February 23, 2014
Will Plug-in Cars Crash the Electric Grid?
Sending data in small packets revolutionized communications, from the radio to the internet.
Now three UVM scientists — Pooya Rezaei, Paul Hines and Jeff Frolik — think packets of power can revolutionize
the way electric companies deal with the coming tide of plug-in cars.
They’ve applied for a patent on their new invention.
(Photo: Sally McCay)
Selecting a Chevy Volt, Tesla Model S, Nissan Leaf — or one of many other new models — shoppers in the United States bought more than 96,000 plug-in electric cars in 2013. That’s a tiny slice of the auto market, but it’s up eighty-four percent from the year before. In Vermont, as of January 2014, there were 679 plug-in vehicles, according to the Vermont Energy Investment Corporation. That’s two hundred percent growth over 2013.
This is good news in terms of oil consumption and air pollution. But, of course, every plug-in has to be, well, plugged in. And this growing fleet will put a lot of new strain on the nation’s aging electrical distribution systems, like transformers and underground cables, especially at times of peak demand — say, six in the evening when people come home from work.
How to manage all these cars seeking a socket at the same time — without crashing the grid or pushing rates to the roof — has some utilities wondering, if not downright worried.
Now a team of UVM scientists have created a novel solution, which they report on in the forthcoming March issue of IEEE Transactions on Smart Grid, a journal of the Institute of Electrical and Electronics Engineers.
Put it in a packet
“The key to our approach is to break up the request for power from each car into multiple small chunks — into packets,” says Jeff Frolik, a professor in the College of Engineering and Mathematical Sciences and co-author on the new study.
By using the nation’s growing network of “smart meters” — a new generation of household electric meters that communicate information back-and-forth between a house and the utility — the new approach would let a car charge for, say, five or ten minutes at a time. And then the car would “get back into the line,” Frolik says, and make another request for power. If demand was low, it would continue charging, but if it was high, the car would have to wait.
“The vehicle doesn't care. And, most of the time, as long as people get charged by morning, they won’t care either,” says UVM’s Paul Hines, an expert on power systems and co-author on the study. “By charging cars in this way, it's really easy to let everybody share the capacity that is available on the grid.”
Taking a page out of how radio and internet communications are distributed, the team’s strategy will allow electric utilities to spread out the demand from plug-in cars over the whole day and night. The information from the smart meter prevents the grid from being overloaded. "And the problem of peaks and valleys is becoming more pronounced as we get more intermittent power — wind and solar — in the system,” says Hines. “There is a growing need to smooth out supply and demand.”
At the same time, the UVM teams' invention — patent pending — would protect a car owner’s privacy. A charge management device could be located at the level of, for example, a neighborhood substation. It would assess local strain on the grid. If demand wasn’t too high, it would randomly distribute “charge-packets” of power to those households that were putting in requests.
“Our solution is decentralized,” says Pooya Rezaei, a doctoral student working with Hines and the lead author on the new paper. “The utility doesn't know who is charging.”
Instead, the power would be distributed by a computer algorithm called an “automaton” that is the technical heart of the new approach. The automaton is driven by rising and falling probabilities, which means everyone would eventually get a turn — but the utility wouldn’t know, or need to know, a person’s driving patterns or what house was receiving power when.
But what if you come home from work and need to charge your plug-in right away to get to your kid’s big basketball game? “We assumed that drivers can decide to choose between urgent and non-urgent charging modes,” the scientists write. In the urgent mode the vehicle requests charge regardless of the price of electricity. In this case, the system gives this car the best odds of getting to the front of the line, almost guaranteeing that it will be charged as soon as possible — but at full market rates instead of the discount rate that would be used as an incentive for those opting-in to the new approach.
Why put plug-in cars on “packetized” demand instead of all the other electric demands in a house? Because the new generation of car chargers, so-called “Level 2 PEV chargers” are likely to be the biggest power load in a home. “The load provided by an electric vehicle and the load provided by a house are basically equivalent,” says Frolik. “If someone gets an electric vehicle it's like adding another house to that neighborhood.”
Imagine a neighborhood where everyone buys a plug-in car. Demand doubles, but it’s over the same wires and transformers. Concern about overload in this kind of scenario has led some researchers and utilities to explore systems where the company has centralized control over who can charge when. This so-called “omniscient centralized optimization” can create a perfectly efficient use of the available power — in theory.
But it also means drivers have to either be willing to provide information about their driving habits or set schedules about when they’ll charge their car. This rubs against the grain of a century’s worth of understanding of the car as a tool of autonomy.
Others have proposed elaborate online auction schemes to manage demand. “Some of the other systems are way too complicated,” says Hines, who has extensive experience working with actual power companies. “In a big city, a utility doesn’t want to be managing millions of tiny auctions. Ours is a much simpler system that gets the job done without overloading the grid and gets people what they want the vast majority of the time.”
February 23, 2014
After 400 years, mathematicians
February 23, 2014
IU mathematician receives $2.7 million to establish center in Russia
Vladimir Touraev | Photo by Indiana University
Feb. 19, 2014
BLOOMINGTON, Ind. -- Vladimir Touraev, a Russian high school math teacher in the mid-1970s who 30 years later would become the first named professor in the Indiana University Department of Mathematics, has been awarded over $2.7 million to establish a new mathematics laboratory in Russia.
The award -- one of 42 megagrants awarded by the Russian government this year to scientists from around the world to conduct research in the country -- will allow a recognized leader in the field of low-dimensional topology to establish a scientific center based in Chelyabinsk, Russia. The new center will include about 20 students and an equal number of experienced mathematicians.
Kevin Zumbrun, chair of the IU Bloomington College of Arts and Sciences’ Department of Mathematics, said the award highlights the quality of Touraev’s work over the years and creates new opportunities for collaboration between IU scientists and their counterparts in Russia.
“The department is both delighted that Vladimir’s excellence is being recognized at this scale and excited about the possibilities this raises for new international collaboration at a number of different levels: undergraduate, graduate, postdoctoral and faculty,” Zumbrun said.
Touraev will be based out of Chelyabinsk State University, but about half of the people he will work with and fund will be based in Moscow, Novosibirsk and St. Petersburg. He’ll be funded to organize and operate the new laboratory through 2016 and will have an opportunity to renew funding for an additional two years.
Touraev will continue in his role as the Boucher Professor of Mathematics at IU, spending summers in Russia.
“The aim of the megagrant is to encourage the development of modern mathematics in Russia: I bring the expertise, they bring the resources and, most importantly, the students and scientists,” Touraev said. “The grant will support these undergrads, grad students, postdocs and experienced mathematicians financially as a complement to their basic salaries and stipends from their home institutions, while also supporting their travel, workshops, conferences and invitations to foreign specialists.”
Touraev’s expertise is in low-dimensional topology, a branch of topology -- the study of the properties of geometric shapes that are unaltered by elastic distortions -- that looks at two-, three- and four-dimensional structures such as knots, braids, tangles, links, surfaces and manifolds. Generally, wherever research involves continuity, equilibria, stability or dynamics, topology comes into play. Areas of modern theoretical physics like string theory, improvements in complex networks like neuron interactions and social networks, and movement planning in automated robots are all areas where topology is relevant.
But Touraev said low-dimensional topology would not be the only focus of the new research center.
“While I will try to promote some directions close to my work, I expect the established researchers to pursue their own lines of research,” he said. “Quantum topology has many aspects including connections with low-dimensional topology, representations of algebras, category theory, mathematical physics. Let us just say that at this stage, the grant creates considerable opportunities for mathematicians working in these fields and excellent possibilities for collaboration.”
A permanent U.S. resident with a dual citizenship in Russia and France, where he worked for 17 years as research director with the French National Center of Scientific Research in Strasbourg, Touraev said the first person he invited to visit the new center was his former Ph.D. advisor Oleg Viro, a Russian topologist and professor at Stony Brook University who is also a senior researcher at Russia’s Steklov Institute of Mathematics, where Touraev received his Ph.D.
Touraev said he already has or will soon extend invitations to a number of other scientists in the U.S., including current and former colleagues of the IU Department of Mathematics.
February 23, 2014
UMass mathematician receives two international prizes
Courtesy of the UMass Amherst College of Natural Sciences
Posted by Katrina Borofski on Monday, February 3, 2014
University of Massachusetts professor Panayotis Kevrekidis recently received two international awards for his contributions to mathematics, specifically in his work involving nonlinear waves.
The first award Kevrekidis received was the J.D. Crawford Prize of the Activity Group on Dynamical Systems of the Society for Industrial and Applied Mathematics (SIAM).
According to Kevrekidis, “The citation for this award reads, ‘for his contributions to our understanding of localized solutions of nonlinear wave equations and for developing these for a variety of applications in nonlinear optics and condensed matter physics including Bose-Einstein condensates and granular crystals.’”
Kevrekidis also received the Aristides F. Pallas Award of the Academy of Athens, Greece. This prize is presented by researchers from Greece, and is awarded to the author of one paper in the areas of mathematical analysis who is presently in Greece or abroad, according to Kevrekidis.
Kevrekidis is originally from Greece and came to Massachusetts for “the status of the University and the culture and quality of life of the area.”
“It’s a really exciting and fun experience,” noted Kevrekidis. “This is my 12th year here and both I and my family have been delighted to have the opportunity to live and work in the Pioneer Valley for this period of time.”
“It’s a wonderful place to live and work at and I hope to continue doing that for a while here.”
The paper in which Panayotis Kevrekidis received this award for is titled, “Nonlinear Waves in Lattices: Past, Present Future,” and was published in the IMA Journal of Applied Mathematics. In addition to receiving international recognition, Kevrekidis is currently a graduate professor at the University.
“My role as a faculty member entails all aspects of teaching, research and service,” said Kevrekidis.
“This semester, I am teaching a graduate Master of Science in Applied Mathematics project course, which is a year-long course whereby nine Master of Science students in Applied Mathematics are concentrating on a series of research-level projects and present their efforts through a written report and an oral public presentation,” Kevrekidis said.
While Kevrekidis has done a remarkable job as a faculty member, his work extends far beyond the classroom.
“Additionally, I am conducting research both with researchers at UMass and at nearby institutions, as well as many colleagues nationally and internationally,” Kevrekidis said.
The research Kevrekidis is currently completing revolves primarily around the theme of nonlinear waves and their applications in physics and elsewhere, said Kevrekidis.
Panayotis Kevrekidis is also one of the two co-chairs for a major dynamical systems conference that will take place in Snowbird, UT in May, 2015.
Despite having worked at the University for over a decade, Kevrekidis has completed research at various other locations in the past.
“I have spent a number of extended stays at the Universities of Heidelberg and Hamburg in Germany,” said Kevrekidis, who did so while completing research as part of a Humboldt research fellowship from Germany.
Kevrekidis has also done research at the University of Minnesota as well as numerous other countries and universities.
Pleased with his time abroad, Kevrekidis noted, “These opportunities have been extremely precious as they have diversified my research and opened up new directions not only for me but also directly or indirectly for many of my students and post-doctoral fellows here at UMass.”
While Kevredikis received his awards primarily for work in the field of nonlinear waves, his research has extended to other realms within the mathematic world.
“I am quite broadly interested in all sorts of different levels and aspects of mathematical modeling and its applications to chemical, biological and physical systems,” said Kevredikis.
For example, Kevredikis has worked on catalytical oxidation of surfaces within chemistry, on angiogenic response of cells to tumors within biology and on many different areas in physics. “My training is generally at the interface of Mathematics and Physics,” he said.
Panayotis Kevrekidis will continue researching these themes and others at the University while collaborating with colleagues from Amherst College, Western New England University and the University of Hartford.
