“The Mathematics Literature Project intends to survey the state of the freely accessible mathematics literature. In particular, it will index freely accessible URLs for mathematics articles. These are legitimately hosted copies of the article (i.e. at publishers, the arXiv, institutional repositories, or authors’ homepages), which are freely available in any browser, anywhere in the world.”
Wes Craven, the famed maestro of horror known for the Nightmare on Elm Street and Scream franchises, died Sunday after a battle with brain cancer. He was 76.
Saddened to hear that filmmaker and fellow Johns Hopkins University alum Wes Craven has passed away this afternoon. He was certainly a scholar and a gentleman and will be missed terribly.
Physicist Stephen Hawking made a splash this week when he announced that he had solved a vexing conundrum that had puzzled generations of leading physicists -- including the 73-year-old scientific superstar himself -- for the better part of a half-century.
Brief book overview of Matthew Cobb's "Life’s Greatest Secret" from The Economist.
For those interested in some of the history behind genetics, evolution, biology and information theory, the following book, which I just saw the attached review in The Economist, is likely to be of interest:
In 1953 James Watson and Francis Crick, with the help of Rosalind Franklin and Maurice Wilkins, described the structure of the molecule at the heart of life. Deoxyribonucleic acid, better known as DNA, was, they said, a double helix, two spirals joined across the middle by pairs of four chemical bases, like a twisted ladder. That work earned Messrs Crick, Watson and Wilkins a Nobel prize and a place in the history books. The image of the double helix now often stands for biology, or even science, itself.
But this was merely the most visible breakthrough in a long struggle to understand the engine of life—how traits are inherited, mutated and weeded out by natural selection, and how the whole mysterious process works at the biochemical level. It is that lesser-known history that Matthew Cobb, a professor of animal behaviour at the University of Manchester, aims to sketch in his book, which has been shortlisted for the Royal Society’s Winton prize for science writing.
The result is a fascinating reminder of just how hard-won are the seemingly obvious facts of modern biology. The development of genetics was a tale of confusion, accident, frustration and the occasional flash of insight. It was, says Dr Cobb, as important as the Manhattan or Apollo projects, but with no government support and little money, carried out by scientists interested in the question for its own sake.
The researchers started from almost total ignorance. William Harvey, better known for describing the circulation of the blood, wondered in the 17th century what could explain why children’s skin colour was often a blend of their parents’, whereas they share a sex with only one, and can have an eye colour different from either.
In the late 19th century a monk, Gregor Mendel, established, through experiments on pea plants, the basic rules of inherited traits. A Danish biologist, Wilhelm Johannsen, coined the term “gene” in 1909 to describe whatever it was that Mendel had found. But as late as 1933 scientists were still debating whether genes were physical things or just useful abstractions, and how they could transmit traits. Scientists knew that DNA existed, but many considered it a boring bit of scaffolding in the cell. Proteins, which come in zillions of different varieties, were seen by many as the only things exciting enough to account for all the diversity seen in life.
After the second world war, ideas from information theory, arising out of wartime work on computers and automation, percolated into biology. Once the structure of DNA had been established, those ideas helped crack the problem of how the four chemical bases do their job. Proteins are built by stringing together 20 different sorts of amino acid. Strings of three bases within a DNA molecule represent these amino acids, but with 64 such triplets, there is much redundancy which information theory alone could not fully explain. Years of painstaking lab-work were needed to reconcile theory with reality.
Dr Cobb is good on the human side of the story, showing science as fuelled by rivalry, jealousy, competitiveness and wonder. The only downside is that he must marshal hundreds of scientists across several disciplines into around 300 pages of narrative. The results can sometimes be dense, and readers without a command of biological jargon will frequently find themselves consulting the glossary for guidance. But the cracking of the code of life is a great story, of which this is an accomplished telling.
It also not coincidentally is the root of the vast majority of the problems the world is currently facing. There are so many great quotes here, I can’t pick a favorite, so I’ll highlight the same one Kimb Quark did that brought my attention to it:
“There’s nothing wrong with an opinion on those things. The problem comes from people whose opinions are actually misconceptions. If you think vaccines cause autism you are expressing something factually wrong, not an opinion. The fact that you may still believe that vaccines cause autism does not move your misconception into the realm of valid opinion. Nor does the fact that many other share this opinion give it any more validity.”
