Monday, December 22, 2014

Math in the Media: Prime Gaps....

I am always amazed at how some of the most vexing, curious and fascinating puzzles in mathematics can be stated so simply, even as they evade solution or even complete understanding for centuries.  It is one of the more alluring aspects of this trade. 

Here's one:  Just how big can the gaps between consecutive pairs of prime numbers get as one traverses the natural numbers out toward infinity? 

One would expect the gaps to get larger and larger and also tend toward infinity in the long run, no?  But showing this, and providing some sort of measure of the growth of the size of the gaps as one goes "out there" has been remarkably elusive. 

I'll let you read this nice article by Erica Klarreich in Quanta Magazine, to "see" that progress has recently been made, and there is promise of more progress coming. 
Mathematicians Make a Major Discovery About Prime Numbers
Will math ever cease to amaze....

Wednesday, December 3, 2014

Math in the Media: Mathematics is Evil?

There is an old saying (really, I just made this up):  If you are looking for immortality, you have two choices.  Become famous, or become infamous.  Either way, you will not soon be forgotten....

A professor here is uncovering an interesting conjecture.  The fictitious, evil mastermind and arch-nemesis of Sherlock Holmes, Professor James Moriarity, was modeled by Sir Arthur Conan Doyle on a mathematics professor here at Hopkins.  The similarities are quite striking, detailed here by current professor Carl McTague

Moriarity at Hopkins

Give it a read.  Nice idea, eh?

Friday, September 26, 2014

Math in the Media: Return on Investment in a Math Degree?

So I would up devoting my life to the study of Mathematics because I absolutely love the subject.  It is inherently beautiful, surprisingly counter-intuitive, and seems to exhibit a logical framework for all that is in a way that I find ever intriguing.

However..., the study of math at a high level is also quite lucrative!!

Here is an article from Bloomberg Business Week, from June:
Undergrad Business Majors Don't Get the Career Payback Math Majors Do
You must love this title from my perspective.  The article highlights a measure of the lifetime worth of different college majors in term of a return on investment of time and effort.  Some majors are harder than others, I am sure.  And why they decided to include math and computer science together is a mystery to me (perhaps that is how the business world sees us?  As the studiers of logic?

In any case, they make a good case for choosing math as a major while here in the Ivory Tower. Call that reason number..., what... 132 in the countably infinite number of reasons why someone can benefit from choosing math as a major?  (BTW, have you heard that over 80% of statistics are made up on the spot?)

Give it a read.  I will await your change-of-major form....  ;-)

Monday, August 18, 2014

Real Mathematics!

I am gearing up for the fall semester here at Hopkins.  This fall, I am teaching our version of vector calculus (aka multivariable calulus), 110.202 Calculus III.  It is a great course, beautifully visual and quite subtle in many ways.  Good stuff! 

I sent out a "hello" email to my 300+ students, inviting them to check out the webpage and generally welcoming them to the course.  In this email, I say near the end:

Even though one may think of calculus as simply a math course where one learns some techniques for solving physics and statistics problems, it actually is much more than this.  Instead of simply learning techniques, we will be learning how and why the techniques even exist, what they say about the structure of mathematics like calculus, and how to think analytically and reason deductively and abstractly.  THIS is the real mathematics.  The techniques will come along for the ride.  You will learn those also. 
Perhaps the best way to drive this point home is the following:  It does not matter what your current and/or future major is or will be.  You are here at Hopkins to train to be a scholar at something.  Part of that training includes proficient and efficient understanding of the abstract logical structural framework found in all complex ideas and constructions.  This is really what mathematics is.  We typically use numbers and operations on those numbers to study and exhibit mathematical ideas because they provide the self-consistent framework needed for the study.  I will say a lot more about this on the first day of class. 
Perhaps this is one of my personal definitions of mathematics.  But I like it.  Make sense?

Thursday, August 14, 2014

Math in the Media: Finally!! A female Fields Medalist....

Well, I am back and in time for a new semester here at Hopkins.  And I am back with some very nice news. 

