
Allows you to cite papers from a.bib file. You can also customize your notes (eg. Title, Authors, Abstract etc..)

Plugin citation from hans/obsidian-citation-plugin
Obsidian Dataview: https://github.com/blacksmithgu/
A powerful plugin that allows you to query your library as a database and generate content automatically. See the MOC section for an example.
Allows you to create notes with specific templates like dates, tags, and headings.

Templater. Obsidian Admonition: https://github.com/valentine195/obsidian-admonition
Blocks allow you to organize your notes.

Plugin warning. Obsidian Admonition (valentine195)
There are many more, but this list should get you started.

3. Workflows: Cool stuff

Here are a few of my workflows for using obsidian for scientific research. This is a list of resources I've found useful for my use-cases. I'll outline and describe them briefly so you can skim them quickly.
3.1 Using Templates to Structure Notes
3.2 Free Note Syncing (Laptop, Phone, Tablet)
3.3 Zotero/Mendeley/JabRef -> Obsidian — Managing Reading Lists
3.4 Projects and Lab Books
3.5 Private Encrypted Diary

3.1 Using Templates to Structure Notes

Plugins: Templater and Dataview (optional).
To take effective notes, you must first make adding new notes as easy as possible. Templates can save you time and give your notes a consistent structure. As an example:

An example of a note using a template.

### [[YOUR MOC]]
# Note Title of your note
**Tags**:: 
**Links**::

The top line links to your knowledge base's Map of Content (MOC) (see previous sections). After the title, I add tags (and a link between the note and the tag) and links to related notes.
To quickly identify all notes that need to be expanded, I add the tag “#todo”. In the “TODO:” section, I list the tasks within the note.
The rest are notes on the topic.
Templater can help you create these templates. For new books, I use the following template:

### [[Books MOC]]
# Title
**Author**:: 
**Date::
**Tags:: 
**Links::

A book template example.

Using a simple query, I can hook Dataview to it.

dataview  
table author as Author, date as “Date Finished”, tags as “Tags”, grade as “Grade”  
from “4. Books”  
SORT grade DESCENDING

using Dataview to query templates.

3.2 Free Note Syncing (Laptop, Phone, Tablet)

No plugins used.

One of my favorite features of Obsidian is the library's self-contained and portable format. Your folder contains everything (plugins included).

Ordinary folders and documents are available as well. There is also a “.obsidian” folder. This contains all your plugins and settings, so you can use it on other devices.
So you can use Google Drive, iCloud, or Dropbox for free as long as you sync your folder (note: your folder should be in your Cloud Folder).

For my iOS and macOS work, I prefer iCloud. You can also use the paid service Obsidian Sync.
3.3 Obsidian — Managing Reading Lists and Notes in Zotero/Mendeley/JabRef
Plugins: Quotes (required).

3.3 Zotero/Mendeley/JabRef -> Obsidian — Taking Notes and Managing Reading Lists of Scientific Papers

My preferred reference manager is Zotero, but this workflow should work with any reference manager that produces a .bib file. This file is exported to my cloud folder so I can access it from any platform.

My Zotero library is tagged as follows:

My reference manager's tags

For readings, I usually search for the tags “!!!” and “To-Read” and select a paper. Annotate the paper next (either on PDF using GoodNotes or on physical paper).
Then I make a paper page using a template in the Citations plugin settings:

An example of my citations template.

Create a new note, open the command list with CMD/CTRL + P, and find the Citations “Insert literature note content in the current pane” to see this lovely view.

Citation generated by the article https://doi.org/10.1101/2022.01.24.22269144

You can then convert your notes to digital. I found that transcribing helped me retain information better.

3.4 Projects and Lab Books

Plugins: Tweaker (required).
PhD students offering advice on thesis writing are common (read as regret). I started asking them what they would have done differently or earlier.

Time management requires planning. Keeping track of multiple projects and lab books is difficult during a PhD. How I deal with it:

• One folder for all my projects
• One file for each project
  I use a template to create each project

### [[Projects MOC]]  
# <% tp.file.title %>  
**Tags**::  
**Links**::  
**URL**::  
**Project Description**::## Notes:  
### <% tp.file.last_modified_date(“dddd Do MMMM YYYY”) %>  
#### Done:  
#### TODO:  
#### Notes

You can insert a template into a new note with CMD + P and looking for the Templater option.

I then keep adding new days with another template:

### <% tp.file.last_modified_date("dddd Do MMMM YYYY") %>  
#### Done:  
#### TODO:  
#### Notes:

This way you can keep adding days to your project and update with reasonings and things you still have to do and have done. An example below:

Example of project note with timestamped notes.

3.5 Private Encrypted Diary

This is one of my favorite Obsidian uses.
Mini Diary's interface has long frustrated me. After the author archived the project, I looked for a replacement. I had two demands:

1. It had to be private, and nobody had to be able to read the entries.
2. Cloud syncing was required for editing on multiple devices.

Then I learned about encrypting the Obsidian folder. Then decrypt and open the folder with Obsidian. Sync the folder as usual.
Use CryptoMator (https://cryptomator.org/). Create an encrypted folder in Cryptomator for your Obsidian vault, set a password, and let it do the rest.
If you need a step-by-step video guide, here it is:

Conclusion

So, I hope this was helpful!
In the first section of the article, we discussed notes and note-taking techniques. We discussed when to use tags and links over folders and when to break up larger notes.
Then we learned about Obsidian, its interface, and some useful plugins like Citations for citing papers and Templater for creating note templates.
Finally, we discussed workflows and how to use Zotero to take notes from scientific papers, as well as managing Lab Books and Private Encrypted Diaries.
Thanks for reading and commenting :)

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