Katrina Borofski can be reached at email@example.com
February 23, 2014
Beauty of mathematics excites emotional brain
Friday 14 February 2014
A new study suggests beauty may have a neurological basis. Using brain scans, researchers in the UK found appreciation of abstract beauty - such as in finding aspects of mathematics beautiful - excites the same emotion centers in the brain as appreciation of beauty that comes from more sensory experience - like listening to music or looking at great art.
They report their findings in the open access journal Frontiers in Human Neuroscience.
Having read reports about how some people compare experiencing the beauty of mathematics to appreciating a fine work of art, the researchers decided to see if the brain's emotion centers are active in the same way for these different experiences of beauty.
Lead author Semir Zeki, a professor at the Wellcome Laboratory of Neurobiology at University College London (UCL), says the amount of activity in a person's brain correlates with how intensely they report their experience of beauty to be - even when the object of their attention is an abstract concept.
"To many of us mathematical formulae appear dry and inaccessible but to a mathematician an equation can embody the quintessence of beauty."The quality that the mathematician finds beautiful may lie in the expression of an immutable truth, or just in the simplicity, symmetry or elegance of a concept.
"For Plato," Prof. Zeki notes, "the abstract quality of mathematics expressed the ultimate pinnacle of beauty."
Beautiful equations activated same part of brain as beautiful music
For their study, the team asked 15 mathematicians to take away and consider 60 mathematical formulae, rate how beautiful they experienced each one to be and to note this as a score between -5 (for ugly) and +5 (for beautiful).
Then 2 weeks later, the researchers invited the participants to rate the formulae again as they took functional magnetic resonance imaging (fMRI) scans of their brains.
The scans showed that when the participants looked at mathematical formulae they consistently rated as beautiful, this activated an area of the emotional brain - the medial orbito-frontal cortex - that is also active when people's experience of beauty comes from a piece of music or great art.
The formulae that the participants most consistently rated as beautiful, both before and during the scans, were Leonhard Euler's identity, the Pythagorean identity and the Cauchy-Riemann equations.
Euler's identity (also known as Euler's equation), links five important mathematical constants (e, i, p, 1 and 0) with three basic arithmetic operations, each occurring only once. Some say the beauty of this equation is equal to that of Hamlet's soliloquy, "To be, or not to be, that is the question..."
The participants rated Srinivasa Ramanujan's infinite series and Bernhard Riemann's functional equation as the ugliest.
Prof. Zeki says as with visual experience of art and listening to music, they found the activity in the brain strongly correlated with how intensely the participants declared their experience of beauty to be, "even in this example where the source of beauty is extremely abstract." He concludes:
"This answers a critical question in the study of aesthetics, one which has been debated since classical times, namely whether aesthetic experiences can be quantified."In 2010, Medical News Today reported a study by scientists at Florida Atlantic University that supported the idea that the brain has a mechanism through which experienced music listeners feel the emotions of the performers, making musical communication a form of empathy.
Written by Catharine Paddock PhD
February 23, 2014
What is it that makes mathematics beautiful?
as artistic masterpieces or music by famous composers
Thursday 13 February 2014
Why do people make scholarly studies of Shakespeare's plays, Rembrandt's paintings, or Beethoven's symphonies? Quite simply, because we recognise in these works a beauty that can be inspirational. They inspire study to those so inclined, as does mathematics despite the fact that most people do not find it at all beautiful. Yet obviously some do, as a recent UCL study confirmed, so what is it about mathematics that makes it beautiful?
Ask this question of different mathematicians and you will get different answers. Some are inspired by geometry, others by algebra. Some think geometrically, others more algebraically, though I know no grounds for supposing any difference in the ways their brains function. My thinking is geometric, and as a child I could see, in a pictorial way why the product of two negative numbers was a positive number. That was beautiful, though a beauty that one could not easily share with others. This is the trouble with beauty — it is in the mind of the beholder, and particularly in mathematics it can be hard to explain it to others.
Sometimes one can explain the beauty of a result, but the other type of beauty — a beautiful way of proving that result, a way that appeals to our sense of symmetry, proportion and elegance — can be harder to get across. A good example is Pythagoras's Theorem for a right-angled triangle, which is usually stated as: the square on the hypotenuse (the slanting side) is equal to the sum of the squares on the other two sides. The most well-known example of such a triangle is one whose sides have lengths 3, 4 and 5, where of course 3 squared+4 squared =5 squared, though there is an infinite array of other examples with whole numbers and different ratios of side lengths.
This is a beautiful result, stated and proved in Euclid's famous treatise on geometry written in about 300 BC, but Euclid's proof is ugly. Since his time, and possibly before, more elegant proofs have been given, and the mere existence of such proofs speaks about the beauty of the result. If it weren't beautiful why would anyone want to find a better proof? But many people have been fascinated by the result, including US President Garfield in 1876. I have my own proof, which is to me a thing of exquisite beauty, and I once looked on the internet to see if it was there — there are zillions of proofs of Pythagoras's Theorem on the internet — but I didn't find it.
Leaving aside geometry, here is an elegant result that usually astonishes students when they first see it. Consider all prime numbers larger than 2. Which ones can be written as a sum of two squares? Each one is an odd number, so its remainder after dividing by 4 is either 1 or 3, and the result is very simply stated. Those having a remainder of 1 are sums of two squares (in a unique way), and those having a remainder of 3 are not. For example, 5 has a remainder of 1 and is equal to 1 squared + 2 squared but 7 which has a remainder of 3 is not a sum of two squares. Try a few examples and you may agree that this is indeed rather beautiful.
February 23, 2014
Why Mathematics Is Beautiful and Why It Matters
David H. Bailey
Jonathan M. Borwein
Scientists through the ages have noted, often with some astonishment, not only the remarkable success of mathematics in describing the natural world, but also the fact that the best mathematical formulations are usually those that are the most beautiful. And almost all research mathematicians pepper their description of important mathematical work with terms like "unexpected," "elegance," "simplicity" and "beauty."
Some selected opinions
British mathematician G. H. Hardy (1877-1947), pictured below, expressed in his autobiographical book A Mathematician's Apology what most working mathematicians experience: "Beauty is the first test; there is no permanent place in the world for ugly mathematics."
Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
Henri Poincaré (1854-1912), often described as a "polymath," wrote, in his essay Mathematical Creation, that ignoring this subjective experience "would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true esthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility."
While a very few very applied mathematicians view such ideas as a waste of time, the mathematics community is almost unanimous in agreeing with Poincare.
Physicists are just as impressed by the beauty of mathematics, and by its efficacy in formulating the laws of physics, as are mathematicians. Mathematical physicist Hermann Weyl (1885-1955) declared, "My work always tried to unite the truth with the beautiful, but when I had to choose one or the other, I usually chose the beautiful." This was fully reflected in his own career, when he first attempted to reconcile electromagnetism with relativity.
His work was initially rejected (by Einstein and others), because it was thought to conflict with experimental results, but the subsequent formulation of quantum mechanics led to a renewed acceptance of Weyl's work. In other words, the "beauty" of Weyl's work anticipated its final acceptance, well before the full scientific facts were known.
Nobel physicist Paul Dirac (1902-1984), shown below and described by Niels Bohr as the strangest man, made his most impressive discoveries or predictions, such as that of the positron, largely from demanding elegant, simple mathematical descriptions.
He further elaborated on mathematical beauty in physics in these terms:
[The success of mathematical reasoning in physics] must be ascribed to some mathematical quality in Nature, a quality which the casual observer of Nature would not suspect, but which nevertheless plays an important role in Nature's scheme...
a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuovo of the Capella Medici and see before me the austere beauty of the four statues representing "Day," "Night," "Evening" and "Dawn" which Michelangelo has set over the tomb of Giuliano de'Medici and Lorenzo de'Medici.
Why is this so?
In February 2014, a team of British researchers, including two neurobiologists, a physicist and a mathematician, published a groundbreaking study in Frontiers in Human Neuroscience on the human experience of mathematical beauty. This study is nicely summarized in a BBC Science report.
These researchers employed functional magnetic resonance imaging (fMRI) to display the activity of brains of 16 mathematicians, at a postgraduate or postdoctoral level, as they viewed formulas that they had previously judged as beautiful, so-so or ugly. The results of this analysis showed that beautiful formulas stimulated activity in same field, namely field A1 of the medial orbito-frontal cortex (mOFC), as other researchers have identified as the seat of experience of beauty from other sources.
This is an entirely satisfactory result. It gives an experimental validation of the mathematicians' intuition that they are experiencing the same qualitative states (qualia) as are experienced in other modalities from architecture and sculpture, to poetry and music.
So what exactly is the source of mathematical beauty? All aesthetic responses seem in part to come from identifying simplicity in complexity, pattern in chaos, structure in stasis. In the arts, "beauty" can be accounted for, at least in part, by well-understood harmonies, distributions of colors or other factors.
But what about mathematics? Aesthetic responses, as Santayana in The Sense of Beauty (1896) has argued, require a certain distance:
When we have before us a fine map, in which the line of the coast, now rocky, now sandy, is clearly indicated, together with the winding of the rivers, the elevations of the land, and the distribution of the population, we have the simultaneous suggestion of so many facts, the sense of mastery over so much reality, that we gaze at it with delight, and need no practical motive to keep us studying it, perhaps for hours altogether. A map is not naturally thought of as an aesthetic object...
This captures the aesthetic in mathematics: balancing form and content, syntax and semantics, utility and autonomy. The 2007 book Mathematics and the Aesthetic is dedicated to exploring "new approaches to an ancient affinity." Formed around nine essays, three by practitioners, three by philosophers and three by mathematical educators, it contains a chapter by one of the present bloggers.
Why it matters
As the Economist put it, in a fine essay on the changing notion of mathematical proof, Proof and Beauty (2005):
Why should the non-mathematician care about things of this nature? The foremost reason is that mathematics is beautiful, even if it is, sadly, more inaccessible than other forms of art. The second is that it is useful, and that its utility depends in part on its certainty, and that that certainty cannot come without a notion of proof.
Some argue that mathematical principles are experienced as "beautiful" because they point directly to the fundamental structure of the universe. Physicist Max Tegmark argues further that the reason that mathematics works so well, and so elegantly, in physics is because the universe (or, more properly, the multiverse) is, ultimately, just mathematics -- mathematical structures and the relations that connect them constitute the ultimate irreducible "stuff" of which our world is made. See our recent article on Tegmark and his new book, Our Mathematical Universe.
Few researchers are willing to go as far as Tegmark. But the widely sensed experiences of mathematical beauty, and the astonishing applicability of sophisticated mathematics in the natural world, still beg to be fully understood.
Understood or not, tapping the aesthetic component of mathematics is a crucial and neglected component of mathematical education. See Simon Fraser mathematical educator Nathalie Sinclair's 2006 book Mathematics and Beauty: Aesthetic Approaches to Teaching Children. Given that basing mathematical education on utility and importance has not worked very well, perhaps introducing the aesthetic is past overdue.
February 23, 2014
The hidden beauty of mathematics
It's time to rethink what you learned in school By Shane Parrish, Farnam Street | February 13, 2014
Most of us are unaware of the hidden world of mathematics. Actually, we'd rather avoid the subject entirely. It's difficult and inaccessible.
A lot of that has to do with the way we're introduced to mathematics as taught in school and university.
Math, however, can be "full of infinite possibilities as well as elegance and beauty," writes mathematician Edward Frenkel in Love and Math: The Heart of Hidden Reality. "Mathematics," he goes on, "is as much part of our cultural heritage as art, literature, and music."