How a mathematical breakthrough from the 1960s now powers everything from spacecraft to cell phones.
Concurrent with the recent Pluto fly by, Alex Riley has a great popular science article on PBS that helps put the application of information theory and biology into perspective for the common person. Like a science version of “The Princess Bride”, this story has a little bit of everything that could be good and entertaining: information theory, biology, DNA, Reed-Solomon codes, fossils, interplanetary exploration, mathematics, music, genetics, computers, and even paleontology. Fans of Big History are sure to love the interconnections presented here.
Reed-Solomon codes correct for common transmission errors, including missing pixels (white), false signals (black), and paused transmissions (the white stripe).
“The Molecular Programming Project aims to develop computer science principles for programming information-bearing molecules like DNA and RNA to create artificial biomolecular programs of similar complexity. Our long-term vision is to establish molecular programming as a subdiscipline of computer science — one that will enable a yet-to-be imagined array of applications from chemical circuitry for interacting with biological molecules to nanoscale computing and molecular robotics.”
An infographic from the South China Morning Post has some interesting statistics about which many modern people don’t know (or remember). It’s very interesting to see the distribution of languages and where they’re spoken. Of particular note that most will miss, even from this infographic, is that 839 languages are spoken in Papua New Guinea (11.8% of all known languages on Earth). Given the effects of history and modernity, imagine how many languages there might have been without them.
For those who are intimidated by the thought of higher mathematics, but are still considering joining our Category Theory Summer Study Group, I’ve just come across a lovely new book by Eugenia Cheng entitled How to Bake Pi: An Edible Exploration of the Mathematics of Mathematics.
Eugenia Cheng’s book How to Bake Pi
It just came out in the U.S. market on May 5, 2015, so it’s very new in the market. My guess is that even those who aren’t intimidated will get a lot out of it as well. A brief description of the book follows:
“What is math? How exactly does it work? And what do three siblings trying to share a cake have to do with it? In How to Bake Pi, math professor Eugenia Cheng provides an accessible introduction to the logic and beauty of mathematics, powered, unexpectedly, by insights from the kitchen: we learn, for example, how the béchamel in a lasagna can be a lot like the number 5, and why making a good custard proves that math is easy but life is hard. Of course, it’s not all cooking; we’ll also run the New York and Chicago marathons, pay visits to Cinderella and Lewis Carroll, and even get to the bottom of a tomato’s identity as a vegetable. This is not the math of our high school classes: mathematics, Cheng shows us, is less about numbers and formulas and more about how we know, believe, and understand anything, including whether our brother took too much cake.
At the heart of How to Bake Pi is Cheng’s work on category theory—a cutting-edge “mathematics of mathematics.” Cheng combines her theory work with her enthusiasm for cooking both to shed new light on the fundamentals of mathematics and to give readers a tour of a vast territory no popular book on math has explored before. Lively, funny, and clear, How to Bake Pi will dazzle the initiated while amusing and enlightening even the most hardened math-phobe.”
Dr. Cheng recently appeared on NPR’s Science Friday with Ira Flatow to discuss her book. You can listen to the interview below. Most of the interview is about her new book. Specific discussion of category theory begins about 14 minutes into the conversation.
August 10-13, 2015 – UC San Diego, La Jolla, California
Application deadline: June 7, 2015
The School of Information Theory will bring together over 100 graduate students, postdoctoral scholars, and leading researchers for four action-packed days of learning, stimulating discussions, professional networking and fun activities, all on the beautiful campus of the University of California, San Diego (UCSD) and in the nearby beach town of La Jolla.
Tutorials by some of the best known researchers in information theory and related fields
Poster presentations by student participants with feedback and discussion
Panel discussion on “IT: Academia vs. Industry Perspectives”
BIRS 5 day worksop, arriving in Banff, Alberta Sunday, June 7 and departing Friday, June 12, 2015
In the years since the first assembly of the human genome, the complex and vital role of RNA and RNA binding proteins in regulation of the genome expression has expanded through the discovery of RNA-binding proteins and large classes of non-coding RNA that control many developmental decisions as part of protein- RNA complexes. Our molecular level understanding of RNA regulation has dramatically improved as many new structures of RNA–protein complexes have been determined and new sophisticated experimental technologies and dedicated computational modeling have emerged to investigate these interactions at the whole-genome level. Further deep insight on the molecular mechanisms that underline genome expression regulation is critical for understanding fundamental biology and disease progression towards the discovery of new approaches to interfere with disease progression.
