|Category Theory Lectures|
I have written and am constantly updating a set of lecture notes for an introductory category theory course aimed at undergraduate students. I am currently compiling them in a book format.
|Becoming a Professor|
My primary career goal is to become a professor in a computer science or mathematics department of a university. Feel free to contact me if you can offer help. :)
|Semantics and Verification Lecture Notes (Winter 2020)|
During the Winter 2020 term, I have typed lecture notes for Semantics and Verification taught by Prof. Colin Riba.
|Higher Algebra 2 Lecture Notes (Winter 2019)|
During the Winter 2019 term, I have typed lecture notes for Higher Algebra 2 taught by Prof. Henri Darmon. David Marcil has also proofread and completed a few parts of the notes.
|SUMM 2019 Conference|
I gave a 45 minute talk at SUMM in January 2019 introducing the P vs. NP problem.
|Lecture Notes for Fall 2018|
I compiled a study document for the content covered in Higher Algebra 1 taught by Prof. Eyal Goren in Fall 2018 and I typed up lecture notes for the first section of a course on Combinatorial Optimization taught by Prof. Bruce Reed.
|Category Theory Study Group (Summer 2018)|
In the summer term of 2018, I organized a reading group on category theory with a few friends, we followed the lecture notes of Prof. Mariusz Wodzicki and tried to produce our own set of notes. The goal of this project was to get familiar with basic category theory concepts.
|SUMM 2018 Conference|
I gave a 35 minute talk at SUMM in January 2018 introducing some concepts in machine learning.
|Graph Theory Lecture Notes (Fall 2017)|
I typed up lecture notes for my MATH 350 class given by Prof. Jan Volec.
|Study Documents (Fall 2017)|
I compiled study documents for MATH 456 taught by Prof. Eyal Goren, MATH 356 taught by Prof. Linan Chen and MATH 350 taught by Prof. Jan Volec.
|Cryptocurrency Browser Extension (Summer 2017)|
While I was learning about cryptocurrencies, I decided to code a browser extension that can display the balance in your cryptocurrency wallets.
|Web Games (Summer 2016)|
|Fractal Design (Winter 2015)|
I have designed a website to present a fractal design project for my calculus 2 class given by Prof. Patrick St-Amant.
|Project Euler (Fall 2015)|
While I was learning Python, I solved the first 50 problems from Project Euler.
Resources I Use
|Friends and Collaborators|
By far, my most reliable and effective resource is the set of friends and collaborators I was lucky to meet. Here is a non-exhaustive enumeration of them.
As you might have inferred from the rest of my website, I really enjoy teaching. Unfortunately, my day-to-day activities involve too many levels of abstraction for me to easily describe them to nonexperts. Thanks to years of practice, I was able to find very good (in my opinion) answers to the question "What do you study?" that help my family and friends better understand what I do. Here are the best relatively short and simple explanations I could come up with to describe my favorite fields of study (in no particular order).
The main goal of complexity theory is to answer the question "What makes a problem hard or easy to solve with a computer?". Of course, there are many angles to attack this question, for example, one needs to define what is a computer, what is a problem, and what it means for a problem to be hard or easy. I will save these discussions for later and only demonstrate the difficulty in answering this question. Let me describe three different tasks varying in complexity.
Your first task is to win at chess against the best artificial intelligence designed to optimally play chess. To make this easier on you, you can control the white pieces (you play first) and it is considered a win for you if the game resolves in a stalemate (tie). You probably know that this is almost impossible to do; all grandmasters of chess (the best human players) cannot compete even with AI players built decades ago.
Your second task is to play two games of chess against the best AI and win (or tie) on at least one board. To make it slightly fairer, in your first game, you control the white pieces, and in your second game, you control the black pieces. I will let you convince yourself that the second task is harder than the first one because the latter is included in the former. Namely, to complete the second task, you must win at chess against the AI and hence complete the first task.
Your third task is to play two games of chess simultaneously against the best AI and win (or tie) on at least one board. Again, there is a board where you play whites and a board where you play blacks. At first sight, this task may look harder than the first one (because you are playing on two boards at once) and just as hard as the second one. In any case, this should still be hopeless.