Mathematics directs the flow of the universe, lurks behind its shapes and curves, holds the reins of everything from tiny atoms to the biggest stars.
Frenkel, who became a professor at Harvard at twenty-one, now teaches at Berkeley. He "hated math" when he was in school. "What really excited me was physics — especially quantum physics."
A reader sent me a pointer to Frenkel's book after reading 17 equations that changed the world. And I'm very grateful.
Math is a way to describe reality and figure out how the world works, a universal language that has become the gold standard of truth. In our world, increasingly driven by science and technology, mathematics is becoming, ever more, the source of power, wealth, and progress.
Frenkel argues that mathematical knowledge can be an equalizer.
Mathematical knowledge is unlike any other knowledge. While our perception of the physical world can always be distorted, our perception of mathematical truths can't be. They are objective, persistent, necessary truths. A mathematical formula or theorem means the same thing to anyone anywhere — no matter what gender, religion, or skin color; it will mean the same thing to anyone a thousand years from now. And what's also amazing is that we own all of them. No one can patent a mathematical formula, it's ours to share. There is nothing in this world that is so deep and exquisite and yet so readily available to all. That such a reservoir of knowledge really exists is nearly unbelievable. It's too precious to be given away to the "initiated few." It belongs to all of us.
One of the key functions of mathematics is the ordering of information.
This is what distinguishes the brush strokes of Van Gogh from a mere blob of paint. With the advent of 3D printing, the reality we are used to is undergoing a radical transformation: Everything is migrating from the sphere of physical objects to the sphere of information and data. We will soon be able to convert information into matter on demand by using 3D printers just as easily as we now convert a PDF file into a book or an MP3 file into a piece of music.
In our information expanding world, the role of mathematics will become even more crucial as a means to organize and order information. (As equations take over we need to be mindful of what is being filtered.)
Frenkel beautifully explains our cultural aversion to math.
What if at school you had to take an "art class" in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. You would probably say something like this: "Learning art at school was a waste of my time. If I ever need to have my fence painted, I'll just hire people to do this for me." Of course, this sounds ridiculous, but this is how math is taught, and so in the eyes of most of us it becomes the equivalent of watching paint dry. While the paintings of the great masters are readily available, the math of the great masters is locked away.
You can appreciate math without studying it.
[M]ost of us have heard of and have at least a rudimentary understanding of such concepts as the solar system, atoms and elementary particles, the double helix of DNA, and much more, without taking courses in physics and biology. And nobody is surprised that these sophisticated ideas are part of our culture, our collective consciousness. Likewise, everybody can grasp key mathematical concepts and ideas, if they are explained in the right way. To do this, it is not necessary to study math for years; in many cases, we can cut right to the point and jump over tedious steps.
The problem is: While the world at large is always talking about planets, atoms, and DNA, chances are no one has ever talked to you about the fascinating ideas of modern math, such as symmetry groups, novel numerical systems in which two and two isn't always four, and beautiful geometric shapes like Riemann surfaces. It's like they keep showing you a little cat and telling you that this is what a tiger looks like.
The mathematician Israel Gelfand once said:
People think they don't understand math, but it's all about how you explain it to them. If you ask a drunkard what number is larger, 2/ 3 or 3/5, he won't be able to tell you. But if you rephrase the question: What is better, two bottles of vodka for three people or three bottles of vodka for five people, he will tell you right away: two bottles for three people, of course.
Perhaps offering some prescient advice to coming generations, Charles Darwin, wrote in his autobiography:
I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.
"Mathematics is the source of timeless profound knowledge," Frenkel writes, "which goes to the heart of all matter and unites us across cultures, continents, and centuries."
My dream is that all of us will be able to see, appreciate, and marvel at the magic beauty and exquisite harmony of these ideas, formulas, and equations, for this will give so much more meaning to our love for this world and for each other.
Love and Math is a book about mathematical love. Frenkel offers the reader a glimpse into the beauty of mathematics with the Langlands Program, "one of the biggest ideas to come out of mathematics in the last fifty years." In so doing he exposes us to the sides of math we don't get to see often: inspiration, profound ideas, and beautiful revelations.
February 23, 2014
Wikipedia-size maths proof too big for humans to check
17:38 17 February 2014 by Jacob Aron
If no human can check a proof of a theorem, does it really count as mathematics? That's the intriguing question raised by the latest computer-assisted proof. It is as large as the entire content of Wikipedia, making it unlikely that will ever be checked by a human being.
"It might be that somehow we have hit statements which are essentially non-human mathematics," says Alexei Lisitsa of the University of Liverpool, UK, who came up with the proof together with colleague Boris Konev.
The proof is a significant step towards solving a long-standing puzzle known as the Erdos discrepancy problem. It was proposed in the 1930s by the Hungarian mathematician Paul Erdos, who offered $500 for its solution.
Imagine a random, infinite sequence of numbers containing nothing but +1s and -1s. Erdos was fascinated by the extent to which such sequences contain internal patterns. One way to measure that is to cut the infinite sequence off at a certain point, and then create finite sub-sequences within that part of the sequence, such as considering only every third number or every fourth.
Adding up the numbers in a sub-sequence gives a figure called the discrepancy, which acts as a measure of the structure of the sub-sequence and in turn the infinite sequence, as compared with a uniform ideal.
Erdos thought that for any infinite sequence, it would always be possible to find a finite sub-sequence summing to a number larger than any you choose - but couldn't prove it.
It is relatively easy to show by hand that any way you arrange 12 pluses and minuses always has a sub-sequence whose sum exceeds 1. That means that anything longer – including any infinite sequence – must also have a discrepancy of 1 or more. But extending this method to showing that higher discrepancies must always exist is tough as the number of possible sub-sequences to test quickly balloons.
Now Konev and Lisitsa have used a computer to move things on. They have shown that an infinite sequence will always have a discrepancy larger than 2. In this case the cut-off was a sequence of length 1161, rather than 12. Establishing this took a computer nearly 6 hours and generated a 13-gigabyte file detailing its working.
The pair compare this to the size of Wikipedia, the text of which is a 10-gigabyte download. It is probably the longest proof ever: it dwarfs another famously huge proof, which involves 15,000 pages of calculations.
It would take years to check the computer's working – and extending the method to check for yet higher discrepancies might easily produce proofs that are simply too long to be checked by humans. But that raises an interesting philosophical question, says Lisitsa: can a proof really be accepted if no human reads it?
Gil Kalai of the Hebrew University of Jerusalem, Israel, says human checking isn't necessary for a proof to stand. "I'm not concerned by the fact that no human mathematician can check this, because we can check it with other computer approaches," he says. If a computer program using a different method comes up with the same result, then the proof is likely to be right.
Kalai was part of a group that decided in 2010 to work on the problem as a Polymath project, an exercise in which mathematicians use blogs and wikis to collaborate on a large scale. Running different software, the group managed to test a sequence of length 1124 – close to the threshold Konev and Lisitsa have now shown was necessary – but gave up when the program wouldn't scale to higher numbers.
When it comes to the Erdos discrepancy problem, there is still some hope for humans, however. Erdos's hypothesis was that a discrepancy of any value can always be found, a far cry from the discrepancies of 1 and 2 that have now been proven. Lisitsa's software has been running for weeks in an attempt to find a result for discrepancy 3. But even if subsequent programs show that higher and higher discrepancies exist for any infinite sequence, a computer cannot check the infinity of all numbers.
Instead, it's likely that computer-assisted proofs for specific discrepancies will eventually enable a human to spot a pattern and come up with a proof for all numbers, says Lisitsa. "The outstanding problems are like lighthouses; they give us targets for our abilities," adds Kalai.
February 23, 2014
Is the Universe a Simulation?
FEB. 14, 2014
By EDWARD FRENKEL
IN Mikhail Bulgakov’s novel “The Master and Margarita,” the protagonist, a writer, burns a manuscript in a moment of despair, only to find out later from the Devil that “manuscripts don’t burn.” While you might appreciate this romantic sentiment, there is of course no reason to think that it is true. Nikolai Gogol apparently burned the second volume of “Dead Souls,” and it has been lost forever. Likewise, if Bulgakov had burned his manuscript, we would have never known “Master and Margarita.” No other author would have written the same novel.
But there is one area of human endeavor that comes close to exemplifying the maxim “manuscripts don’t burn.” That area is mathematics. If Pythagoras had not lived, or if his work had been destroyed, someone else eventually would have discovered the same Pythagorean theorem. Moreover, this theorem means the same thing to everyone today as it meant 2,500 years ago, and will mean the same thing to everyone a thousand years from now — no matter what advances occur in technology or what new evidence emerges. Mathematical knowledge is unlike any other knowledge. Its truths are objective, necessary and timeless.
What kinds of things are mathematical entities and theorems, that they are knowable in this way? Do they exist somewhere, a set of immaterial objects in the enchanted gardens of the Platonic world, waiting to be discovered? Or are they mere creations of the human mind?
This question has divided thinkers for centuries. It seems spooky to suggest that mathematical entities actually exist in and of themselves. But if math is only a product of the human imagination, how do we all end up agreeing on exactly the same math? Some might argue that mathematical entities are like chess pieces, elaborate fictions in a game invented by humans. But unlike chess, mathematics is indispensable to scientific theories describing our universe. And yet there are many mathematical concepts — from esoteric numerical systems to infinite-dimensional spaces — that we don’t currently find in the world around us. In what sense do they exist?
Many mathematicians, when pressed, admit to being Platonists. The great logician Kurt Gödel argued that mathematical concepts and ideas “form an objective reality of their own, which we cannot create or change, but only perceive and describe.” But if this is true, how do humans manage to access this hidden reality?
We don’t know. But one fanciful possibility is that we live in a computer simulation based on the laws of mathematics — not in what we commonly take to be the real world. According to this theory, some highly advanced computer programmer of the future has devised this simulation, and we are unknowingly part of it. Thus when we discover a mathematical truth, we are simply discovering aspects of the code that the programmer used.
This may strike you as very unlikely. But the Oxford philosopher Nick Bostrom has argued that we are more likely to be in such a simulation than not. If such simulations are possible in theory, he reasons, then eventually humans will create them — presumably many of them. If this is so, in time there will be many more simulated worlds than nonsimulated ones. Statistically speaking, therefore, we are more likely to be living in a simulated world than the real one.
Very clever. But is there any way to empirically test this hypothesis?
Indeed, there may be. In a recent paper, “Constraints on the Universe as a Numerical Simulation,” the physicists Silas R. Beane, Zohreh Davoudi and Martin J. Savage outline a possible method for detecting that our world is actually a computer simulation. Physicists have been creating their own computer simulations of the forces of nature for years — on a tiny scale, the size of an atomic nucleus. They use a three-dimensional grid to model a little chunk of the universe; then they run the program to see what happens. This way, they have been able to simulate the motion and collisions of elementary particles.
But these computer simulations, Professor Beane and his colleagues observe, generate slight but distinctive anomalies — certain kinds of asymmetries. Might we be able to detect these same distinctive anomalies in the actual universe, they wondered? In their paper, they suggest that a closer look at cosmic rays, those high-energy particles coming to Earth’s atmosphere from outside the solar system, may reveal similar asymmetries. If so, this would indicate that we might — just might — ourselves be in someone else’s computer simulation.
Are we prepared to take the “red pill,” as Neo did in “The Matrix,” to see the truth behind the illusion — to see “how deep the rabbit hole goes”? Perhaps not yet. The jury is still out on the simulation hypothesis. But even if it proves too far-fetched, the possibility of the Platonic nature of mathematical ideas remains — and may hold the key to understanding our own reality.