The proposed workshop will bring together experts in RNA biology with structural biologists that focus on RNA-protein complexes, as well as computational biologists who seek to model and develop predictive tools based on the confluence of these experimental advances. The workshop intends to foster new collaborations between experimental and computational biologists and catalyze the development of new and improved technologies (such as single cell binding methods) that merge experimental analysis with novel mathematical and computational techniques to better understand the rules of protein-RNA recognition and RNA-based biological regulation.
The organizers of the workshop are both leaders in the field of protein-RNA recognition and interactions: Yael Mandel-Gutfreund has been working in the field of protein-Nucleic Acids interactions since 1994. Her main research interest is protein-RNA recognition and regulation. She has developed several tools and web servers for predicting RNA binding proteins and RNA binding motifs. Yael is the head to the computational molecular laboratory at the Technion and the president of the Israeli society of Bioinformatics and Computational Biology. Gabriele Varani has been working in the field of RNA structure and protein-RNA interactions since 1987. His main research interest is the structural basis for protein-RNA recognition and the prediction and design of RNA-binding proteins. He determined some of the first few structures of protein-RNA complexes and developed computational tools to analyze and predict the specificity of RNA -binding proteins. His group applies these tools to design RNA-binding proteins with new specificity to control gene expression. Our invitation to participate in the workshop has been met with great enthusiasm by the researchers. More than 20 principle investigators have already confirmed their interest in attending. Six of the confirmed participants are female scientists including the organizer Yael Mandel-Gutfreund as well as Traci Hall, Lynne Maquat, Elena Conti, Susan Jones, Drena Dobbs. We also have invited and confirmed the participation of young and promising researchers including Markus Landthaler, Gene Yeo, Jernej Ule, Uwe Ohler and others. Our confirmed participants come from many different countries: US, Canada, UK, Scotland, Germany, Spain, Switzerland, Poland and Israel. Two confirmed participants as well as the organizer have attended the BIRS workshops in the past.
A key objective of the workshop is to bring together researchers with experimental, mathematical and computational background to share results and discuss the main advances and challenges in the prediction, analysis and control of RNA-protein recognition and RNA-based regulation of gene expression. Towards this aim, we plan to adopt the format of previous BIRS meetings in which invited participants (including selected students) will present relatively short presentations of 20 minutes plus 10 minutes of active discussions. This format will leave aside ample time for informal discussions to foster exchanges between participants. To stress the collaborative, multidisciplinary nature of the workshop, we plan to dedicate each of the workshop sessions to a specific topic that will comprise presentations of structural, experimental and computational approaches, rather than create session focused on a particular approach. Each session we will include at least one lecture from a young scientist/postdoctoral fellow/student to be chosen among attendees by the organizers.
Suggested preliminary schedule:
Day 1: Modeling and high throughput approaches to genome-wide analysis of protein-RNA interactions
Day 2: Predicting and designing new RNA binding proteins
Day 3: Generating and modeling RNA-based regulatory networks
Day 4: Principles of RNA regulation by RNA binding proteins
Day 5: Conclusion round table discussion on the present and future challenges of the field
“My vision of life is that everything extends from replicators, which are in practice DNA molecules on this planet. The replicators reach out into the world to influence their own probability of being passed on. Mostly they don’t reach further than the individual body in which they sit, but that’s a matter of practice, not a matter of principle. The individual organism can be defined as that set of phenotypic products which have a single route of exit of the genes into the future. That’s not true of the cuckoo/reed warbler case, but it is true of ordinary animal bodies. So the organism, the individual organism, is a deeply salient unit. It’s a unit of selection in the sense that I call a “vehicle”. There are two kinds of unit of selection. The difference is a semantic one. They’re both units of selection, but one is the replicator, and what it does is get itself copied. So more and more copies of itself go into the world. The other kind of unit is the vehicle. It doesn’t get itself copied. What it does is work to copy the replicators which have come down to it through the generations, and which it’s going to pass on to future generations. So we have this individual replicator dichotomy. They’re both units of selection, but in different senses. It’s important to understand that they are different senses.”
