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.

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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)

My teaching advice and resources (so far)

As I general rule I find thinking about math pedagogy deeply rewarding. Teaching a technical and beautiful discipline like math is difficult to do well. Students come from all sorts of backgrounds, the material can be challenging, and there are tons of moving parts in a course. It’s a challenge that I find exhilarating.

On the other hand, I find the act of reading the scholarship of math education to be dreadful and unpleasant. It is filled with jargon and hero-worship.

That being said, I’ve been extremely lucky to have great mentors and colleagues to bounce ideas off of. I’ve collected some of this advice in a Reddit post, which I’ll recreate here.

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How does modern AI work? – Math for my mom

This is part of a series of posts aimed at helping my mom, who is not a scientist, understand what I’m up to as a mathematician.

 

Lately, Artificial Intelligence (AI) has made some remarkable milestones. There are computers that are better than humans at the strategy board game GO and at Poker. Computers can turn pictures into short moving clips and can “enhance” blurry pictures as in television crime shows. They can also produce new music in the style of Bach or customized to your tastes. It’s all very exciting, and it feels pretty surreal; remember back when Skype video calling felt like the future?

I’m going to give you a broad overview for how these types of AI work, and how they learn. There won’t be any equations or algebra.

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