Edward Frenkel, a professor of mathematics at the University of California, Berkeley, is the author of “Love and Math: The Heart of Hidden Reality.”
February 23, 2014
Geophysicist teams with mathematicians to describe how river rocks round
A new study by the University of Pennsylvania's Douglas Jerolmack, working with mathematicians at
Budapest University of Technology and Economics, has found that rocks traveling down a riverbed
follow a distinct pattern, first becoming rounder, and then smaller. Credit: University of Pennsylvania
For centuries, geologists have recognized that the rocks that line riverbeds tend to be smaller and rounder further downstream. But these experts have not agreed on the reason these patterns exist. Abrasion causes rocks to grind down and become rounder as they are transported down the river. Does this grinding reduce the size of rocks significantly, or is it that smaller rocks are simply more easily transported downstream?
A new study by the University of Pennsylvania's Douglas Jerolmack, working with mathematicians at Budapest University of Technology and Economics, has arrived at a resolution to this puzzle. Contrary to what many geologists have believed, the team's model suggests that abrasion plays a key role in upholding these patterns, but it does so in a distinctive, two-phase process. First, abrasion makes a rock round. Then, only when the rock is smooth, does abrasion act to make it smaller in diameter.
"It was a rather remarkable and simple result that helps to solve an outstanding problem in geology," Jerolmack said.
Not only does the model help explain the process of erosion and sediment travel in rivers, but it could also help geologists answer questions about a river's history, such as how long it has flowed. Such information is particularly interesting in light of the rounded pebbles recently discovered on Mars—seemingly evidence of a lengthy history of flowing rivers on its surface.
Jerolmack, an associate professor in Penn's Department of Earth and Environmental Science, lent a geologist's perspective to the Hungarian research team, comprised of Gábor Domokos, András Sipos and Ákos Török.
Their work is to be published in the journal PLOS ONE.
Prior to this study, most geologists did not believe that abrasion could be the dominant force responsible for the gradient of rock size in rivers because experimental evidence pointed to it being too slow a process to explain observed patterns. Instead, they pointed to size-selective transport as the explanation for the pattern: small rocks being more easily transported downstream.
The Budapest University researchers, however, approached the question of how rocks become round purely as a geometrical problem, not a geological one. The mathematical model they conceived formalizes the notion, which may seem intuitive, that sharp corners and protruding parts of a rock will wear down faster than parts that protrude less.
The equation they conceived relates the erosion rate of any surface of a pebble with the curvature of the pebble. According to their model, areas of high curvature erode quickly, and areas of zero or negative curvature do not erode at all.
The math that undergirds their explanation for how pebbles become smooth is similar to the equation that explains how heat flows in a given space; both are problems of diffusion.
"Our paper explains the geometrical evolution of pebble shapes," said Domokos, "and associated geological observations, based on an analogy with an equation that describes the variation of temperature in space and time. In our analogy, temperature corresponds to geometric (or Gaussian) curvature. The mathematical root of our paper is the pioneering work of mathematician Richard Hamilton on the Gauss curvature flow."
From this geometric model comes the novel prediction that abrasion of rocks should occur in two phases. In the first phase, protruding areas are worn down without any change in the diameter of the pebble. In the second phase, the pebble begins to shrink.
"If you start out with a rock shaped like a cube, for example," Jerolmack said, "and start banging it into a wall, the model predicts that under almost any scenario that the rock will erode to a sphere with a diameter exactly as long as one of the cube's sides. Only once it becomes a perfect sphere will it then begin to reduce in diameter."
The research team also completed an experiment to confirm their model, taking a cube of sandstone and placing it in a tumbler and monitoring its shape as it eroded.
"The shape evolved exactly as the model predicted," Jerolmack said.
The finding has a number of implications for geologic questions. One is that rocks can lose a significant amount of their mass before their diameter starts to shrink. Yet geologists typically measure river rock size by diameter, not weight.
"If all we're doing in the field is measuring diameter, then we're missing the whole part of shape evolution that can occur without any change in diameter," Jerolmack said. "We're underestimating the importance of abrasion because we're not measuring enough about the pebble."
As a result, Jerolmack noted that geologists may also have been underestimating how much sand and silt arises because of abrasion, the material ground off of the rocks that travel downstream. "The fine particles that are produced by abrasion are the things that go into producing the floodplain downstream in the river; it's the sand that gets deposited on the beach; it's the mud that gets deposited in the estuary," he said.
With this new understanding of how the process of abrasion proceeds, researchers can address other questions about river flow—both here on Earth and elsewhere, such as on Mars, where NASA's rover Curiosity recently discovered rounded pebbles indicative of ancient river flow.
"If you pluck a pebble out of a riverbed," Jerolmack said, "a question you might like to answer, how far has this pebble traveled? And how long has it taken to reach this place?" Such questions are among those that Jerolmack and colleagues are now asking.
"If we know something about a rock's initial shape, we can model how it went from its initial shape to the current one," he said. "On Mars, we've seen evidence of river channels, but what everyone wants to know is, was Mars warm and wet for a long time, such that you could have had life? If I can say how long it took for this pebble to grind down to this shape, I can put a constraint on how long Mars needed to have stable liquid water on the surface."
February 23, 2014
Forest model predicts canopy competition
February 20, 2014 | Media Contact: David Orenstein | 401-863-1862
Scientists use measurements from airborne lasers to gauge changes in the height of trees in the forest. Tree height tells them things like how much carbon is being stored. But what accounts for height changes over time — vertical growth or overtopping by a taller tree? A new statistical model helps researchers figure out what’s really happening on the ground. PROVIDENCE, R.I. [Brown University] — Out of an effort to account for what seemed in airborne images to be unusually large tree growth in a Hawaiian forest, scientists at Brown University and the Carnegie Institution for Science have developed a new mathematical model that predicts how trees compete for space in the canopy.
What their model revealed for this particular forest of hardy native Metrosideros polymorpha trees on the windward slope of Manua Kea, is that an incumbent tree limb greening up a given square meter would still dominate its position two years later a forbidding 97.9 percent of the time. The model described online in the journal Ecology Letters could help generate similar predictions for other forests, too.
Why track forest growth using remote sensing, pixel by pixel? Some ecologists could use that information to learn how much one species is displacing another over a wide area or how quickly gaps in the canopy are filled in. Others could see how well a forest is growing overall. Tracking the height of a forest’s canopy reveals how tall the trees are and therefore how much carbon they are keeping out of the atmosphere — that is, as long as scientists know how to interpret the measurements of forest growth.
James Kellner, assistant professor of ecology and evolutionary biology at Brown University, the paper’s lead and corresponding author, noticed what seemed like implausibly large canopy growth in LIDAR images collected by the Carnegie Airborne Observatory over 43 hectares on the windward flank of Manua Kea. In the vast majority of pixels (each representing about a square meter) the forest growth looked normal, but in some places the height change between 2007 and 2009 seemed impossible: sometimes 10 or 15 meters.
The data were correct, he soon confirmed, but the jumps in height signaled something other than vertical growth. They signaled places where one tree had managed to overtop another or where the canopy was filling in a bare spot. The forest wasn’t storing that much more carbon; taller trees were growing a few meters to the side and creating exaggerated appearances of vertical growth in the overhead images.
Turning that realization into a predictive mathematical model is not a simple matter. Working with co-author Gregory P. Asner at the Carnegie Institution for Science in Stanford, Calif., Kellner created the model, which provides a probabilistic accounting of whether the height change in a pixel is likely to be the normal growth of the incumbent tree, a takeover by a neighboring tree, or another branch of the incumbent tree.
The model doesn’t just work for this forest but potentially for different kinds of forests, Kellner said, because its interpretation of the data is guided by the data itself. The model uses what seems to be the forest’s normal rate of growth to determine when evidence of vertical growth is more than plausible — and therefore a possible signal of lateral overtopping.
“While we can all agree that a 20-meter increase over two years is definitely not vertical growth, where you put the boundary, is a necessarily subjective decision,” Kellner said. “The neat thing about the analytical framework is you have the data choosing for you. The data arbitrate when a given height change is judged to be vertical rather than lateral, and that is based on the unique neighborhood around that position and what we’ve observed in the rest of the data.”
So even in an area where growth is quite uniform, the model can still predict whether a height change is due to growth or a takeover. Accounting for several neighborhoods, including some with more variance, can delineate trends such as how close trees have to be before one could overtop another.
Using the model, Kellner and Asner gained a number of insights beyond the huge incumbency advantage. They found that a tree’s height was a poor predictor of whether it would evade rivals. Very short trees (less than 11 meters) were clearly in some trouble, but beyond 11 meters tallness was not much of a factor. Instead, they saw, proximity to taller neighbors was a tree’s biggest threat.
“When a position in the canopy was lost to a neighbor, it was almost exclusively due to competition among the immediate neighbors (the 3-by-3 pixel neighborhood), which represented locations that were less than 1.77 meters away,” Kellner and Asner wrote. “Neighbors at greater distances accounted for just two of the 3,906 episodes of lateral capture inferred to have occurred in our data.”
But in a forest with trees capable of more dramatic lateral growth, that distance might end up being bigger. The model would illuminate that.
“There’s definitely basic ecological interest in understanding what might be called the rules of the game,” Kellner said. “If you think of the trees as competing for access to space in the canopy and we can infer what those rules are by analyzing data like these.”
The National Science Foundation (DEB-0715674) and the Carnegie Institution for Science funded the study.
Editors: Brown University has a fiber link television studio available for domestic and international live and taped interviews, and maintains an ISDN line for radio interviews. For more information, call (401) 863-2476.
February 23, 2014
Sound, light and water waves and how
February 23, 2014
Ray Steiner, 1941-2014: BGSU mathematician prolific scholar, author
BY MARLENE HARRIS-TAYLOR
Ray Steiner, 72, a retired Bowling Green State University professor and internationally known mathematician and scholar, died Feb. 13 at Toledo Hospital, where he was taken after suffering a massive brain hemorrhage at home.
Mr. Steiner did not come to breakfast that morning, and one of his cats alerted his wife, Carol, that something was wrong.
“He always said that he wanted to go in his sleep,” and that is essentially what happened, Mrs. Steiner said.
He was a quiet and gentle man who loved teaching and solving complex mathematical equations, his wife said. He retired from BGSU in 1998 after 30 years in the math department.
Mr. Steiner had several offers after he received his doctorate at Arizona State University, but chose to join the faculty at BGSU.
“He kind of had his pick. He said there was something about Bowling Green that seemed like him,” Mrs. Steiner said.
The Steiners met in 1971 on a blind date and married less than a year later at the Monroe Street Methodist Church in Toledo. They were introduced by a friend who thought they were a good intellectual match.
“We clicked right away. We knew in three weeks that this was it. He said, ‘I think we ought to get married,’ and I said, ‘I think it sounds like a pretty good idea.’ I told him the only requirement is that he had to give me a heart-shaped diamond ring,” Mrs. Steiner said.
Mr. Steiner did give her that ring and the two had what their friends describe as an ideal marriage. They were both teachers, Mrs. Steiner taught high school music, a passion they both shared.
“He played both violin and the viola and I accompanied him on the piano," she said.
“They had one of the best marriages I have ever seen. They are inseparable. I have never met any couple like them. When he married Carol, he really came out of his shell,” said Tom Hern, a close friend and former BGSU colleague.
The two joined the math department a year apart and retired at the same time.
“He was one of the smartest people I have ever known, and to other people that made him a little weird and nerdy,” Mr. Hern said.
He said Mr. Steiner taught undergraduate and graduate courses and was a prolific scholar who published 35 publications and numerous books and papers. He was highly respected in the department and internationally known by the particular group of mathematicians known as number theorists.
