While I'm positive that stories about math at the level of sophistication described here require some technical underpinning which the writer and editors may potentially lack and which therefore make it both more difficult to be aware of them (to know to report on them in the first place) but also to objectively lay out the topic, from a journalistic perspective, this particular story, which references prior work by Benedict Gross, should have at least mentioned, as a caveat, the relationship between Quanta as a publisher and Gross's position as an advisory board member to it.

While Gross's bona fides are certainly not in question given the topic or his distinguished career, what, if any, was his involvement with the piece? Did he suggest it? Did he provide background? (I notice he wasn't directly quoted, which makes me wonder even further). Was he paid? There's certainly the possibility that something potentially not quite kosher may be going on here (even if it is just free publicity for him and his past work, particularly given that the final result discussed here hasn't been published or independently verified), and not providing at least a mention of the relationship from an editorial standpoint is a glaring error.

Aside from this small journalistic oversight, I otherwise laud Quanta for consistently providing interesting, entertaining, and timely coverage of the world of math and science in such a fantastic fashion. Helping to humanize and even idolize mathematicians and scientists should be a more common effort in our society.

Given the somewhat broad nature of the readership and range of backgrounds, for those who may not have the high level of background for reading the related papers (once they're publicly available) yet, many may be interested in knowing that Dr. Gross has an excellent and free online series of lectures on basic abstract algebra available (http://www.extension.harvard.edu/open-learning-initiative/abstract-algebra) which may go a reasonable part of the way for helping the not-so-technical readers to better understand some of these areas of mathematics and how they interrelate.

Cross posted as: https://www.quantamagazine.org/20151208-four-mathematicians/#comment-363094

A general foothold into the overlap of maximum entropy methods and biology:

John Harte's work on applying the mathematical theory of maximum entropy to ecology is certainly one of the better known examples of the application of this area of mathematics to science, in part because he literally wrote the textbook: [Maximum Entropy and Ecology: A Theory of Abundance, Distribution, and Energetics (Oxford Series in Ecology and Evolution)][1]

To be clear, maximum entropy (also known as MaxEnt in some of the literature, though most/all researchers use the longer form in publications) is a mathematical tool stemming from the fields of probability theory, statistics, and information theory. It's use is classically most often seen in thermodynamics, statistical thermodynamics, physics, and information theory, primarily because these were the areas in which E.T. Jaynes was working when he posited the idea. [Wikipedia has links to his two seminal papers.][2]But because of it's mathematical form, it can be applied in a multitude of areas, typically where one can utilize probabilistic methods.

If you're looking for additional areas of application, simply google the phrase "applied maximum entropy" and you'll find a [wealth of areas][3] including: econometrics, natural language processing, nuclear medicine, queuing systems, mass spectrometry, image processing, machine learning, and many others.

For ecology related work, a cross search on maximum entropy and "genetics", "evolution", "species", and similar words will provide a wealth of papers like ["A maximum entropy approach to species distribution modeling"][4].

Given the generic nature of your question, I might suggest that you'll find E.T. Jaynes' paper ["On the Rationale of Maximum-Entropy Methods" (IEEE, 1982)][5] useful.

Those generally interested in the broader applications of information theoretic methods to biology will likely appreciate some of the work that came out of last year's [NIMBioS Workshop on Information and Entropy in Biological Systems][6] (which Harte both attended and presented at), the [BIRS Workshop Biological and Bio-Inspired Information Theory][7], and the 2014 [CECAM Entropy in Biomolecular Systems][8]. The NIMBios Workshop was organized by John Baez, a physicist, who has worked with MaxEnt methods and explored them on his blog "[Azimuth][9]".

Those with a more sophisticated mathematical background (including measure theory, functional analysis, etc.) may appreciate Henryk Gzyl's text [The Method of Maximum Entropy (World Scientific: Series on Advances in Mathematics for Applied Sciences, Vol 29, 1995)][10].


[1]: http://amzn.to/1SnNB2c
[2]: https://en.wikipedia.org/wiki/Principle_of_maximum_entropy
[3]: https://scholar.google.com/scholar?q=applied%20maximum%20entropy
[4]: http://dl.acm.org/citation.cfm?id=1015412
[5]: ftp://129.240.33.108/pub/outgoing/IMN/Prediction%20modelling%20artikler%20fra%20Anders%20K%20W/Jaynes%201982,%20On%20the%20rational%20of%20Maximum-Entropy%20models.pdf
[6]: http://boffosocko.com/2015/05/20/videos-from-nimbios-workshop-on-information-and-entropy-in-biological-systems/
[7]: http://www.birs.ca/events/2014/5-day-workshops/14w5170
[8]: http://www.cecam.org/workshop-1014.html
[9]: https://johncarlosbaez.wordpress.com/?s=maximum%20entropy
[10]: http://amzn.to/1MFrrIC

David Dylan Thomas asked, "When was the last time in human history when you could have possibly consumed every piece of culture than had ever been created?"