Richard Dawkins
RICHARD DAWKINS is an evolutionary biologist; Emeritus Charles Simonyi Professor of the Public Understanding of Science, Oxford; Author, The Selfish Gene; The Extended Phenotype; Climbing Mount Improbable; The God Delusion; An Appetite For Wonder; and (forthcoming) A Brief Candle In The Dark.
Watch the entire video interview and read the transcript at Edge.org.
Category theory looks set to become the dominant foundational basis for all mathematics. It could, in fact, already have achieved that status through stealth.
Beauty, even in Maths, can exist in the eye of the beholder. That might sound a little surprising, when, after all, what could be more objective than mathematics when thinking about truth, and what, therefore, could be more natural than for beauty and goodness, the twin accomplices to truth, to be co-joined ?
In the 70 odd years since Samuel Eilenberg and Saunders Mac Lane published their now infamous paper “A General Theory of Natural Equivalences“, the pursuit of maths by professionals (I use here the reference point definition of Michael Harris – see his recent publication “Mathematics without Apologies“) has become ever more specialised. I, for one, don’t doubt cross disciplinary excellence is alive and sometimes robustly so, but the industrially specialised silos that now create, produce and then sustain academic tenure are formidable within the community of mathematicians.
Beauty, in the purest sense, does not need to be captured in a definition but recognised through intuition. Whether we take our inspiration from Hardy or Dirac, or whether we experience a gorgeous thrill when encountering an austere proof that may have been confronted thousands of times before, the confluence of simplicity and beauty in maths may well be one of the few remaining places where the commonality of the “eye” across a spectrum of different beholders remains at its strongest.
Neither Eilenberg nor Mac Lane could have thought that Category theory, which was their attempt to link topology and algebra, would become so pervasive or so foundational in its influence when they completed and submitted their paper in those dark days of WW 2. But then neither could Cantor, have dreamt about his work on Set theory being adopted as the central pillar of “modern” mathematics so soon after his death. Under attack from establishment figures such as Kronecker during his lifetime, Cantor would not have believed that set theory would become the central edifice around which so much would be constructed.
Of course that is exactly what has happened. Set theory and the ascending magnitude of infinities that were unleashed through the crack in the door that was represented by Cantor’s diagonal conquered all before them.
Until now, that is.
In an article in Science News, Julie Rehmeyer describes Category Theory as “perhaps the most abstract area of all mathematics” and “where math is the abstraction of the real world, category theory is an abstraction of mathematics”.
Slowly, without fanfare, and with an alliance built with the emergent post transistor age discipline of computer science, Category theory looks set to become the dominant foundational basis for all mathematics. It could, in fact, already have achieved that status through stealth. After all, if sets are merely an example of a category, they become suborned without question or query. One might even use the description ‘subsumed’.
There is, in parallel, a wide ranging discussion in mathematics about the so called Univalent Foundation that is most widely associated with Voevodsky which is not the same. The text book produced for the year long univalence programme iniated at the IAS that was completed in 2013 Homotopy type theory – Univalent Foundations Programme states:
“The univalence ax-iom implies, in particular, that isomorphic structures can be identified, a principle that mathematicians have been happily using on workdays, despite its incompatibility with the “official”doctrines of conventional foundations..”
before going on to present the revelatory exposition that Univalent Foundations are the real unifying binding agent around mathematics.
I prefer to think of Voevodsky’s agenda as being narrower in many crucial respects than Category Theory, although both owe a huge amount to the over-arching reach of computational advances made through the mechanical aid proffered through the development of computers, particularly if one shares Voevodsky’s view that proofs will eventually have to be subject to mechanical confirmation.
In contrast, the journey, post Russell, for type theory based clarificatory approaches to formal logic continues in various ways, but Category theory brings a unifying effort to the whole of mathematics that had to wait almost two decades after Eilenberg and Mac Lane’s paper when a then virtually unknown mathematician, William Lawvere published his now much vaunted “An Elementary Theory of the Category of Sets” in 1964. This paper, and the revolutionary work of Grothendieck (see below) brought about a depth and breadth of work which created the environment from which Category Theory emerged through the subsequent decades until the early 2000’s.