“Through him I met Louis Mordell, one of the most famous number theorists of the 20th century,” Mr. Hern said.
Mr. Steiner's graduate thesis was an explanation of how to solve Mordell's Equation, a complex mathematical problem studied by theorists around the world.
He was a gentle giant in his field, but many of his friends are just learning about all his accomplishments because he did not brag or boast, Mr. Hern said.
In 1987, Mr. Steiner traveled to Budapest to present a paper at the International Numbers Theory Conference.
There he showed mathematicians from around the world an easier and more complete solution to solving a very well-known equation.
In a Blade article about his travels, Mr. Steiner said: “Nothing's unsolvable. Some things are just unsolved.”
Born in New York City to the late Joseph and Clara Finkelstein, he later changed his name to Steiner because he did not want any children to be teased and called Frankenstein, his wife said.
When he was just 8 years old, his family moved to Arizona because of his asthma.
His father, who had worked in the Brooklyn Navy Yard, found work as an electrician after the move, Mrs. Steiner said.
He received a bachelor of science in electrical engineering from the University of Arizona in 1963, followed by a master's degree in 1965.
His mother, she said, was brilliant with languages and shared that talent with her son.
Mr. Steiner became fluent in nine languages and was a Hebrew scholar.
Surviving are his wife, Carol Steiner, and sister, Sharon Perry.
Visitation will be from 5 to 7 p.m. March 21 at the Deck-Hanneman Funeral Home, Bowling Green. A Celebration of Life Service will be at 11 a.m. March 22 at the First Christian Church, Bowling Green.
The family suggests tributes to the church or to a scholarship fund in his name for students entering the mathematical field, in the care of the mortuary.
Contact Marlene Harris-Taylor Marlene Harris-Taylor at: firstname.lastname@example.org or 419-724-6091.
February 6, 2014
Juggling Math Professor Receives MAA National Teaching Award
Juggling. Weird numbers. Mozart. What do these things have in common? They have Dominic Klyve [KLEE-vee], an uncommonly gifted mathematics professor at Central Washington University.
Klyve recently received the national Henry L. Alder Award for Distinguished Teaching from the Mathematics Association of America. Klyve is the first winner from Washington State in the 18-year history of the award. The Alder Award goes to individuals whose teaching has been extraordinarily successful and who have had influence beyond their own classrooms.
“I’m absolutely thrilled,” said Klyve. “It gives national recognition to my department and my students, as well as being a tremendous honor for me. This wouldn’t have been possible if I had been at an institution other than Central—because our math department is so good, and is so deeply engaged in teaching.”
In his first four years at Central, he has taught a surprising variety of classes, developed new ones, and founded a Mathematics Honors Program. He has engaged first-year statistics students in analyzing a countywide nutrition survey, where they discovered a statistically significant link between freezers and hunger. His undergraduate students have recently discovered the largest weird number* in the world. And he has developed a Math 101 class for music majors that explores regression and correlation in Mozart’s sonatas.
“I have to give credit to my colleagues for so much of this,” he enthused. “Ian Quitadamo [science education professor] encouraged me to involve my students in community-based inquiry. The math and music class was created by Todd Shiver [music chair and professor. I’m especially grateful to my chair, Tim Englund for nominating me for this award.”
In this collegial atmosphere, Klyve has flourished as a teacher who, according to one undergraduate, instills “a lingering love for mathematics” in his students. He is a regular speaker on mathematics (and a teacher of juggling!) at Wisconsin's Suzuki music festival. In fact, he is so intrigued with juggling that he formulated an equation that mathematically determines how juggling balls fall (“A Zeta Function for Juggling Sequences,” Journal of Combinatorics and Number Theory, 01/2012).
He has already built a reputation as a popular lecturer. He has been invited to schools throughout the United States to speak about his work. In 2013, the University of Canterbury brought him to New Zealand to teach a five-week course in the History of Mathematics.
He is currently working on a grant to teach math using original, historical sources and has recruited dozens of mathematicians across the country to develop projects to teach math using the history of mathematics.
Klyve was also cited for combining teaching with scholarship.
“Undergraduate research is really important to me,” he said, “and I’m lucky that so many CWU students are interested in pursuing research projects.”
Since coming to Central four years ago, he has written six papers with students for publication in international research journals. He is a Councilor on the national “Council on Undergraduate Research,” and will serve as the first president of the Math Association of America’s Special Interest Group on Undergraduate Research.
In addition to his college coursework, he is active in working with high school students. In addition to bringing the American Mathematics Competition to Ellensburg two years ago, he teaches under-represented and first-generation high school students during the summers as part of the Upward Bound Program.
“I’ve known for a long time that I love math, and I love explaining things to people,” said Klyve. “I’m grateful to have a supportive department and chair that allow me the freedom to pursue what I love to do.”
*Weird numbers are those in which no combination of its divisors adds up to the original number. For example, the smallest weird number is 70; its divisors are 1, 2, 7, 10, 14, and 35. No combination of any or all of its factors equal 70.
Media Contact: Valerie Chapman-Stockwell, Public Affairs, 509-963-1518, email@example.com
February 6, 2014
Algebra-cadabra! Here's a classic magic trick, and the mathematical secret behind it
You only need a cursory familiarity with the work of magicians like Derren Brown and David Blaine to realise that at the heart of many illusions lies a piece of rock-solid mathematics.
Sometimes tricks require fooling people with probability, as Brown expertly did in The System, his classic show about predicting the results of horse races.
And sometimes it relies on genuinely surprising and clever theory.
The literature on mathematical magic overwhelmingly concerns tricks using playing cards.
Here's one that's in the book.
The good thing about this trick is that you need an attractive assistant, which is one of the best things about being a magician.
Okay, the assistant doesn't need to be attractive. But I like to aim high, and will assume that she is for the remainder of this post.
The trick was invented by William Fitch Cheney Jr, a US mathematics professor, in 1951. It was originally called Telephone Stud since it could be done over the phone. Mulcahy calls it Fitch Cheney's Five-Card Twist.
First the magician leaves the room, leaving the attractive assistant with the audience. She gives a full deck of cards to an audience member, and asks him or her to shuffle it and then to choose any five cards.
The assistant takes the cards, looks at them, places one face down, and places the four others face up and side by side.
The magician is allowed back in. He glances at the table and – abracadabra – names the hidden card. The audience gasps in awe, since there was no way he knew which cards had been chosen.
So how did he do it?
What the magician has done is to deduce the hidden card from the four visible cards.
Or rather, the assistant has placed the four visible cards in a certain way that communicates the value of the hidden card. There is a code that the magician and the assistant have agreed on beforehand. Before I explain the exact method it is worth having a think about what this code might be. How can four randomly chosen cards always identify any of the 48 other cards in the deck?
It's done like this:
The assistant sees the five cards and has the choice of which four to reveal and which one to keep hidden.
Since there are five cards, but only four suits, it must be the case that at least two cards have the same suit.
So, the assistant choses one of these cards to be the hidden one, and places the other in a fixed position on the table, say first in the line.
Just say there are at least two hearts. The assistant hides one of the hearts and puts another of the hearts first in the line. When the magician returns he looks at the four cards lined side by side, sees a heart card in position 1 and knows instantly that the missing card is a heart.
So far, so good.
The cleverer part is how the other three cards determine the value of the hidden card.
There are 13 values in a deck. In order, they are: A,2,3,4,5,6,7,8,9,10, J, Q, K.
Consider these values as repeating, as if they are numbers round a clock face, so after K comes A, 2, 3 and so on.
If you choose any two cards, their values can be at most six positions apart. For example, 3 and 10 are six positions apart since we count up from 10: J, Q, K, A, 2, 3.
We established before that there are at least two cards of the same suit in the five that the assistant saw. The assistant has a choice about which one to keep hidden and which one to place in the first position
The rule is that the lowest card is revealed, so that the hidden card must be either 1, 2, 3, 4, 5, or 6 values above it.
The positions of the other three visible cards, therefore, must be placed in such a way as to convey a number between 1 and 6.
Once the magician has this number all he does is he counts up from the value of the card in position 1 and he knows which card is hidden.
The trick relies on having an agreed ordering of all the cards in the deck. Let our ordering be this one: the lowest card is the ace of clubs and then we go up the clubs to king,
then move on to the ace of hearts, and move up the hearts to king, and then continue with the spades and then diamonds.
The assistant looks at the three visible cards and works out which is the lowest card in the ranking, which is the middle card, and which is the highest. Let's call these cards L, M and H.
There are six arrangements of the cards when they are laid side by side.
Let the rule be: LMH = 1, LHM = 2, MLH = 3, MHL = 4, HLM = 5, HML = 6.
The assistant places the cards in the correct arrangement that encodes the correct number.
So, when the assistant places the four revealed cards in a row, the one in the first position gives the suit of the hidden card, and the cards in the next three positions encode a number which is the number the magician must count up from the value of the first card in order to deduce the value of the hidden card.
Now thank your lovely assistant. You really couldn't have done it without her.
February 6, 2014
Kazakh mathematician may have solved $1 million puzzle
Becoming a millionaire mathematician?
15:09 22 January 2014 by Jacob Aron and Katia Moskvitch
Mathematics is a universal language. Even so, a Kazakh mathematician's claim to have solved a problem worth a million dollars is proving hard to evaluate – in part because it is not written in English.
Mukhtarbay Otelbayev of the Eurasian National University in Astana, Kazakhstan, says he has proved the Navier-Stokes existence and smoothness problem, which concerns equations that are used to model fluids – from airflow over a plane's wing to the crashing of a tsunami. The equations work, but there is no proof that solutions exist for all possible situations, and won't sometimes "blow up", producing unrealistic answers.
In 2000, the Clay Mathematics Institute, now in Providence, Rhode Island, named this one of seven Millennium Prize problems offering $1 million to anyone who could devise a proof.
Otelbayev claims to have done just that in a paper published in the Mathematical Journal, also based in Kazakhstan. "I worked on the problem on and off, for 30 years," he told New Scientist, in Russian – he does not speak English.
Mathematical Babel fish
However, the combination of the Russian text and the specialist knowledge needed to understand the Navier-Stokes equations means the international mathematical community, which usually communicates in English, is having difficulty evaluating it. Although mathematics is expressed through universal symbols, mathematics papers also contain large amounts of explanatory text.
"Over the years there have been several alleged solutions to the Navier-Stokes problem that turned out to be wrong," says Charles Fefferman of Princeton University, who wrote the official formulation of the problem for Clay. "Since I don't speak Russian and the paper is not yet translated, I'm afraid I can't say more right now."
Otelbayev is a professional, so mathematicians are paying more attention to his proof than is typical for amateur efforts to solve Millennium Prize problems, which are regularly posted online.
The Russian-speaking Misha Wolfson, a computer scientist and chemist at the Massachusetts Institute of Technology is attempting to spark an online, group effort to translate the paper. "While my grasp on the math is good enough to enable translation up to this point, I am not qualified to say anything about whether or not the solution is any good," he says.
Stephen Montgomery-Smith of the University of Missouri in Columbia, who is working with Russian colleagues to study the paper, is hopeful."What I have read so far does seem valid," he says "but I don't feel that I have yet got to the heart of the proof."
Otelbayev says that three colleagues in Kazakhstan and another in Russia agree that the proof is correct.
Burden of proof
Understandably, a high burden of proof is required to claim the $1 million prize. Clay's rules say the solution must be published in a journal of "worldwide repute" and remain unchallenged for two years before it can even be considered. Nick Woodhouse, president of the Clay Mathematics Institute, declined to comment on Otelbayev's proof.
