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This story of discovery begins a century ago with Albert Einstein, who realized that space is not an immutable stage on which events play out, as Isaac Newton had envisioned. Instead, through his general theory of relativity, Einstein found that space, and time too, can bend, twist and warp, responding much as a trampoline does to a jumping child. In fact, so malleable is space that, according to the math, the size of the universe necessarily changes over time: the fabric of space must expand or contract — it can’t stay put. For Einstein, this was an unacceptable conclusion. He’d spent 10 grueling years developing the general theory of relativity, seeking a better understanding of gravity, but to him the notion of an expanding or contracting cosmos seemed blatantly erroneous. It flew in the face of the prevailing wisdom that, over the largest of scales, the universe was fixed and unchanging. Einstein responded swiftly. He modified the equations of general relativity so that the mathematics would yield an unchanging cosmos. A static situation, like a stalemate in a tug of war, requires equal but opposite forces that cancel each other. Across large distances, the force that shapes the cosmos is the attractive pull of gravity. And so, Einstein reasoned, a counterbalancing force would need to provide a repulsive push. But what force could that be? Remarkably, he found that a simple modification of general relativity’s equations entailed something that would have, well, blown Newton’s mind: antigravity — a gravitational force that pushes instead of pulls. Ordinary matter, like the Earth or Sun, can generate only attractive gravity, but the math revealed that a more exotic source — an energy that uniformly fills space, much as steam fills a sauna, only invisibly — would generate gravity’s repulsive version. Einstein called this space-filling energy the cosmological constant, and he found that by finely adjusting its value, the repulsive gravity it produced would precisely cancel the usual attractive gravity coming from stars and galaxies, yielding a static cosmos

There are three formal AIMS undertakings: a master’s degree program in Mathematical Sciences, research, and teacher training. The master’s program offers free tuition to accepted students and trains them in both general principles – problem formulation, the scientific method, communication – and cutting-edge math in subjects including computer science, biomathematics, and financial mathematics. Research will allow for international collaborations and advanced student training.

Traditionally, classrooms were led by an authoritative teacher who disseminated information to silent students, but Zomahoun hopes to turn that paradigm on its head. “We train people who can challenge the status quo,” he explains, “not just people who learn from books, listen to lectures, and just repeat it.” Rather, he hopes to instill qualities like “critical thinking, independent thinking, and problem solving” in order to prepare students for real-world problems.

Corrypt, upon proper exploration, revealed itself to be a brilliantly designed puzzle game, unforgiving and unwilling to accommodate players who refuse to give it their full attention. Peel back one layer, and it reveals another more surprising one.

After completing a degree in math and computer science at the University of Auckland, Brough moved to London and began working towards a Ph.D. He landed a decent-paying programming job while continuing his scholastic work, but continued making games, including the beautiful, abstract strategy game Vertex Dispenser, which even Brough admits may have been too esoteric. It combined elements of shooters and real-time strategy games with a complex puzzle system, and many players felt overwhelmed. “I just could not get my head around those concepts at the same time,” said one.

Well-designed games, he believes, can teach people how to do some things better. By simulating challenging situations, games can teach us about “managing unexpected situations… making good decisions, thinking about the costs of our actions and dealing with the consequences,” he says.

He studied math and physics at Yale, got a masters in physics and was working on his Ph.D. at UC Berkeley. Then he realized his real love was music, and his Ph.D. turned into the study of music perception and cognition.

In the 38 years since the Hungarian architecture professor Erno Rubik invented his cube, it has alternately been regarded as an object of fun, art, mathematics, nostalgia and frustration

“You can use Rubik’s Cube to teach engineering, you can use it to teach mathematics, and you can use it to talk about the interplay between design and engineering and mathematics and creativity,”

Nor is it clear that the math we learn in the classroom has any relation to the quantitative reasoning we need on the job. John P. Smith III, an educational psychologist at Michigan State University who has studied math education, has found that “mathematical reasoning in workplaces differs markedly from the algorithms taught in school.”

It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar. Demanding algebra across the board actually skews a student body, not necessarily for the better.

Instead of investing so much of our academic energy in a subject that blocks further attainment for much of our population, I propose that we start thinking about alternatives. Thus mathematics teachers at every level could create exciting courses in what I call “citizen statistics.”

The impact of data abundance extends well beyond business. Justin Grimmer, for example, is one of the new breed of political scientists. A 28-year-old assistant professor at Stanford, he combined math with political science in his undergraduate and graduate studies, seeing “an opportunity because the discipline is becoming increasingly data-intensive.” His research involves the computer-automated analysis of blog postings, Congressional speeches and press releases, and news articles, looking for insights into how political ideas spread.

But the computer tools for gleaning knowledge and insights from the Internet era’s vast trove of unstructured data are fast gaining ground. At the forefront are the rapidly advancing techniques of artificial intelligence like natural-language processing, pattern recognition and machine learning.

“We’re trying to take kids who haven’t been engaged in school and hook them to an expanded vision of what their future might be,” he said. When they return to their own schools, the hope is that they’ll be more interested in history, math and English — and have a sense of environmental stewardship as well.

“The most important thing really is teaching kids through hands on, experiential learning,” said Rees. “Our kids do this because they’re inspired to be there every week, to work with adults and do hands on things.”

non-­scientist gamers developed more-­complex proteins than biochemists did

As harder concepts are introduced, students who need more time on a level get additional problems; those who understand it move on.

93 percent of K–12 students successfully mastered concepts after only 90 minutes of gameplay, and they didn’t want to stop

This approach can even be useful for applications that are not, strictly speaking, compressed sensing problems, such as the Netflix prize.

Given the enormous popularity of Netflix, even an incremental improvement in the predictive algorithm results in a substantial boost to the company’s bottom line. Recht found that he could accurately predict which movies customers might be interested in purchasing, provided he saw enough products per person. Between 25 and 100 products were sufficient to complete the matrix.

Across every discipline, data sets are getting bigger and more complex, whether one is dealing with medical records, genomic sequencing, neural networks in the brain, astrophysics, historical archives, or social networks. Alessandro Vespignani, a physicist at Northeastern University who specializes in harnessing the power of social networking to model disease outbreaks, stock market behavior, collective social dynamics, and election outcomes, has collected many terabytes of data from social networks such as Twitter, nearly all of it raw and unstructured. “We didn’t define the conditions of the experiments, so we don’t know what we are capturing,” he said.

Then, in November, she posted on YouTube a video about doodling in math class, which married a distaste for the way math is taught in school with an exuberant exploration of math as art .

At first glance, Ms. Hart’s fascination with mathematics might seem odd and unexpected. She graduated with a degree in music, and she never took a math course in college.

The ensuing attention has come with job offers and an income. In one week in December, she earned $300 off the advertising revenue that YouTube shares with video creators. She is also happy that, unlike in her early efforts, which drew an audience typical of mathematics research — older and male, mostly — the biggest demographic for her new videos, at least among registered users, are teenage girls.