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.

Wednesday, March 19, 2014

Math in the Media: TEDx and me....

Late last year, I was asked to give a TEDx talk (the 'x' means locally organized) for the inaugural TEDx event here at Hopkins (called TEDxJohnsHopkinsUniversity).  I gladly accepted, seeing it as a chance to say something I've been wanting to say for a while:  I wanted to give a talk on what mathematics means to me and why I chose it as a lifestyle.  On February 22, 2014, here on campus, I gave the talk, entitled "Why Mathematics?".

Here it is in full:
Why Mathematics?
It was a great experience, and the organizers did an excellent job.  I hope you find the talk interesting.

Monday, February 17, 2014

Math in the Media - Perhaps the Matrix is....

Here is an article filed under the category "Thoughts to Ponder":  Edward Frenkel, a mathematician from Berkeley, posits that the university perhaps is just a giant simulation and we are simply participants.  How would we know?  Can we detect if we were?  The article is an OpEd in the New York Times, and can be found here:
Is the Universe a Simulation?
Frenkel gives some sense to this idea by differentiating (no pun intended) mathematical ideas (manuscripts, really) from literary ones in the following way:  Mathematical ideas are somewhat universal.  The laws and constructions of Pythagorus, Euclid, Newton, etc., would surely have been created (discovered?) even if these greats had never existed.  It may have taken longer for someone else to develop them.  But the structure of mathematics (its logical framework) exists as it is whether we discover it or not.  Try that with a sonnet sans Shakespeare....

It is a very nice read, this article, and again, gives a sense for how mathematics seems different from other disciplines of study.  Frenkel mentions that many mathematicians consider themselves Platonists, believers that everything exists in the ideal, and what we perceive in this world is simply real versions of that ideal.  It works for me.  I would believe that it would work for most all mathematicians, really.

Frenkel even goes so far as to say that the giant computer simulation that we exist in is, like all computer simulations, not entirely without anomalies, coding inaccuracies that render the coding conspicuous.  Perhaps all of our logic in mathematics is simply facets of the coding that can be detected "from within"?

Certainly a "thought to ponder"....     

Friday, February 14, 2014

Beauty in Math

Mathematicians often talk about their craft in emotional terms.  We get excited by elegant, beautiful, clever constructions and the hidden insight in the logical relationships we uncover.  We can easily be stunned into awe when our intuition leads us astray, and something we did not expect pops out of our reasoning.  And when we see a formula or other type of mathematical construction that not only looks aesthetically pleasing, but contains meaning far beyond its simple symbolic patterns, we treat it as something that should be hanging in the Louvre....

Beauty has profound meaning in mathematics, at least to us.

Not sure you believe me?  Well, a paper just published in the journal Frontiers in Human Neuroscience may just change your mind.  Researchers use Functional Magnetic Resonance Imaging (fMRI) to map the brain activity of mathematicians as they viewed various mathematical formulae and constructions that they have rated on an ugly-beautiful scale.  That part of the brain that is activated when people see beautiful art, or hear beautiful music (yes, there is a specific place)?  Evidently, that place lights up when we view math that we see as beautiful.  At least to us, it is real. 

Give it a read (H/T to CG!!): 

Mathematical beauty activates same brain region as great art or music

BTW, the most beautiful formula of the study:  Why Euler's Identity, of course!  Can you see the beauty?

Monday, February 3, 2014

The importance of mathematical "error"?

I recently participated in an intersession course here at Hopkins (a short, three week course offerd between our fall and spring semesters) called "Thinking though the Fields, A Round Table on Bridging Science and the Humanities at Hopkins"  (sound like a title in need of an acronym, eh?) The course was run by Dr. Kristin Cook-Gailloud.  It was a wonderful experience, with a topic of the day presented in a short talk by a few academics in diverse fields, and then a general discussion about the different interpretations and experiences.  I talked about the importance of puzzles and game playing in mathematics and mathematical research.  Neat.