"It is currently being translated by my students, and will be available soon," says Otelbayev. He says that he will publish it again once it is translated into English – initially in a second Kazakh journal, and then perhaps abroad.
To date, only one Millennium Prize problem has been officially solved. In 2002, Grigori Perelman proved the Poincaré conjecture, but later withdrew from the mathematical community and refused the $1 million prize.
A possible solution for another problem, known as P vs NP, caught mathematicians' attentions in 2010, but later proved to be flawed. Whether Otelbayev's proof will share the same fate remains to be seen.
February 6, 2014
Modeling buildings by the millions: Building codes in China tested for energy savings
Building energy codes — which regulate factors such as building insulation —
could play a major role in reducing China’s building energy consumption, according to a study
led by the Department of Energy’s Pacific Northwest National Laboratory.
January 28, 2014
Eric Francavilla, PNNL, (509) 372-4066
Changes to China’s building codes could cut building energy use by 22%
RICHLAND, Wash. – China can build its way to a more energy efficient future — one house, apartment and retail store at a time — by improving the rules regulating these structures, according to a study by the Department of Energy's Pacific Northwest National Laboratory.
PNNL scientists at the Joint Global Change Research Institute, a partnership with the University of Maryland in College Park, Md., have created a unique model that projects how much energy can be saved with changes to China's building energy codes.
Already home to almost one-fifth the world's population, China is not only growing, but rapidly developing. And it's consuming more energy along the way. Reducing energy consumption through building codes is a win-win for China and the rest of the world, by reducing fossil fuel use and carbon dioxide emissions while still promoting economic growth and energy security.
The study focused on realistic improvements to codes that regulate building aspects like insulation and lighting. Improvements to these codes could reduce building energy consumption by up to 22 percent by the end of this century, compared to a no-change scenario, the researchers found.
"A 22-percent cut is a large change in China's trajectory," said Meredydd Evans, the PNNL scientist who managed the project. "More energy could be saved with additional standards and policies, but this study shows that a distinct set of codes can have great impact."
Findings from the study were published in Energy Policy.
Before foundations, buildings start with codes
Since China implemented its first building energy codes in the 1980s, the country has expressed a commitment to reducing energy consumption and carbon dioxide emissions through improved codes, Evans said. In fact, China's codes are not radically different than those in the U.S., though significant gaps remains, she said.
Among China's strengths is a high compliance rate, which has been achieved through private, third-party inspectors that oversee construction on a routine basis, and government oversight. And in December 2012, China began closing a gap in codes for rural buildings by instating a voluntary code. About half of China's population lives in rural buildings, which often lack proper insulation, air-tightness and energy-efficient cooking methods. The voluntary codes are the first step in raising rural China to the same, mandatory standards as the rest of the country.
Given that China continues to grow and evolve, policy makers and researchers alike face a challenge of determining which regulations to improve.
"China won't find one golden policy that solves its energy and pollution problems," said Sha Yu, lead scientist and principle author for the study. "They need policies that are comprehensive and feasible."
This study focuses on a set of building energy codes, most of which involve the building envelope. As the barrier between the interior and outside elements, the envelope includes walls, the roof, windows and other items that maintain a building's structure and climate control. The codes in this study dealt with insulation, heating, ventilation, cooking and lighting.
Part of the upgrade to these codes will increase the need for efficient, high quality building materials. This transition will be an opportunity for both China and the U.S. to grow business in the energy efficiency industry.
Improving China's building energy codes is a feasible goal, Yu said, but assessing the impacts of those changes is easier said than done. That's where the model comes in.
Modeling buildings by the millions
When calculating the impacts of building codes over nine decades and across one-billion-plus people, a simple model won't do.
The researchers in this study used the PNNL-developed Global Change Assessment Model to carry out their analysis. Also known as GCAM, the model takes into account an exhaustive list of human and ecological variables.
For example, the model factors in population growth, which is assumed to peak in China in 2035. Urbanization level, or the percentage of people living in urban buildings, will continue to increase through the end of the century. This is important, because urban buildings — filled with electronic appliances — consume more energy, Evans said.
A building's performance changes in different climates, which is why the model divides China into four climate regions. The model even accounts for climate change projections.
Other variables considered in the model include changes in building technology, energy supply and climate policy.
The model uses these variables to test codes in three building types: urban residential, rural residential and commercial. Furthermore, it assesses codes that only apply to new construction and codes that require retrofitting existing buildings.
Overall, no other study has included these important dynamics in an integrated way — making the results a valuable resource to inform policymakers.
The results: Codes play a major role in energy efficiency
With proper enforcement and education, better building codes will lead to more efficient buildings. In this study, three improved-code scenarios yielded decreases in net building energy demand, compared to a scenario where buildings codes remained at 2010 levels. In other words, more energy will go toward powering buildings by the end of the century, but improving codes will slow that trend.
China has much to gain from improving codes for new urban-residential and commercial buildings-a 13 percent cut in building energy demand by the end of the century. China can accomplish this goal if it continues its current rate of improvements, Evans said.
China could cut another 9 percent by adding rural buildings to mandatory new-building codes and retrofit requirements for all buildings. Altogether, that's a 22 percent reduction in energy used by buildings by the end of the century.
Developed countries use more energy for buildings than developing ones-and China will be no exception. But this study shows that changes to building codes don't have to be radical to make a difference. Additional changes, such as appliance standards, could add to these energy savings, Evans said.
Funding for this study came from the DOE Office of Energy Efficiency and Renewable Energy and the Global Technology Strategy Program. Research by PNNL staff at the Joint Global Change Research Institute focuses on developing dialogues across disciplines and national boundaries to address global change issues.
The findings have been presented to the Chinese government, China Academy of Building Research, DOE, members of industry, and at the U.S.-China Energy Efficiency Forum.
Reference: Yu, S., et al., A long-term integrated impact assessment of alternative building energy code scenarios in China. Energy Policy (2013). DOI: j.enpol.2013.11.009.
February 6, 2014
Tracing unique cells with mathematics
Fluorescence- in-situ-hybridization shows mRNA-activity in a tissue sample.
Blue: low, red: high activity –
Image: S. S. Bajikar / University of Virginia, Charlottesville (USA)
23.01.2014, Research news
Stem cells can turn into heart cells, skin cells can mutate to cancer cells; even cells of the same tissue type exhibit small heterogeneities. Scientists use single-cell analysis to investigate these heterogeneities. But the method is still laborious and considerable inaccuracies conceal smaller effects. Scientists at the Technische Universitaet Muenchen (TUM), the Helmholtz Zentrum Muenchen and the University of Virginia (USA) have now found a way to simplify and improve the analysis by mathematical methods.
Each cell in our body is unique. Even cells of the same tissue type that look identical under the microscope differ slightly from each other. To understand how a heart cell can develop from a stem cell, why one beta-cell produces insulin and the other does not, or why a normal tissue cell suddenly mutates to a cancer cell, scientists have been targeting the activities of ribonucleic acid, RNA.
Proteins are constantly being assembled and disassembled in the cell. RNA molecules read blueprints for proteins from the DNA and initiate their production. In the last few years scientists around the world have developed sequencing methods that are capable of detecting all active RNA molecules within a single cell at a certain time.
At the end of December 2013 the journal Nature Methods declared single-cell sequencing the "Method of the Year." However, analysis of individual cells is extremely complex, and the handling of the cells generates errors and inaccuracies. Smaller differences in gene regulation can be overwhelmed by the statistical "noise."
Easier and more accurate, thanks to statistics
Scientists led by Professor Fabian Theis, Chair of Mathematical modeling of biological systems at the Technische Universitaet Muenchen and director of the Institute of Computational Biology at the Helmholtz Zentrum Muenchen, have now found a way to considerably improve single-cell analysis by applying methods of mathematical statistics.
Instead of just one cell, they took 16-80 samples with ten cells each. "A sample of ten cells is much easier to handle," says Professor Theis. "With ten times the amount of cell material, the influences of ambient conditions can be markedly suppressed." However, cells with different properties are then distributed randomly on the samples. Therefore Theis's collaborator Christiane Fuchs developed statistical methods to still identify the single-cell properties in the mixture of signals.
Combining model and experiment
On the basis of known biological data, Theis and Fuchs modeled the distribution for the case of genes that exhibit two well-defined regulatory states. Together with biologists Kevin Janes and Sameer Bajikar at the University of Virginia in Charlottesville (USA), they were able to prove experimentally that with the help of statistical methods samples containing ten cells deliver results of higher accuracy than can be achieved through analysis of the same number of single cell samples.
In many cases, several gene actions are triggered by the same factor. Even in such cases, the statistical method can be applied successfully. Fluorescent markers indicate the gene activities. The result is a mosaic, which again can be checked to spot whether different cells respond differently to the factor.
The method is so sensitive that it even shows one deviation in 40 otherwise identical cells. The fact that this difference actually is an effect and not a random outlier could be proven experimentally.
This work has been funded by the American Cancer Society, the National Institutes of Health, the German Research Foundation (DFG), the German Academic Exchange Service (DAAD), the Pew Scholars Program in the Biomedical Sciences, the David and Lucile Packard Foundation, the National Science Foundation and the European Research Council.
February 6, 2014
Stuff symphony: Beautiful music makes better materials
05 February 2014 by Markus J. Buehler
NATURE is rich in structure, which defines the properties not only of the tiniest pieces of matter, but of galaxies and the universe itself. That structure explains both the sound of music, and what is embodied in our DNA.
Our world consists of complex hierarchies of about 100 different chemical elements, and it is the arrangement of these elements into molecules that gives rise to the rich set of materials around us – from the sugar molecules in the food we eat to the oxides in the Earth's crust. In the living world, a limited set of building blocks of DNA (with four distinct letters) and amino acids (with around 20 distinct types) creates some of the most functionally diverse materials we know of, the stuff that builds our bones, skin and complex organs such as the brain.
The properties of a piece of matter are defined not by the basic building blocks themselves but by the way they are organised into hierarchies. This paradigm – where structure defines function – is one of the overarching principles of biological systems, and the key to their innate ability to grow, self-repair, and morph into new functions. Spider silk is one of the most remarkable examples of nature's materials, created from a simple protein spun into fibres stronger than steel.
As we begin to appreciate the universal importance of hierarchies, engineers are applying this understanding to the design of synthetic materials and devices. They can gain inspiration from a surprising source: music.
In the world of music, a limited set of tones is the starting point for melodies, which in turn are arranged into complex structures to create symphonies. Think of an orchestra, where each instrument plays a relatively simple series of tones. Only when combined do these tones become the complex sound we call classical music. Essentially, music is just one example of a hierarchical system, where patterns are nested within larger patterns – similar to the way characters form words, which form sentences, then chapters and eventually a novel.
Composers have exploited the concept of hierarchies for thousands of years, perhaps unknowingly, but only recently have these systems been understood mathematically. This maths shows that the principles of musical composition are shared by many seemingly diverse hierarchical systems, suggesting many exciting avenues to explore. From the basic physics of string theory to complex biological materials, different functions arise from a small number of universal building blocks. I call this the universality-diversity-paradigm.
Nature uses this paradigm to design its materials, creating new functions via novel structures, built using existing building blocks rather than fresh ones. Yet through the ages humans have relied on a totally different approach to construct our world, introducing a new building block, or material, when a new function is required. For example, an aeroplane consists of thousands of different materials that originate from very different sources, such as plastics, metals or ceramics.
It is not the building block itself that is limiting our ability to create better, more durable or stronger materials, but rather our inability to control the way these building blocks are arranged. To overcome this limitation, I am trying to design new materials in a similar way to nature. In my lab we are using the hidden structures of music to create artificial materials such as designer silks and other materials for medical and engineering applications. We want to find out if we can reformulate the design of a material using the concept of tones, melodies and rhythms. Can a composer come up with a radically different approach to design?