The next recipients of one of our field's top honors, the Fields Medal, includes Maryam Mirzakhani, an Iranian born mathematician at Stanford University.  She is a dynamicist (a mathematician whose field of study is dynamical systems) and the first woman to receive this prize since its inception in 1936.  She shares the prize this year (the prize is given out every four years) with three other mathematicians, listed in this article in the New York Times
Top Math Prize Has Its First Female Winner.
And while there should be nothing special about a woman receiving the prize (math is hardly a gender-specific endeavor), I do have sort of a glass-ceiling-breaking-moment feeling here.  Congratulations, Professor and Professors!  Here's to more outstanding math research.

And here is another nice write up of this event and her contributions to mathematics.

Tuesday, June 24, 2014

Math in Film: NUMB3RS

So I finally got around to watching he pilot for the series NUMB3RS.  Yeah, I know, the show is way old and quite dead now (it ran from 2005-2010).  But I tend to avoid shows that have mathematicians as main characters.  Hollywood (and environs) understand so little about our practice that they rely on stereotypes rather than seek to educate or play straight.  My son, however, found the pilot, watched it, and promptly told me that the show actually gets some genuine features of mathematicians.  So, armed with his endorsement, I jumped in.

For those who do not know, the show is a crime drama centering around an FBI agent who winds up using his brother's help and expertise to solve very complicated crimes in LA.  The brother is said to be a young, genius, mathematician (professor at Stanford).  The brother's mathematical insight and ideas are a central aspect of the show.  I suspect that in each episode, they are crucial to the solution of the case.  I have only watched the one introductory episode so far.  But they is some merit here.

For the most part, mathematicians are considered brilliant but weird, fascinating but off-putting, playful but socially awkward to the point that people do not really know what to do with them.  I must admit that this is a fairly accurate portrayal even from the inside.  NUMB3RS gets this part right, and the character mathematician has the right zest for life and obsession with the logical structure of everything that he can easily make his way around a conference unnoticed. 

What works is (1) "his work is his life is his work" aspect of how he approaches new puzzles, (2) the idea that there is an elegant solution to every problem and the trick is to simply find it, (3) the notion that everything is mathematical in that everything has a logical structure which, once understood, can be exploited, (4) the absolute certainty of results once proved, and lastly (5) the idea that mistakes are merely foundations for building more enlightened theory.  All seemingly fresh, given other depictions I have seen in film and TV.  I guess this idea in future episodes, like in this one, would be a single big lesson taught in each case, and each lesson would be different.

What didn't work for me?  Well..., the acting was generally very wanting.  The pilot was a bit like the many CSI-type crime dramas where a team is working together to solve a crime.  Every scene with the team has each member saying one line which is crucial to the case (so that they all contribute), and there is little wasted banter. Too unrealistic for me.  Also, the idea that mathematicians only deal with equations, and to them everything is an expression.  In this idea, mathematicians can only work when their ideas are rendered into equations.  This is not true at all.  That was, I suspect a simplification to mesh with the stereotype. 

In any case, it was refreshing to see a depiction much closer to reality than is usual.  Give it is shot.

Tuesday, April 1, 2014

Playfully Serious Math: A glimpse at Vi Hart and Fibonacci

I am often struck by just how repulsive mathematics is to some people when, in my eyes, it is all an absolute kaleidoscope of color, art and logical splendor.  But it is not always easy to get someone else to see what you see.  This is what education is all about, I guess.  One step at a time....

I was recently turned on to an absolutely wonderful math and science educator whose videos would do well to provide the backbone of the next generation of the Common Core, at least in math education.  Vi Hart is a videographer (is that what one would call someone who makes videos) who specializes in a playful, though very serious approach to expose and illustrate complicated science concepts and techniques.  One of her series, entitled Doodling in Math Class, exposes the rich, playful and beautiful structure inherent in every math class but lost in the tedium of sterile, and solely utile function.  Below is a three part video explaining why and how the Fibonacci Sequence (not a series, really) shows up so often in nature.  It is mind-boggingly well done, IMHO:

Doodling in Math: Spirals, Fibonacci, and Being a Plant

The other two parts follow immediately from this one.  Give them a look!

BTW, THIS is what mathematics is really about.  Vi's money quote (at the end of the third part):
This is why science and math are so much fun.  You discover things that seem impossible to be true, and then you get to figure out why it is impossible for them not to be true.