# AA II Taking Over Math Club

This Wednesday, April 27th, Dr. Savic’s Abstract Algebra II students will host an exhibition of children’s books and mobile apps about Abstract Algebra! Come and learn from their creative endeavors!  The art exhibition will run from 6-7.  From 5:30-6 there will be officer elections for next year.

As always, there will be pizza.  Hope to see you all there!

Here are the details:

When: Wednesday, April 27, at 5:30

Where: PHSC1105

# You and I are no more than 4.57 friendships away from one another.

Over the past hundred years social relationships have evolved, making us increasingly more connected with those around the world in a wide variety of ways.  In this post, I will tell you a story about the connectedness of social relationships, taking us from an early 20th century Hungarian author to an 18th century Swiss mathematician, to one of the largest tech giants of our era, and touch on some cool mathematics along the way.

In an attempt to articulate his observation of the growing interconnectedness within society, in 1929, Hungarian author Frigyes Karinthy famously suggested that any two individuals were connected by 6 friendships, which have been coined “degrees of separation.”  In other words, given any other person, say famous actor Will Smith, you should be able to find 5 other people such that the 7 of you are connected by 6 friendships, as depicted below.

I bring Will Smith up since he starred in the 1993 theatrical version of the famous play “Six Degrees of Separation”, which was loosely based on this notion of social interconnectedness.

Philosophy and media aside, how can we actually mathematically encode and measure social relationships?  The answer: graphs.  First studied by the famous mathematician Leonhard Euler, a graph is simply a collection of vertices and the edges that connect them.  For example, the above picture connecting you to Will Smith is a graph with 7 vertices and 6 edges.  Euler famously used graphs to solve the Bridges of Konigsberg problem.  Since then, graph theory has proven to be a powerful tool to solve many problems.  Google has used graphs to encode the connectedness of websites across the internet, and Facebook has encoded the friendships of their users with their Facebook Friendship Graph.  This Facebook Friendship Graph is pretty straight forward:

A vertex <=>  A Facebook user

An edge connects two vertices <=> The two users are Facebook friends

To celebrate its 12th birthday, Facebook recently set about to actually compute the average degrees of separation between two Facebook users.  In terms of the Facebook Friendship Graph, this amounts to finding the average distance (i.e. minimal edge-path length) between two vertices.  This kind of computation can be done by brute force methods for small graphs.  However, as of the writing of this post, there are about 1.6 billion Facebook users, and they are highly connected.  That means there are over 1018 pairs of vertices whose distance needs to be checked, something that is simply too computationally intensive to do directly.  So Facebook instead used a variety of techniques to statistically analyze this enormous data set, including hash functions, the Flajolet–Martin algorithm, and a powerful graph processing program Apache Giraph.  Check out their blog post below where they explain their methods in greater detail.

Facebook found that the average distance between any two vertices in the Facebook Friendship Graph is approximately 4.57 so, in their language, there is an average of 3.57 degrees of separation between two users.

You can go to their blog and see the average distance between you and a random other Facebook user.  Personally, I am an average distance of 4.15 away (i.e. 3.15 degrees of separation away) from anyone else, confirming my longtime suspicion that I am highly connected.

# Happy 200th Birthday George Boole!

Today (November 2, 2015) is the 200th Birthday of the mathematician George Boole.  To celebrate, here is a fantastic song and video created specially for this occasion.

“The Mathematician – The Bould Georgie Boole”
Performed by The Arthur Céilí Band, featuring Jim Flanagan & Mike Simpson

You can also download the song so you can listen to it all the time here:

Besides being a lot of fun, this song and video is a great little biography.  Of course, if you want more info, you can check out George Boole’s Wikipedia page:
https://en.wikipedia.org/wiki/George_Boole

That being said, there is one story I would like to point out here.  George Boole worked in many areas, but perhaps the most important area today is what we now call Boolean algebra  (https://en.wikipedia.org/wiki/Boolean_algebra).

Introduced by Boole in the around 1850, it was not until the 1930’s that an undergraduate at the University of Michigan named Claude Shannon realized this hitherto abstract theory could be applied to electromechanical relays.   Shannon went on to write a master’s thesis on this topic at MIT.  Since then Boolean algebra has become the basis of digital circuit design.

Here is my challenge to all you undergraduates out there: can you make a connection like Shannon and use a seemingly totally abstract mathematical theory to model some real world phenomena?  Maybe you too can make some revolutionary discovery!

# OU Math Movie Night: A Beautiful Mind

I am excited to announce the first OU Math Movie Night.  This Thursday, we will be showing the critically acclaimed “A Beautiful Mind” in honor of John Nash who passed away earlier this year.  This event is open to everyone, so invite your friends to come watch this movie.  Afterwards we will have an informal discussion of game theory.  Here are the details:

When: Thursday, October 22, 5:30-8:00pm

Where: PHSC 1105

What: Movie, Food, and Math!

# Gabriel’s Horn — an oldie but still a goodie!

Here is a “paradox” that one encounters in second semester Calculus. Start with the function y = 1/x in the first quadrant, which is a connected piece of the usual rectangular hyperbola. Consider the curve in the range x ≥ 1. Rotate this curve about the X-axis to generate a 2 dimensional “horn” or infinite “bucket” in 3 dimensional space i..e, something like this:

Now using standard Calculus 2 techniques one may compute separately the volume of this object (how much stuff it can hold) and its surface area (how large an area to paint). And the calculation looks this:

Hmm! You see the problem — an object with finite volume but infinite surface area!! So if you were to fill it with paint there is only so much paint it can hold. But if you were to paint it, then no amount of paint would be enough. Can you explain the paradox??

# Free Books!

No one is more sensitive to the ever growing cost of textbooks than students. And it may surprise students to know that faculty are also pained by publishers selling the 8th edition of a book for \$200. While we are happy when publishers offer great books and make money from that effort, something probably needs to change in this model. Here is a list of “free” books approved by the American Institute of Mathematics. Note this is free as in beer, not free as in freedom (what is the distinction? You can find that here)

Approved textbooks from AIM.

In particular, note the multiple Calculus and ODE offerings. If nothing else you can use these as references or as sources of problems for further study.