Lawvere’s work has, at times, been seen as an attempt to simply re-work set theory in Category theoretic terms. This limitation is no longer prevalent, indeed the most recent biographical reviews of Grothendieck, following his death, assume that the unificatory expedient that is the essential feature of Category theory (and I should say here not just ETCS) is taken for granted, axiomatic, even. Grothendieck eventually went much further than defining Category theory in set theoretic terms, with both Algebraic Topology and Mathematical Physics being fields that now could not be approached without a foundational setting that is Category theory. The early language and notation of Category Theory where categories ‘C’ are described essentially as sets whose members satisfy the conditions of composition, morphism and identity eventually gave way post Lawvere and then Lambek to a systematic adoption of the approach we now see where any and all deductive systems can be turned into categories. Most standard histories give due credit to Eilenberg and Mac Lane as well as Lawvere (and sometimes Cartan), but it is Grothendieck’s ‘Sur quelques points d’algebre homologique’ in 1957 that is now seen as the real ground breaker.
My own pathway to Category theory has been via my interest in Lie Groups, and more broadly, in Quantum Computing, and it was only by accident (the best things really are those that come about by accident !) that I decided I had better learn the language of Category theory when I found Lawvere’s paper misleadingly familiar but annoyingly distant when, in common with most people, I assumed that my working knowledge of notation in logic and in set theory would map smoothly across to Category theory. That, of course, is not the case, and it was only after I gained some grounding in this new language that I realised just how and why Category theory has an impact far beyond computer science. It is this journey that also brings me face to face with a growing appreciation of the natural intersection between Category theory and a Wittgensteinian approach to the Philosophy of Mathematics. Wittgenstein’s disdain for Cantor is well documented (this short note is not an attempt to justify, using Category theory, a Wittgensteinian criticism of set theory). More specifically however, it was Abramsky and Coecke’s “Categorical Quantum Mechanics” that helped me to discern more carefully the links between Category Theory and Quantum Computing. They describe Category Theory as the ‘language of modern structural mathematics’ and use it as the tool for building a mathematical representation of quantum processes, and their paper is a thought provoking nudge in the ribs for anyone who is trying to make sense of the current noise that surrounds Quantum mechanics.
Awodey and Spivak are the two most impressive contemporary mathematicians currently working on Category Theory in my view, and whilst it is asking for trouble to choose one or two selected works as exemplars of their approach, I would have to say that Spivak’s book on Category Theory for the Sciences is the standout work of recent times (incidentally the section in this book on ‘aspects’ bears close scrutiny with Wittgenstein’s well known work on ‘family resemblances’).
Awodey’s 2003 paper is as good a recent balance between a mathematical and philosophical exposition of the importance of category theory as exists whilst his textbook is often referred to as the standard entry point for working mathematicians.
Going back to beauty, which is how I started this short note. Barry Mazur wrote an article in memory of Saunders Mac Lane titled ‘When is one thing equal to another‘ which is a gem of rare beauty, and the actual catalyst for this short note. If you read only one document in the links from this article, then I hope it is Mazur’s paper.
Imagine you had to take an art class in which you were taught how to paint a fence or a wall, but you were never shown the paintings of the great masters, and you weren't even told that such paintings existed. Pretty soon you'd be asking, why study art?
That's absurd, of course, but it's surprisingly close to the way we teach children mathematics. In elementary and middle school and even into high school, we hide math's great masterpieces from students' view. The arithmetic, algebraic equations and geometric proofs we do teach are important, but they are to mathematics what whitewashing a fence is to Picasso — so reductive it's almost a lie.
Most of us never get to see the real mathematics because our current math curriculum is more than 1,000 years old. For example, the formula for solutions of quadratic equations was in al-Khwarizmi's book published in 830, and Euclid laid the foundations of Euclidean geometry around 300 BC. If the same time warp were true in physics or biology, we wouldn't know about the solar system, the atom and DNA. This creates an extraordinary educational gap for our kids, schools and society.
An interesting train of thought to be sure. I should post in response to this, or at least think about how it could be structured. I definitely want to come back to write more about this topic.
While browsing through some textbooks and researchers today, I came across a fantastic looking title: Probability Models for DNA Sequence Evolution by Rick Durrett (Springer, 2008). While searching his website at Duke, I noticed that he’s made a .pdf copy of a LaTeX version of the 2nd edition available for download. I hope others find it as interesting and useful as I do.
I’ll also give him a shout out for being a mathematician with a fledgling blog: Rick’s Ramblings.
Probability Models for DNA Sequence Evolution by Richard Durrett
Alex Riley (NOVA Next | PBS)
Ilyas Khan (LinkedIn Pulse)