Our brains have a natural capacity for dealing with the hierarchical structure of music, a talent that may unlock a greater creative potential for understanding and designing artificial materials. For example, in recent work we designed different sequences of amino acids based on naturally occurring ones, introducing variations to create our own materials with better properties. However, the way in which the different sequences of amino acids interact to form fibres is largely a mystery and is difficult to observe in an experiment. To gain more understanding, we translated the process by which sequences of amino acids are spun into silk fibres into musical compositions.
In this translation from silk to music, we replaced the protein's building blocks (sequences of amino acids) with corresponding musical building blocks (tones and melody). As the music was played, we could "listen" to the amino acid sequences we had designed, and deduce how certain qualities of the material, such as its mechanical strength, appear in the musical space. Listening to the music improved our understanding of the mechanism by which the chains of amino acids interact to form a material during the silk-spinning process. The chains of amino acids that formed silk fibres of poor quality, for example, translated into music that was aggressive and harsh, while the ones that formed better fibres sounded softer and more fluid, as they were derived from a more interwoven network. In future work we hope to improve the design of the silk by enhancing those musical qualities that reflect better properties – that is, to emphasise softer, more fluid and interwoven melodies.
This approach has implications far beyond the design of new materials. In future we might be able to translate melodies to design better sequences of DNA or amino acids, or even to reinvent transportation systems for cities. Underlying this approach is a branch of mathematics called category theory, which describes the character of the links between different objects within a system. For example, it describes the way a material works by categorising each of the building blocks (atoms and molecules) with respect to each other, the way they interact and how their interactions create a certain function such as toughness. Category theory finds common descriptions of seemingly distinct systems, creating the possibility to translate back and forth from one field to another.
Universal patterns Using category theory we can discover universal patterns that form the blueprints of our world. We may be able to derive all the things we know – molecules, living tissues, music, the universe – by applying universal patterns in different physical manifestations. For example, a pattern of building blocks might be represented as music, to create a certain melody, or might be represented as DNA to create a certain protein. Both manifestations share the same basic rules for connecting up their building blocks, but the actual building blocks are physically different. In both representations, we choose properties according to the criteria we want to achieve: "beautiful" music or "strong" protein, for example.
Using music as a tool to create better materials may seem like an unusual proposal, but when we appreciate that the underlying mathematics of the structure of music are shared across many fields of study, it begins to make sense. Nature does not distinguish between what is art and what is material, as all are merely patterns of structure in space and time.
This article appeared in print under the headline "More materials, Maestro"
Markus J. Buehler is a materials scientist and heads the department of civil and environmental engineering at the Massachusetts Institute of Technology. His research focuses on the science of natural and synthetic materials, and he asks fundamental questions about how properties of matter derive from their elementary structures
February 6, 2014
In the End, It All Adds Up to – 1/12
This is what happens when you mess with infinity.
You might think that if you simply started adding the natural numbers, 1 plus 2 plus 3 and so on all the way to infinity, you would get a pretty big number. At least I always did.
So it came as a shock to a lot of people when, in a recent video, a pair of physicists purported to prove that this infinite series actually adds up to ...minus 1/12.
To date some 1.5 million people have viewed this calculation, which plays a key role in modern physics and quantum theory; the answer, as absurd as it sounds, has been verified to many decimal places in lab experiments. After watching the video myself, I checked to make sure I still had my wallet and my watch.
Even the makers of the video, Brady Haran, a journalist, and Ed Copeland and Antonio Padilla, physicists at the University of Nottingham in England, admit there is a certain amount of “hocus-pocus,” or what some mathematicians have called dirty tricks, in their presentation. Which has led to some online grumbling.
So what’s going on with infinity?
“This calculation is one of the best-kept secrets in math,” said Edward Frenkel, a mathematics professor at the University of California, Berkeley, and author of “Love and Math: The Heart of Hidden Reality,” (Basic Books, 2013), who was in town recently promoting his book and acting as an ambassador for better math education. “No one on the outside knows about it.”
The great 18th-century mathematician Leonhard Euler, who was born in Switzerland but did most of his work in Berlin and St. Petersburg, Russia, was the first one down this road. Euler wanted to know if you could find an answer to endless sums of numbers like 1 plus 1/2 plus 1/3 plus 1/4 on up to infinity, or the squares of those fractions..
These are all different versions of what has become known as the Riemann zeta function, after Bernhard Riemann, who came along about a century after Euler. The zeta function is one of the more mysterious and celebrated subjects in mathematics, important in the theory of prime numbers, among other things. It was one of the plot threads, for example, in Thomas Pynchon’s 2006 novel, “Against the Day.”
In 1749, Euler used a bag of mathematical tricks to solve the problem of adding the natural numbers from 1 to infinity, a so-called divergent series because the terms keep growing without limit as you go along. Clearly, if you stop adding anywhere along the way — at a quintillion (1 with 18 zeros after it), say, or a googolplex (10^100 zeros ) — the sum will be enormous. The problem with infinity is that you can’t stop. You never get there. It’s more of a journey than a destination. As Dr. Padilla says to Mr. Haran at the end of their video, “You have to face infinity, Brady.”
The method in the video is essentially the same as Euler’s. It involves nothing more complicated than addition and subtraction (although the things being added and subtracted were more infinite series) and a small piece of algebra that my sixth-grade daughter would breeze through.
You are not alone in wondering how this can make sense. The Norwegian mathematician Niels Henrik Abel, whose notion of an Abel sum plays a role here, once wrote, “The divergent series are the invention of the devil, and it is a shame to base on them any demonstration whatsoever.”
In modern terms, Dr. Frenkel explained, the gist of the calculations can be interpreted as saying that the infinite sum has three separate parts: one of which blows up when you go to infinity, one of which goes to zero, and minus 1/12. The infinite term, he said, just gets thrown away.
And it works. A hundred years later, Riemann used a more advanced and rigorous method, involving imaginary as well as real numbers, to calculate the zeta function and got the same answer: minus 1/12.
“So Euler guessed it right,” Dr. Frenkel said.
Those of us who are not mathematicians probably wouldn’t care so much about infinity except that it crops up again and again in calculations of things, like the energy of the electron, that we know are finite, or in string theory, which physicists would like to hope is finite.
In this case, our current understanding of the very solidity of reality depends on coming up with a consistent way to assign values to infinite sums.
In the process known as regularization, which is a part of many calculations in quantum theory, physicists do something similar to what Euler did, arriving at a real number that corresponds to the quantity they want to know and an infinite term, which they throw away. The process works so well that theoretical predictions in quantum electrodynamics, the fancy version of the familiar force of electromagnetism, agree with experiments to a precision of one part in a trillion.
Which is remarkable given that infinite quantities have been thrown away, or “swept under the rug,” in the words of the California Institute of Technology physicist Richard Feynman, who helped invent a lot of this stuff but thought it was more than faintly scandalous.
Likewise, it is no surprise that the factor 1/12 shows up a lot in string theory equations, Dr. Frenkel said. Why it all works is still a mystery.
“Quantum physics needs its own Riemann to come and give a rigorous explanation of these mysteries,” Dr. Frenkel said.
To him and others, this is just another example of what the eminent physicist Eugene Wigner called the “unreasonable effectiveness of mathematics.” Why should such woolly and abstract concepts as zeta functions or imaginary numbers, the products of a chess game in our minds, have such relevance in describing the world?
Riemann’s explorations of the geometry of curved spaces in 1854 laid the foundation for Einstein’s theory of gravity, general relativity, half a century later.
There were mathematicians and philosophers who were ready to jump out the window later in the 1800s when Georg Cantor, a Russian-born mathematician, set out to classify the kinds of infinity. In a speech in 1908, the French mathematician Henri Poincaré compared “Cantorism,” as he called it, to a disease.
Mathematicians today agree that there is an infinite number of natural numbers (1, 2, 3 and so on) on the bottom rung of infinity. Above that, however, is another rung of so-called real numbers, which is bigger in the sense that there is an uncountable number of them for every natural number. And so it goes.
Cosmologists do not know if the universe is physically infinite in either space or time, or what it means if it is or isn’t. Or if these are even sensible questions. They don’t know whether someday they will find that higher orders of infinity are unreasonably effective in understanding existence, whatever that is.
Here is where we sprain our imaginations, and perhaps check to see that we still have our wallets.
February 6, 2014
Anand Kumar, founder of Super30 classes, gets Ramanujan Mathematics award
Mathematician and founder of Super30 classes Anand Kumar (file pic)
Updated: January 28, 2014 20:08 IST
Rajkot: Leading mathematician and founder of Super30 classes Anand Kumar was conferred this year's prestigious Ramanujan Mathematics Award at the Eighth National Mathematics Convention organised by a Rajkot-based school.
Scientists Pitamber Patel and JJ Rawal presented the cash award of Rs. 20,000 to Mr Kumar. The annual award is given to a person who has at least three papers published in international journals and is involved in teaching mathematics.
The 41-year-old mathematician from Patna is better known as the founder of Super30 - an institute that offers free residential mentoring on how to crack the entrance test for IITs to 30 students selected from underprivileged families each year..
"I am planning to write a mathematics book to project the subject in an interesting way and help students learn the subject easily," Mr Kumar said after receiving the award.
Mr Kumar's papers have been published in a number of international journals including Mathematical Spectrum and Mathematical Gazette of the UK. He has been invited many times by the Mathematical Association of America and the American Mathematical Society to present his papers.
February 6, 2014
Retirement a new equation for USQ mathematician
Newly retired USQ Fraser Coast mathematician John Murray
PLAYING golf or bowls is not part of John Murray's new equation.
Instead, the now retired University of Southern Queensland Fraser Coast mathematician says, tongue-in-cheek, that he plans to solve global warming this semester.
"Although, there is a lot of cricket on TV at the moment so there will be some delays," Mr Murray said on Tuesday, the first day of his latest retirement.
Mr Murray, 74, has made several attempts to retire in recent years but his passion for teaching keeps drawing him back into classrooms.
"I'll take a few weeks' break now then look for some voluntary tutoring work, perhaps with apprentices or in secondary schools," he said.
"I ought to be using my skills, not playing golf or bowls."
Mr Murray has devoted most of his life to teaching students of all ages and was honoured with the dedication of a seat in his name at USQ Fraser Coast in June 2012.
The seat is located at the entrance to C Block.
Riding his bicycle to work most days, he tutored mathematics, engineering and education students.
He also made weekly visits to the Maryborough Correctional Centre to tutor prisoners who were studying courses through USQ.
Mr Murray's career began in electronics in the United Kingdom where he worked as a physicist building parts for missile guidance systems.
"It was a lonely life so I gave it away to become a teacher," he said.
"When Australia put out a call for teachers in the early 1970s, I answered and moved across the globe.
"My first Australian posting was at the tiny Queensland town of Kilcoy.
"I then moved to Hervey Bay to teach at the Wide Bay TAFE Senior College and joined the staff at USQ in 2000."
Six years later he received an Australia Day Award for his work as a lecturer and teacher.
Mr Murray also tutored students who had endured troubled upbringings.
He helped them believe they could do anything to which they set their minds.
Mr Murray especially encouraged women to believe they were capable of entering the traditionally male-dominated fields of mathematics and science.
"What's really important is the students have confidence in themselves," he said.
"I just give them confidence and they do the rest.
"The big thing for me is when they say: 'I can do it.'
"At the end of the time they actually like algebra and I like teaching them."
February 6, 2014
Examining the Square Root of D’oh!