In a realistic sense, you'd have to name a "sphere" of culture to do this as the link between even the occidental and the oriental was not well established such that a western scholar, for example, may have possibly known about "all" culture until modern times - and then being able to consume it all would be nearly impossible.

Within the much smaller realm of mathematics, the last mathematician who was said to have known "all" of mathematics was David Hillbert around 1900-1910. No modern mathematician functions in all of the various branches the way he and many of his predecessors did.

Given that there are currently over 7,000 "distinct" languages on the planet and they're dying at an increasing rate, if we surmise that there were at most 2x this number at the peak number of languages (which is a laughable guess at best), just knowing all these languages fluently would be a massive hurdle at best, much less all the culture raining down from each of them.

There's still plenty of time to register for Algebraic Number Theory @uclaextension #LosAngeles #Mathematics http://boffosocko.com/2015/07/27/algebraic-number-theory-ucla-extension/

Category Theory certainly isn't useless, and has some really beautiful and interesting things going on. One of the bigger issues it has is that it is relatively so young as a discipline within math. (Imagine what people said about set theory in 1910.) Your toughest hurdle as an undergraduate is finding a university that is offering it as an accessible class depending on your level of mathematical background. I haven't noticed a lot of universities even offering it at the graduate level yet either.

That being said, David Spivak's new book Category Theory for the Sciences which came out of MIT Press last year is the closest textbook I've seen to attempt making Category Theory easily accessible to not only an undergraduate audience, but also to a broader range of scientists who are not necessarily steeped in mathematics. For your particular purposes, it actually has a nice handful of biology related examples in the early chapters.

Because it expounds on some of the beauty of category theory and it's application to science, I'll draw your attention to an essay that Ilyas Khan wrote recently: https://www.linkedin.com/pulse/category-theory-bedrock-mathematics-ilyas-khan

I'd recommend category theory more highly, if you plan on pursuing a Ph.D. and research work, but in the meanwhile you might also find some interesting material in John Carlos Baez's Azimuth blog or on the n-Category Cafe.

This seems like an extension of Colin McNamara's post a while back, which Eric Waldstein is quietly referencing by copying the two of us: https://www.yammer.com/bhpteachercommunity/#/Threads/show?threadId=493899391

I'm also reminded of some of the simple examples that Warren Weaver and John Robinson Pierce presented shortly after the publication of Claude Shannnon's seminal work "The Mathematical Theory of Communication" which, because it underpins ALL of the modern day communications revolution (digital communication, computers, satellites, cell phones, etc.) since the 1940's, is often cited (along with religious texts like the Bible, the Koran, etc.) as one of humanity's most influential written works. [http://worrydream.com/refs/Shannon%20-%20A%20Mathematical%20Theory%20of%20Communication.pdf]

Shannon had previously taken George Boole's new "algebra" from the late 1800's (now known as Boolean Algebra, which underpins most of modern logic) and applied it to electronic circuits. Though somewhat technical most advanced high school algebra students should be able to read and understand most of Claude Shannon's MIT master's thesis, which itself is often cited as one of the most influential theses ever written. [http://www.cs.virginia.edu/~robins/Shannon_MS_Thesis.pdf]

In any case, Pierce in particular does some simple exercises relating to language in his book "An Introduction to Information Theory: Symbols, Signals and Noise" [http://www.amazon.com/gp/product/0486240614/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0486240614&linkCode=as2&tag=bighistory-20&linkId=ITZN6S4HHWRWDZZB] Amongst a bevy of interesting topics, he has a chapter on "Language and Meaning" as well as examples of the built in redundancy of languages and "error correcting" which mathematically allows us to have crossword puzzles as word games.

This also to some extent, underpins our ability to look at the following paragraph and understand what it says. All the letters have been jumbled (mixed). Only the first and last letter of ecah word is in the right place:

I cnduo't bvleiee taht I culod aulaclty uesdtannrd waht I was rdnaieg. Unisg the icndeblire pweor of the hmuan mnid, aocdcrnig to rseecrah at Cmabrigde Uinervtisy, it dseno't mttaer in waht oderr the lterets in a wrod are, the olny irpoamtnt tihng is taht the frsit and lsat ltteer be in the rhgit pclae. The rset can be a taotl mses and you can sitll raed it whoutit a pboerlm. Tihs is bucseae the huamn mnid deos not raed ervey ltteer by istlef, but the wrod as a wlohe. Aaznmig, huh? Yaeh and I awlyas tghhuot slelinpg was ipmorantt! See if yuor fdreins can raed tihs too.

