Machine Learning Observations 1 – Turning a picture into a vector doesn’t lose information!

This term I’m auditing CSC311 Introduction to Machine Learning. I’ve tried learning this topic before, but it wasn’t at the right level for me. This time I’m optimistic because my colleague Sonya Allin is teaching the course, and I feel comfortable bombarding her with naive questions.

As I was sitting in the first class, I had my mathematician hat on and I noticed some things. I’m not saying these are deep, or unknown things, but they were interesting to me. Maybe they’ll be interesting to you too!

Observation 1: When we turn a picture into a vector we seem to lose a lot of geometric data

A standard way of storing a (greyscale) picture as data is to first write it as a matrix of data (an nxn table) where the entries are intensities (on a scale of 0-255). Then it cuts chopped up and reconstituted into a vector.

Continue reading Machine Learning Observations 1 – Turning a picture into a vector doesn’t lose information!

How many sizes of infinity are there?

“O God, I could be bound in a nutshell, and count myself a king of infinite space – were it not that I have bad dreams.”

Hamlet, Act 2, Scene 2. Lines 252-254.

Recently, a colleague asked me:

I know that there are different sizes of infinity, but what I want to know is how many different sizes of infinity there are?

-Curious Colleague

This is a great question! I tried to explain my answer at the time, but it came out garbled and I think I confused him more than I helped. So this post is an effort to remedy that and answer his question.

Continue reading How many sizes of infinity are there?

Review of Ungrading (Blum 2020) for use in UTM math courses

My brief, unfair summary

At its best, the book Ungrading (Blum, 2020) and its authors

  • Provide many blueprints, implementations, and reflections on ungrading practices.
  • Address many practical difficulties of implementing these practices.
  • Repeatedly make calls for revolution, and provide tools and recipes for making that happen.
  • Encourage critical pedagogy.
  • Come from a variety of disciplines and settings (K-12 vs post-secondary).

At its worst, the book and its authors

Continue reading Review of Ungrading (Blum 2020) for use in UTM math courses

Aligned teaching and punishing student success

This term I’m teaching (and coordinating) a 1200 person Integral Calculus course at UTM. Term Test 1 went surprisingly well for students, and Term Test 2 was an unqualified success (very high average, lots of people got 100% on the test).

This lead to the following (paraphrased) question on our course message board:

Is the exam going to be a lot more difficult because the averages on the term tests were so high?

I took the time to think about exactly why I don’t like adjusting the difficulty of exams based on students doing well previously. Here’s the answer I posted on our message board.

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How to organize an equitable pre-CUMC conference for students

Every summer, Canadian undergraduate students in mathematics meet at the Canadian Undergraduate Mathematics Conference (CUMC). Hundreds of students attend, and it gives them a chance to meet other people excited by mathematics. Students are also encouraged to give a short presentation on a math topic that interests them.

I attended the 2007 CUMC at Simon Fraser university and the 2008 CUMC at the University of Toronto (where I would go on to complete my PhD and then eventually work at).

In the summer of 2018, while I was a Post Doc at the University of Calgary, we hosted a “mini pre-CUMC conference” for undergrads to give their presentations ahead of time. It was so successful that I ran an expanded version of this at the University of Toronto for CUMC 2019.

I think these events and workshops are important for all students, but in particular it helps break down barriers to entry for marginalized students. With that in mind, I’m sharing my resources, thoughts and experiences about our pre-CUMC conference with the hope that other universities and colleges in Canada will benefit.

exercise-86200_640

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Unifying themes in Ramsey Theory – BIRS 2018

In November 2018, 41 of the top researchers in Ramsey theory met at the BIRS in Banff for the Unifying Themes in Ramsey Theory conference. By all measures the conference was a big success. What makes Ramsey theory so special is that it has wide ranging impacts in diverse fields in mathematics. The participants gave talks showing how Ramsey theory has impacted fields like graph theory, topological dynamics, set theory, model theory, operator algebras, logic and statistics.

Since I have a somewhat broad base of knowledge in Ramsey theory, I tried my best to give a short description of each of the speakers in language that makes sense to me. My view is biased, and my intent is always to show off the amazing work everyone is doing. I hope nothing comes across as negative or critical; that is not my intent.

You can find all the abstracts here, and all the videos of their talks here.

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The participants of the 2018 Unifying Themes in Ramsey Theory conference at BIRS, Banff, Alberta, Canada. November 18-23, 2018. Full list of participants.

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How does the size of a cookie depend on the size of the ball of dough?

This term I’m teaching Calculus 3 which involves learning about the concept of curvature. This is a measurement of how bendy or curvy something is. The flatter something is, the less curvature it has.

We learn in class that a circle or sphere of radius r has curvature inversely proportional to its radius, that is it has curvature \frac{1}{r}.

In this class we used baking cookies to illustrate how the curvature of an object can change over time. Seen from over top, a ball of cookie dough flattens out as it bakes.

giphy

This got me thinking about how exactly is the size of the ball of cookie dough related to the size of the cookie you get in the end? So I did some science.

Continue reading How does the size of a cookie depend on the size of the ball of dough?

IMO resources for Graph Theory

I will be participating as a trainer for Canada’s 2018 IMO Summer Training camp. I’m giving a session on graph theory. As I prepared my notes I found many resources online that already cover some aspects of graph theory. So here are those resources:

“IMO Training 2008: Graph Theory” by Adrian Tung.

This is an in-depth description of the basic combinatorial and geometric techniques in graph theory. It is a very thorough and helpful document with many Olympiad level problems for each topic. (No solutions are given.)

Topics include:

  1. Trees and Balancing
  2. Friends, Strangers and Cliques
  3. Directed Graphs and Tournaments
  4. Matchings
  5. Hamiltonian/Eulerian Paths/Cycles

“Graph Theory” by Po-Shen Lo. (2008)

A large collection of problems and topics almost all of which have solutions or hints.

Topics include:

  1. Basic facts
  2. Extremal Graph Theory
  3. Matchings
  4. Ramsey Theory
  5. Planarity

“Graph Theory” by Matthew Brennan. (Canada Winter Camp, 2014)

Contains a concise list of important results together with a guided discussion to five example problems that use graph theory.

“Probabilistic Method/Graph Theory” by James Rickards. (Canada Summer Camp, 2015)

An introduction to the probabilistic method in graph theory along with 10 problems.

“SIMO Graph Theory Training”. (SIMO training 2003)

A list of about 30 problems and solutions in graph theory.

Topics:

  1. Graph Theory
  2. Coloring problems

“Ramsey Theory and the IMO” by Ben Green. (2008)

This is a 4 page article that introduces Ramsey Theory for graphs and arithmetic progressions and its historical relation to the IMO.

“Coloring Points” at Cut-the-knot

A collection of 12 topics about coloring graphs and planes. There are many problems with solutions.

“Equivalence of seven major theorems in combinatorics” by Robert Borgersen (2004).

This series of slides states 7 results in extremal combinatorics that are really the same.

Topics:

  1. Dilworth’s Theorem
  2. Konig’s Bipartite Theorem
  3. Hall’s Marriage Theorem
  4. Menger’s Theorem
  5. (Others)