Surprisingly, the third task is the easiest to complete and anyone could do it after being told the strategy. You simply have to copy the moves done by the AI on one board on the other board. That is, you let the AI play its first move on the board where it plays whites and you do the same move on the board where you play whites, then you wait for the AI's answer to that move and you replicate it on the other board. Whatever happens during the game, if the AI wins (or ties) on one board, you automatically win (or tie) on the other board, thus you complete the task.
It is very fascinating to me (and I guess to most complexity theorists) how these three tasks with a very similar description have very different intrinsic difficulties. To rephrase the ultimate goal of complexity theory, it would be wonderful if we could tell how hard it is to solve a problem by looking only at its statement.
As I have mentioned, there is a lot more to complexity theory that was not advertised in this toy example. Still, I hope this makes you wonder more deeply about the original question as many computer scientists do every day.
My main field of research is called categorical semantics. I have yet to come up with a quick layman explanation for the term 'categorical', but I think I can give you the gist of what studying the semantics of programming languages means.
It is infamously hard to predict the result of executing a computer program by looking at its code. Of course, humans (with programming experience) are exceptionally good at this task; if you ask a programmer to look at some simple code written by another human, they will surely be able to describe what it does. However, this takes time and as the code gets longer and more complex, humans are more likely to make mistakes. This is the main reason why bugs exist — ironically, the origin of the word "bug" suggests that the error is not the programmer's fault. The principal objective of program semantics is to help design programs that can understand how other programs behave by looking at their code.
The term semantics comes from a central step in the analysis of programs where we give meaning to each part of the code before bringing it together into information about the whole program. The details vary wildly depending on the method used, how the program is specified (e.g. what programming language), and the properties of the program that are considered. Anyway, you can basically think of this step as a translation of the code into abstract mathematical objects which themselves have to be interpreted to understand the result.
Let me use a cooking analogy to give more intuition about the process described above. It is also hard to predict the result of a recipe without following its every step. This time, even a human (with cooking experience) will have a hard time being able to tell with good accuracy what the end result is supposed to look like, if it will taste good to them, etc. — I concede that a very good chef who has seen similar recipes before will fare better, but if they have never seen how, say, puff pastry is made, I doubt they can foresee the transformation from layers of dough and butter into flaky, crusty and light pastries.
One could imagine a program that reads a recipe and generates the outcome of each step as a sequence of chemical reactions (these replace the abstract mathematical objects in the programming story) that will happen during the cooking. The program could then extract some useful information to tell us whether there is too much compounds that taste like burnt food, there is not enough chocolate flavor compounds, the Maillard reaction has produced enough brown color, enough water evaporated to get a crispy crust, and so on...
It is hard to say with certainty which of analyzing code or recipes is easiest (see my explanation of complexity theory), but I would certainly love it if we are able to do either in my lifetime.
|Hodgepodge of Ideas|
Here are hand-picked ideas (they maximize some function depending on originality and potential) from my much longer list of ideas.
Crowdsourcing YouTube Captions in Class
I would like to build a platform where language teachers can give as homework to their students the task of captioning YouTube videos. Initially, the teacher can assemble a big playlists with videos in their desired language which have no captioning yet (or maybe the video has caption in one language but not in another). Then they can ask their students to choose a video in this playlist and caption it (or translate the existing captions). Finally, the teacher can go over the work done by the students and correct it before submitting the captions to YouTube.
I think that many people (teachers, creators and students) could benefit from a platform like this. I trust the great diversity in the content published to YouTube will make it possible for every teacher to find videos they think are suitable for their class and for every student to find a video they can engage with. The creators will also reach a larger audience through this platform.
I am also pretty sure this could easily be implemented as part of Google Classroom. As of November 2020, I have not found such a project and even worse, YouTube has recently removed the community contributions.
|No Rights Reserved|
The following is a copy & paste from Toby Bartels' page, but it applies to me as well.
I maintain no copyright or patent restrictions on any of my creative work. As far as copyright and patent law are concerned, I grant you irrevocable permission to do anything with my work that you wish. This is a perfectly free, noncopyleft licence. For example, you may:
Some of the creative work that I produce is a modification derived from other persons' previous work. Since the other persons may reserve legal rights to their work, you might not have legal permission to do what you want with my work, even as far as copyright and patent law are concerned. But that could only be because you did not have permission from the other persons; you already have permission from me.
I request that you give me credit for my work, and that you offer any derivative works and so forth to other people with no conditions, as I have offered this work to you with no conditions. But I will not enforce these requests legally; they are not requirements. You will have to decide for yourself.