Pierce's book, while it does have some small amount of math, should be generally readable by most high school students. He also provides a simple model of language in his chapter "A Mathematical Model" which many may find interesting, or be able to pull some interesting examples out of. This particular book is also an excellent example of the intertwining of the ideas of physics, mathematics, engineering, psychology, thermodynamics as well as the ideas of emergence, convergence, and allows for the increased complexity and collective learning that underpins the philosophy of Big History.

I've just come across two Coase references in as many days... The first was by way of Cesar Hidalgo's new book "Why Information Grows: The Evolution of Order, from Atoms to Economies" which posits a relatively simple framework for economies based on our favorite topic of information theory. http://www.amazon.com/gp/product/0465048994/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0465048994&linkCode=as2&tag=itbio-20&linkId=AJJXNTLI3JJ6577T Though I wish it had some significant mathematics in the book to further validate the examples, on it's face his thesis has a lot of general merit and seems logically sound. I'll hope to have a short review of it up shortly.
Thanks for the further references and the podcast.

My answer to Question on @Quora: Would it be worth it to study category theory? http://qr.ae/R7u9ME

The sad answer for most in your situation is: probably not yet. There just isn't a broad layer of applied use of category theory in your area yet. (I'll note that I'm also an electrical engineer with a background in information theory.) HOWEVER, this certainly doesn't mean it's not important and that the future won't find lots of applications for it, particularly given its abstraction and ability to connect disparate areas of mathematics. On the bleeding edge of research, however, it's another story. (For a flavor of some of this, take a look at John Baez's blog Azimuth: https://johncarlosbaez.wordpress.com/?s=category+theory). Given my diverse mathematical background, I have no doubt that category theory will be exceedingly important over the next century, and particularly in the area of communications.

My suggestion to you, particularly as you mention that it seems very abstract, is that you do some preliminary background work in abstract algebra, combinatorics, and even set theory, which will likely be much more immediately applicable to your present area of study. These will give you a far better preparation for attacking category theory in the future. (I can't recall having seen or heard any EE departments even suggesting category theory courses recently, much less requiring them, even at the master's level. Further, very few universities are offering coursework directly in the topic at the moment, so it's far more likely that you'll only see it at the graduate level.

Those looking to dip their feet in the water might consider taking a look at How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics by Eugenia Cheng [http://www.amazon.com/gp/product/0465051715/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0465051715&linkCode=as2&tag=boffosocko-20&linkId=3QHVSGPXXWMTWGE2] which just came out in May in the US. It's a popular science text which gives some good overview of the subject and is the only book in that space at the moment to my knowledge.

I'll recommend that, if you're impatient, you take a peek at David Spivak's book Category Theory for the Sciences [http://www.amazon.com/gp/product/0262028131/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0262028131&linkCode=as2&tag=boffosocko-20&linkId=BW3G2IMA3JGCU2FC] which is currently one of the best introductory books on the market for those without more advanced background in mathematics. It was released last year by MIT Press and most of it can be found for free on the web - see the study site mentioned below for details. While it doesn't jump right in with heavy theorems/proofs the way most other popular texts in the area do, it does require a reasonable amount of mathematical sophistication, and it does provide a wealth of introductory material in chapters 1-4 on functions, sets, homomorphisms, isomorphisms, basic group theory, for those who don't have any background in set theory, basic topology, or abstract algebra.

As you're roughly in the same area I am, I'll mention that my interest in the topic was recently piqued by an article by Ilyas Khan: Category Theory – the bedrock of mathematics? [http://boffosocko.com/2015/04/30/category-theory-the-bedrock-of-mathematics-via-ilyas-khan/] It prompted me to drop my prior summer plans and create a category theory study group with a group of friends and many strangers. [see: Category Theory Study Group [http://cat.boffosocko.com/], which has a HUGE list of resources (including videos) in the area and is meant for students and self-study; feel free to join us if you'd like.]

As someone currently studying the topic, I will note that it isn't very difficult so far, and is relatively much easier than topics like advanced algebraic topology or Lie groups, so it could certainly be done at an advanced undergraduate level, but I'd certainly recommend diving into Spivak's book before any/many of the others which require a much higher level of mathematical sophistication typically required at the master's level or above. I would suggest that if you've done coursework in calculus and linear algebra and know most of the material in a book like A Primer of Abstract Mathematics (Classroom Resource Materials) by Robert B. Ash [http://www.amazon.com/gp/product/0883857081/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0883857081&linkCode=as2&tag=math01-20&linkId=J4TBWXWBEVTPHDU7http://www.amazon.com/gp/product/0883857081/ref=as_li_tl?ie=UTF8&camp=1789&creative=390957&creativeASIN=0883857081&linkCode=as2&tag=math01-20&linkId=J4TBWXWBEVTPHDU7], you should be able to tackle Spivak's book without too much difficulty. Having one or more of the other prerequisites I mentioned before would certainly help you move through the material much faster.