Shall we (again) begin...?

Going into hibernation is a lot like falling asleep.  You never really know when it is happening unless you wake up out of it.  And then at some point, you do wake up, and reflect on that sleep (in the past tense).  

It has been a while since I wrote here, and will try yet again to start up and keep writing.  

Let's see how I do this time, shall we?

Finding your Passion?

As faculty (advisors, mentors), we are often telling undergraduates that they are in a good position to explore their future and "find their passion".  You know, "Definitely take chemistry courses if you love the subject, but now is the time to also take that one class in a subject you were always curious about", or "well, if you are not happy with your studies, perhaps your focus is not what you would really love to pursue".  We tend to strongly encourage students to use their undergraduate years as a means to "find" themselves.  Tara Bahrampour of the Washington Post in the article

'Find your passion'?  That's bad advice, scientists say.

says that this may, in fact, be bad advice.  Well, not so much bad advice, and incomplete advice.  Developing a mindset of general growth, nurturing multiple interests, at least initially, may lead to a much more fulfilling and successful future path.  Moreover, fixing the mindset on only a singular topic of study may stunt the pursuit of other interests.

I like the analysis.  You should give it a read.

Math in the Media: Arguing on Pi-Day

I cannot say that we, as mathematicians, do not have our fair share of math-arguments and inside jokes and math puns and such.  You know, stuff that the "outside" world would either groan at or simply walk away from in a head-shaking fashion.  But Pi-Day, March 14, or 3/14, does seem to bring things like this to the surface....

Here are two articles that have leaked out into the "real" world.  The first is not a real debate or controversy, really..., but it is kinda fun in a strange sort of way.  It is an argument for a better way to generally represent the constant that arises from comparing the diameter or radius of a circle to its circumference.  Since pi radians represents only half a turn around a circle, why not have the universal constant simply be 2pi, representing a full turn around the circle.  Call this number tau = 2pi.  The article, in the Verge, is kind of a rant on pi's fame:

Stop Celebrating Pi Day and embrace Tau as the true circle constant

I am not sure about this one, but the accompanying "Tau Manifesto" is a pretty good read. 

The other is really more of a comedy routine, designed to educate and highlight some real math.  The sort of sweetened medicine you were forced to take as a child.  Broadcast via Mother Jones, the interview/debate

What is the greatest number of all time?

is an argument between two mathematicians Tom Garrity and Colin Adams.  Clever....

Enjoy Pi-Day!!!

...Back from hibernation....

Sorry to all for the long lapse in communication.  With other projects on the front burner (a textbook, now due on the shelf this summer), and local distractions right and left, this forum went dormant for a long while.  It is now to be considered alive....

Talk to you soon about math in teaching, education, recreation, etc. 

Math in the Media: The 2016 Abel Prize

Andrew Wiles, (well..., actually, Sir Andrew John Wiles), the British mathematician, currently at Oxford who is credited with solving the famous "Last Theorem" of Pierre de Fermat, has been awarded the 2016 Abel Prize for outstanding contributions to the field.  From the Abel Committee of the Norwegian Academy of Science and Letters, 

[t]he prize is meant to recognize contributions of extraordinary depth and influence to the mathematical sciences. Such work may have resolved fundamental problems, created powerful new techniques, introduced unifying principles or opened up major new fields of research. The intent is to award prizes over the course of time in a broad range of fields within the mathematical sciences.

One can easily say Professor Wiles is qualified, no? 

Congratulations, Professor!  Read up on this amazing accomplishment here:

Fermat's Last Theorem proof secures mathematics' top prize for Sir Andrew Wiles

Math in the Media: The importance of Recreational Math?

Math as a plaything is something most mathematicians take for granted.  It is one of the main reasons we choose to study math as a discipline:  Many of us "played" a lot when we were kids.  We marveled at logic puzzles, looked for patterns and clues in complicated word problems, played tricks with the double meanings of words and phrases and generally devoured games that rewarded the player when one can uncover the proper strategy.  Martin Gardner was a wonder at finding the mathematics hidden in playful puzzles and tricky games.  His "Mathematical Games" series in Scientific American, something on the order of 300 columns from 1956 to 1986, is a true treasure trove of beautiful, yet playful mathematics.

Now, Manil Suri, a professor in the Mathematics Department of the University of Maryland, Baltimore County, writes in the New York Times of this importance of the recreational side of mathematics, especially at the primary and secondary levels, citing Gardner's work.  The piece is

The Importance of Recreational Math

It's a worthy topic to illuminate.  Do take a look....

Viral Math....

So....  Have you figured out Cheryl's birthday yet? 

It's funny how so few people regard this puzzle as real math (as opposed to calculating a derivative or something.)  In my mind, THIS is real math;  taking a complicated situation, digging out and exposing the logical skeletal structure underlying it, studying that structure and abusing it to say something conclusive about the situation.  Problem solving, in a nutshell, is mathematics.  Or perhaps real math is learning how to synthesize and generalize the logical structure so that it can be used again in other puzzles, or situations. 

Either way....  This is real math. 

Math in the Media: The Best jobs of 2015?

I keep telling you....  Math pays!! 

In the Business section of the online newspaper The Huffington Post sits an article by Jennie Che detailing the Best Jobs of 2015, a report prepared each year by CareerCast, ranking the top 200 jobs by work environment, income, stress and hiring outlook. 