A short time afterwards, I received a request from some of the students in the class.  They were doing a class project to compile a booklet on topics of a similar capacity.  They asked me to answer a few questions about how mathematicians use error in their work.  My answers are below.  Enjoy:

In relation to your field, how do you define error?
  • In Mathematics, we define or use error in many ways.  Perhaps two important ones are:  (1) As a means to study levels of inaccuracy in estimation and approximation, and (2) as a means to address falsity in claims of truth, like proofs.  For (1), mathematics is the study of the logical structure of complicated things.  Many times, these complicated things are systems defined by equations involving numbers.  When used to model something physical, we must accept that our model might not be completely accurate, due to the fact that we cannot properly account for some influential effects in our model.  Think how a model of a pendulum may take into air resistance when predicting its position at some future time, but possibly not that the humidity of the air may affect the constant that we use for air resistance.  Instead of trying to account for everything, we make an approximation and hope that we are fairly accurate in the end, accepting the errors that will accrue, but hoping that they are small.  Also, when modeling mathematics on a computer, another kind of error we see is the fact that computers cannot be precise in the way that we are when doing arithmetic.  For example, there is really no such thing as zero on a computer.  When defining arithmetic on a computer, and assigning numbers to variables we must determine a level of precision (number of bits to devote to a number.) This works well for normal calculations, but when doing high precision work, if one were to multiply an extremely small number to a very very large one, the result may be inaccurate, since the very small number may only be accurate to a finite degree and the multiplication may bring some of the inaccuracies up to the range of what we consider normal numbers.  There is a field of mathematics that studies errors in calculation like these, call numerical analysis.  For (2) , any new mathematical structure or concept or theorem is an abstract idea that must be proven to be consistent with all other mathematical ideas.  Many times, a new idea is claimed to be proven, but under scrutiny by other mathematicians, it is shown to not be proven completely.  There is an error in the proof.  Either the claim is wrong, or the claim is not fully justified as proven.  At this point, the idea is NOT a fact, and dangerous to try to use to help prove other possible facts.  All mathematical ideas claimed to be proven are scrutinized extremely carefully by the mathematical community, either in research paper review, or by other independent verification.  It is a strength of the field that nothing is really proven until verified fully.       

How do you deal with and interpret error in your field of work
  • Mostly, the above answer works here also.  For (1), we deal with errors in accuracy by trying desperately to manage it and/or minimize it.  Typically, on a computer, decreasing error means increasing computational time and effort.  Hence there is often a trade off between how accurate you want your answer to be and how long you want the computer (or you) to spend trying to compute the answer.  For (2), when a new idea seems to be proven, a mathematician will immediately go to colleagues and collaborators to have them assess the value, correctness and completeness of the proof.  Errors are often found and arguments (statements of the proof) are changed to address the criticisms.  Once a research paper with some new result (proof) is submitted, there is a formal review process where independent mathematicians with knowledge in a particular field assess the correctness of the proof.  Papers are deemed unacceptable for publication when not correct, and must be reworked or abandoned, depending on the nature of the errors.  Sometimes, when a paper is published with an error, the error must be fixed either with an addendum to the original paper, or with withdrawal of the paper from the journal.  There are no instances where errors are tolerated in mathematical proof.
During your career has erroneous findings led to any key or luminous findings?
  • In my work personally, no.  Although some work has not been published due to errors unseen in the original drafts.  However, so much beautiful, amazing mathematical ideas have come from initial errors.  The famous Fermat's Last Theorem, proved only recently but stated 300 years ago, was a simply stated idea that was claimed to have a simple proof by Pierre de Fermat.  Alas, he never wrote down his proof, and the community has been trying to find it for 3 centuries.  The idea is now a fact (theorem), but the recent proof is not simple at all.  However, two things come out of this:  (a) So much beautiful math has been developed in the search of this proof, and (b), it is now basically universally believed in the mathematics community that if Fermat indeed had an idea for a simple proof, it had an error.  We will never know, but....  And in the 70's, Stephen Smale claimed to prove that Chaos (the theory of unpredictability in deterministic mathematical models) does not exist in mathematics.  His proof was in error, and this was shown by another mathematician who produced a counterexample (a single example of something that shows that a supposed fact is incorrect.)  Smale set out to prove he was indeed correct, and in doing so, developed a new branch of mathematics called hyperbolic dynamics, centered around his famous "Smale Horseshoe".  Alas, he only really proved he was originally mistaken, but the error is considered a beautiful one due to what came out of it!

I'm back....

I must apologize for my lack of communication over the last few months.  I enjoy my time here at Hopkins and try to take an active role in the community.  However, adding activities tends to take attention away from other activities.  And this blog took a hit, unfortunately. 

I recently was inspired to jump back in and write on the Chalkboard (so here I am..., H/T to AE!)

Talk to all of you soon.