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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.

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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

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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.

groupphoto
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.

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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)

Stevo’s Forcing Class Fall 2012 – Class 9

(This is the ninth lecture in Stevo Todorcevic’s Forcing class, held in the fall of 2012. You can find the eighth lecture here. Quotes by Stevo are in dark blue; some are deep, some are funny, some are paraphrased so use your judgement. As always I appreciate any type of feedback, including reporting typos, in the comments below.)

Summary

  • We define the notion of properness.
  • Show that the poset \in -Collapse is proper.
  • Justin Moore gives some intuition.
  • Explain the Open Colouring Axiom.
  • Give a related proper poset.

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