And lo and behold, mathematics permeates most of the list, with the actual job of Mathematician, sitting at number 3 (Actuarial Scientist tops the list, with Statistics sitting at number 4).  The HuffPost article

These Are The 10 Best Jobs Of 2015

is a summary of the longer article posted directly in the CareerCast website:

The Best Jobs of 2015

(always go to the source, right?)   Personally, I've held position both in academia and in industry (NASA Goddard and Lincoln Laboratory, as well as in a private tech firm.)  All were great work environments, with interesting people, work conditions, and colleagues.  I certainly cannot argue with the conclusion that math is a great field to play in.

Give the article a read!  It's good stuff.

Math in the Media: NFL Mathematician??

I am often asked what kinds of careers can a mathematician construct for themselves.  There are many answers to this question, and in full generality, the list is long and very diverse.  However, I recently found a new one:  Professional American football player!!  

It seems that our very own John Urschel, a guard of the Baltimore Ravens ("our very own" because we are here in Baltimore after all), is a mathematician whose recent paper "A Cascadic Multigrid Algorithm for Computing the Fielder Vector of Graph Laplacians” has been accepted for publication in the Journal of Computational Mathematics.  Mr. Urschel received his Masters Degree in Mathematics from Penn State and was drafted by the Ravens last year.

One can find mathematicians in the oddest of places, no?  One just has to look carefully....

...although in this case, it is not hard to see.  Mr. Urschel's is on Twitter.  His name??

....@MathMeetsFball 

Math in the Media: Gender Bias in Teachers?

Yes, it does take a village to raise a child....  (H/T to The Honorable Hillary Rodham Clinton.)  But it seems it also takes a village to discourage girls from achieving their full potential in math and math-related fields....  (Sigh!)

In a new study, headed by Edith Sand, an economist at the Bank of Israel and an instructor at the Tel-Aviv University’s Berglas School of Economics, teachers themselves contribute to the problem of too many female students shying away from higher-level math courses as they progress in their education.  The study found a gender bias in performance evaluation;  Teachers who knew the identities of their students tended to grade girls more harshly and boys less so on exams than teachers who did not have any information about the students.   Unconscious or not, our influence as teachers on students always goes far beyond the content of our lectures and exercises.  But this influence may not always be constructive.  What care we need to always take....

The article, reported by Linda Carroll for Today, is  here:

Teacher Bias May Help Discourage Girls from Math, Study Finds

The study is published by the National Bureau of Economic Research in Cambridge, Mass.

Depressing...?  Yes.  Hopeful?  Also.  Knowing of an unconscious bias can contribute to its cure, eh? 

Remember School House Rock?  "Knowledge is Power"

Math in the Media: Homer vs. Pierre?

I just had the pleasure of watching a neat 8 minute video detailing some of the mathematics injected in to the Simpsons animations.  Apparently, there are mathematicians among the creative staff who cannot help themselves throwing in a little math humor into the background every so often. 

The video, listed here on YouTube by Numberphile is titled

Homer Simpson vs Pierre de Fermat

 Do give it a watch.  It is always good to know where the subliminal messages about how cool math really is are lurking, no?

What is Mathematics? Passages....

So I was asked during the break between the semesters to take part in an Intersession course designed to bridge the void between the sciences and the humanities (mind the gap!) by having professors from all stripes discuss common topics to a diverse audience.  The idea is that each professor sees the topic from the perspective of their chosen field and the same topic often looks quite different to different people.  It is a wonderful idea hatched and developed by Dr. Kristin Cook-Gailloud, the Director of the Program in French Language and Culture here at Hopkins.  This is the second year running this course and I thoroughly endorse it.  Alas, due to scheduling issues, I did not participate this year.  But a topic in this year's course, Passages, stuck in my head.  The idea of movement from one state to another is something innate to a mathematician, if regarded as movement from a state of ignorance and confusion to clarity and enlightenment. 

So I wrote an essay to clarify my idea of passage in mathematics.  It is here:

Passages in Mathematics

Enjoy and do let me know what you think....

Math in the Media: Stereotype Threat?

How often we are in Mathematics faced with the fact that our profession was, and still is, quite female-starved;  that boys and men are thought to be better at math than girls and women (an amazingly ridiculous thought, given my view up here!).  So many recent studies seem to point decidedly at the dangerous effects of a person's perceptions of ability at the moment of evaluation and how easily they can affect performance.  For example, reminding students of a stereotype they conform to just before taking a math test tends to degrade performance.  This slow drip of research exposing the damaging effects of culture bias and preconceptions on lack of ability can only have a good effect in the long run.  And I do see here at Hopkins some light in the form of a general welcoming attitude and positive outreach to students studying higher mathematics regardless of gender.  But it does seem that this huge ship turns only very slowly.

Over at the online newspaper, the Huffington Post, Cailin O’Connor, a Professor of the Philosophy of Science at the University of California, Irvine, details some of the evidence that runs contrary to the notion of an innate gender bias in mathematical ability here:

Are Women Worse at Math?  It's Time to Stop Asking

Her take?   Perhaps it is time to stop focusing on looking for innate differences between the genders in mathematical ability and start simply addressing the cultural barriers that keep the gender balances way too tilted to one side. 

I agree, but still love seeing the rising tide of evidence condemning the idea that math is more a male thing.  Geez!

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....

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?

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....  ;-)

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?

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.

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.

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.