The episode "Bart the Genius" included a calculus problem.
JAN. 27, 2014
Did you know that Homer Simpson disproved Fermat’s last theorem? He did, or so it seemed, when he scribbled “398712+436512=447212” on a blackboard in a 1998 episode of “The Simpsons.”
If Homer is right, then he has proved that the great 17th-century mathematician Pierre de Fermat was wrong in stating that the equation xn+yn=zn has no solution when x, y and z are positive whole numbers and n is a whole number greater than 2.
(That would also mean the British mathematician Andrew Wiles was wrong when he finally proved the theorem in 1995. Fortunately, he was not; a fuller explanation can be found at Wordplay, The Times’s crossword blog.)
Nor is Homer the only mathematical prodigy on “The Simpsons”: In a 1993 episode we learn that Apu Nahasapeemapetilon, proprietor of the Kwik-E-Mart in Springfield, has memorized pi to the 40,000th digit, which he correctly informs us is 1; and in 2010, Lisa becomes nearly unbeatable as a baseball manager by mastering complex statistics.
All this may come as a surprise to most “Simpsons” fans, who presumably do not tune in to ponder the mysteries of higher mathematics. But as the British author Simon Singh shows in “The Simpsons and Their Mathematical Secrets,” math is everywhere in the Simpsons’ world, from references that flash across the screen in an eye blink (such as Springfield’s Googolplex movie theater) to entire segments that explore deep mathematical concepts (like “Homer3” in 1995). Math is built into the show’s DNA.
Not content merely to point out the mathematical references, Mr. Singh uses them as a starting point for lively discussions of mathematical topics, anecdotes and history. Even someone with no mathematical background will enjoy his accounts of the nature of infinity and the meaning of the number e, the life of the tragic genius Srinivasa Ramanujan and the obsessions of Bill James, the oracle of baseball statistics.
Perhaps the most surprising revelation is the composition of “The Simpsons” creative team, which over the years has included J. Stewart Burns, who has a master’s degree in math from Harvard; David X. Cohen (master’s in computer science, University of California, Berkeley); and Ken Keeler (Ph.D. in applied math, Harvard). Most astonishing is Jeff Westbrook (Ph.D., computer science, Princeton), who was an associate professor at Yale before he joined the team.
It is understandable that a group with such credentials would enjoy sprinkling the show with tidbits from their previous line of work. As Mr. Cohen explains, “It cancels out those days when I’ve been writing those bodily function jokes.”
This might explain the prevalence of mathematics in the show, but it also cries out for an explanation: Why would so many talented mathematicians forsake academia to write outrageous stories and gags for an animated TV show? After all, the contrast between the elegant abstractions of higher mathematics and the foibles of the imbecilic Homer Simpson could hardly be greater.
But perhaps the similarities are closer than it appears. Think of it this way: To write an episode of “The Simpsons,” one begins with a known set of characters — Homer, Bart, Lisa, Marge — and confronts them with a problem. The rest of the episode follows the characters through a complicated series of moves until the problem is resolved.
And while the show certainly allows for a wide range of improbable turns (Homer disappears into the third dimension, Lisa is rescued from an angry mob by Stephen Hawking), not everything is allowable: The characters must remain true to their personalities and the stories must follow their own inner logic, for a story free of the constraints of personality, logic and motivation is no story at all.
Now think of proving a geometric theorem: Once again, one has a certain set of elements — points, lines, triangles, circles — and is confronted with a problem. What is the sum of the angles of a triangle? What is the area of a polygon? The proof then consists of a series of moves that leads to a resolution of the problem.
Just as in a story, there are constraints. All geometric objects must remain true to their unique characteristics, and each step in the proof must follow the strict rules of logical deduction. A proof that breaks these conditions is no proof at all.
In their basic structure, then, mathematical reasoning and storytelling might not be so different after all. In a series like “The Simpsons,” the similarities are particularly strong. In the end, it turns out that the improbable world of Springfield is in some ways not that different from the equally surprising world of higher mathematics.
This seems to me to go a long way in explaining why so many mathematicians found a welcome home on “The Simpsons” writing team, and why they have left such a strong imprint on the show. It also points to the deeper implication of Mr. Singh’s highly entertaining book: that every mathematical theory or theorem, however difficult and inaccessible it may seem, always has a story to tell.
Amir Alexander’s next book, “Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World,” will be published in April.
February 6, 2014
Heini Halberstam, 1926-2014
As head of the mathematics depart at the University of Illinois at Urbana-Champaign,
Heini Halberstam was generous in support of his students.
(Halberstam family photo, Halberstam family photo)
His mother's foresight saved mathematician Heini Halberstam from the Holocaust and brought him to England, where chance led to a career in number theory and a focus on the puzzle of prime numbers, whole numbers like seven and 13 that can be divided only by one and themselves.
Mr. Halberstam, 87, died of congestive heart failure Saturday, Jan. 25, in his Champaign home, according to his daughter Judith.
"He was a distinguished researcher," said Harold Diamond, a former colleague at the University of Illinois at Urbana-Champaign, where Mr. Halberstam headed the mathematics department from 1980 until 1988, and retired as emeritus professor in 1996.
"It's an intellectual and academic pursuit," Diamond said of the work he, Mr. Halberstam and others did in the field of number theory.
But this theoretical work has some real-world applications in fields like cryptography, for example, Diamond said. Prime numbers are an important ingredient in an encryption protocol called public key cryptography, used around the world for secure communications in some financial and defense-related applications.
Mr. Halberstam, an only child, spent his early years in Most, a small town in an area called the Sudetenland that is now part of the Czech Republic. His father, a rabbi, died when he was 10 and he moved with his mother to Prague.
His mother recognized the rising danger of anti-Semitism and sent him to England on one of the Kindertransport trains that carried thousands of mostly Jewish children to the relative safety of the United Kingdom. None were accompanied by parents.
Mr. Halberstam was 12 when he arrived in England. His mother died three years later of typhus in a Nazi work camp.
Although he had relatives in England, it was the foster parent who took him in and with whom he stayed for the duration of the war, Anne Welsford, who recognized his intellectual abilities. She encouraged him to go to university and paid for the portion of his education that wasn't covered by scholarships.
He studied number theory at the University of London, getting a doctorate there in 1952, according to Diamond.
"His research was in a field his adviser was working on and it seemed like an interesting area," Diamond said. "When he took it up, number theory was a field of pure study done by academics, just pure mathematics with no obvious applications."
After two years at Trinity College Dublin, Mr. Halberstam moved to the mathematics department of the University of Nottingham in 1964. His time there included two stints as head of the department and one period as dean of the faculty.
He left Nottingham in 1980 to come to the U. of I. as head of the mathematics department.
"The school was known for number theory, and it was a really good career move," his daughter said.
Mr. Halberstam relished the intellectual challenge of studying prime numbers. "There are many, many unsolved problems. It's a premier area of study in theoretical mathematics," his daughter said.
Diamond, who collaborated with Mr. Halberstam on a book as well as a number of articles, said his colleague's specialty was what is called sieve theory, a way of attacking some problems involving prime numbers.
In 1974, with another colleague, Mr. Halberstam published "Sieve Methods," which remains a major book in the field. He was a fellow of the American Mathematical Society and of University College, London.
In addition to his research, Mr. Halberstam was an effective teacher, generous in his support for students who met his high standards, according to his son Michael.
"My father made a point of helping students he believed in and helping them to advance in securing positions and jobs and getting their papers published," his son said.
In a short memoir he wrote for his children and grandchildren, Mr. Halberstam reflected on the strange twists and turns his life had taken, recalling a quote from a Vladimir Nabokov novel about the "mysterious thing, this branching structure of life" being as mysterious as the prime numbers he studied.
Mr. Halberstam's first wife, Heather, died in an auto accident in 1971. His second wife, Doreen, survives him; as do three other daughters, Lucy, Naomi Strachan and Jean Smith; another son, John Smith; and eight grandchildren.
Plans for a memorial service are pending.
February 6, 2014
Paul J. Sally, mathematics professor at U. of C., 1933-2013
January 16, 2014|By Bravetta Hassell, Special to the Tribune
To some students, University of Chicago professor Paul J. Sally Jr. was "mathematics, personified," said Shmuel Weinberger, chairman of the school's mathematics department.
Known for his contributions to the field of harmonic analysis, Mr. Sally had taught at the U. of C. since 1965. He was chairman of the math department from 1977 to 1980 and since 1984 had been the department's director of undergraduate studies.
Mr. Sally, 80, died of congestive heart failure Monday, Dec. 30, at the U. of C. Hospital, his family said.
During his years at the U. of. C., Mr. Sally oversaw the work of 19 Ph.D. students. He was also the first director of the curriculum effort called the University of Chicago School of Mathematics Project and in 1992 founded Seminars for Elementary Specialists and Mathematics Educators, a program for teachers in Chicago Public Schools.
Diane Herrmann, co-director of undergraduate studies for the U. of C.'s math department, worked with Mr. Sally for nearly 40 years and said his colleague was always open to trying new things, He was supportive when she expressed a desire give women and minorities more opportunities to do math on the high school level and in 1988 joined her in starting the Young Scholars Program for high school students who show proficiency in math.
Mr. Sally was an energetic lecturer and a patient teacher who knew just how much information to give students before sending them off to solve a problem, Herrmann said. His approach toward undergraduate teaching provided a model for many graduate students and colleagues, she said.
Mr. Sally was born in Boston, one of four children. He received bachelor's and master's degrees in math from Boston College and a doctorate in the field from Brandeis University in Waltham, Mass. He was an instructor at Washington University in St. Louis before coming to the U. of C.
Mr. Sally was diagnosed with Type 1 diabetes as a teenager and dealt with complications associated with the disease throughout his life, his family said. Over time, he lost both of his legs and sight in one eye. But it didn't slow him down much, colleagues and family said.
Mr. Sally "was incredibly demanding of everyone, but from a guy like that, everyone understood he wasn't any less demanding of himself," Weinberger said.
He handled his disabilities with grace and good humor. Because of the eye patch he wore, he was nicknamed "The Math Pirate" and "Professor Pirate."
In the classroom, he was not above grand displays when he felt his students were less than focused — he is rumored to have on occasion thrown a distracted student's cellphone out the window. True or not, David Sally, the eldest of Mr. Sally's three sons, said his father loved people thinking he was capable of such an act.
"He was one who believed in the importance of mathematics," Weinberger said. "And he believed in people's abilities to accomplish enormous things if only they would try hard enough."
"So he didn't take excuses because he felt that everyone had the power to make superhuman efforts and that they would pay off."
At home, David Sally said there was a rhythm to the way his father did math. After the last newscast of the evening, Mr. Sally would set up in the study, with a blackboard or pencil and pen and paper, and work on his research into the early morning. The discussions he'd have often with his wife, Judith, herself once a research mathematician, had a "beautiful purity" to them, David Sally said.
Weinberger said Mr. Sally was dedicated to teaching and mathematics.
"He would be lecturing in the evenings. He would be lecturing on weekends — and he had all of these programs," Weinberger said. "Whenever the university wasn't scheduling classes, it seemed that he had arranged for some outreach programs to bring in some community."
Mr. Sally won the U. of C.'s Quantrell Award for Excellence in Undergraduate Teaching in 1967 and the Provost's Teaching Award in 2005.
Mr. Sally was notified last month that he was the first recipient of the American Mathematical Society's Award for Impact on the Teaching and Learning of Mathematics, Weinberger said. The award was to be announced in April.
Survivors include his wife; two other sons, Stephen and Paul III; a sister, Elizabeth Massey; a brother, Frank; and eight grandchildren.
Services were held.