Posts Tagged ‘topology’
“Topology is precisely the mathematical discipline that allows the passage from local to global”*…
Jordana Cepelewicz on two new topographical results that bring some order to the confoundingly difficult study of four-dimensional shapes…
The central objects of study in topology are spaces called manifolds, which look flat when you zoom in on them. The surface of a sphere, for instance, is a two-dimensional manifold. Topologists understand such two-dimensional manifolds very well. And they have developed tools that let them make sense of three-dimensional manifolds and those with five or more dimensions.
But in four dimensions, “everything goes a bit crazy,” said Sam Hughes, a postdoctoral researcher at the University of Oxford. Tools stop working; exotic behavior emerges. As Tom Mrowka of the Massachusetts Institute of Technology explained, “There’s just enough room to have interesting phenomena, but not so much room that they fall apart.”
In the early 1990s, Mrowka and Peter Kronheimer of Harvard University were studying how two-dimensional surfaces can be embedded within four-dimensional manifolds. They developed new techniques to characterize these surfaces, allowing them to gain crucial insights into the otherwise inaccessible structure of four-dimensional manifolds. Their findings suggested that the members of a broad class of surfaces all slice through their parent manifold in a relatively simple way, leaving a fundamental property unchanged. But nobody could prove this was always true.
In February, together with Daniel Ruberman of Brandeis University, Hughes constructed a sequence of counterexamples — “crazy” two-dimensional surfaces that dissect their parent manifolds in ways that mathematicians had believed to be impossible. The counterexamples show that four-dimensional manifolds are even more remarkably diverse than mathematicians in earlier decades had realized. “It’s really a beautiful paper,” Mrowka said. “I just keep looking at it. There’s lots of delicious little things there.”
Late last year, Ruberman helped organize a conference that created a new list of the most significant open problems in low-dimensional topology. In preparing for it, he looked at a previous list of important unsolved topological problems from 1997. It included a question that Kronheimer had posed based on his work with Mrowka. “It was in there, and I think it was a little bit forgotten,” Ruberman said. Now he thought he could answer it…
Read on for the details: “Mathematicians Marvel at ‘Crazy’ Cuts Through Four Dimensions,” from @jordanacep in @QuantaMagazine.
* Rene Thom
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As we savor surprising shapes, we might send carefully-modeled birthday greetings to William Bowie; he was born on this date in 1872. A geodetic engineer who joined the United States Coast and Geodetic Survey in 1895, he investigated isostasy (a principle that dense crustal rocks to tend cause topographic depressions and light crustal rocks cause topographic elevations).
Bowie was the first President of the American Geophysical Union from 1920 to 1922 and served as president a second time from 1929 to 1932. The William Bowie Medal, the highest honor of the AGU, is named in his honor.
“Everybody wants to build and nobody wants to do maintenance”*…
The most unappreciated and undervalued forms of technological labour are also the most ordinary: those who repair and maintain technologies that already exist, that were ‘innovated’ long ago. This shift in emphasis involves focusing on the constant processes of entropy and un-doing – which the media scholar Steven Jackson calls ‘broken world thinking’ – and the work we do to slow or halt them, rather than on the introduction of novel things…
We can think of labour that goes into maintenance and repair as the work of the maintainers, those individuals whose work keeps ordinary existence going rather than introducing novel things. Brief reflection demonstrates that the vast majority of human labour, from laundry and trash removal to janitorial work and food preparation, is of this type: upkeep. This realisation has significant implications for gender relations in and around technology. Feminist theorists have long argued that obsessions with technological novelty obscures all of the labour, including housework, that women, disproportionately, do to keep life on track. Domestic labour has huge financial ramifications but largely falls outside economic accounting, like Gross Domestic Product. In her classic 1983 book, More Work for Mother, Ruth Schwartz Cowan examined home technologies – such as washing machines and vacuum cleaners – and how they fit into women’s ceaseless labour of domestic upkeep. One of her more famous findings was that new housekeeping technologies, which promised to save labour, literally created more work for mother as cleanliness standards rose, leaving women perpetually unable to keep up.
Nixon, wrong about so many things, also was wrong to point to household appliances as self-evident indicators of American progress. Ironically, Cowan’s work first met with scepticism among male scholars working in the history of technology, whose focus was a male pantheon of inventors: Bell, Morse, Edison, Tesla, Diesel, Shockley, and so on. A renewed focus on maintenance and repair also has implications beyond the gender politics that More Work for Mother brought to light. When they set innovation-obsession to the side, scholars can confront various kinds of low-wage labour performed by many African-Americans, Latinos, and other racial and ethnic minorities. From this perspective, recent struggles over increasing the minimum wage, including for fast food workers, can be seen as arguments for the dignity of being a maintainer…
Entire societies have come to talk about innovation as if it were an inherently desirable value, like love, fraternity, courage, beauty, dignity, or responsibility. Innovation-speak worships at the altar of change, but it rarely asks who benefits, to what end? A focus on maintenance provides opportunities to ask questions about what we really want out of technologies. What do we really care about? What kind of society do we want to live in? Will this help get us there? We must shift from means, including the technologies that underpin our everyday actions, to ends, including the many kinds of social beneficence and improvement that technology can offer. Our increasingly unequal and fearful world would be grateful…
Capitalism excels at innovation but is failing at maintenance, and for most lives it is maintenance that matters more: “Hail the maintainers.”
[image above: source]
* Kurt Vonnegut
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As we invest in infrastructure, we might send carefully-calculated birthday greetings to Jules Henri Poincaré; he was born on this date in 1854. A mathematician, theoretical physicist, engineer, and a philosopher of science, Poincaré is considered the “last Universalist” in math– the last mathematician to excel in all fields of the discipline as it existed during his lifetime.
Poincaré was a co-discoverer (with Einstein and Lorentz) of the special theory of relativity; he laid the foundations for the fields of topology and chaos theory; and he had a huge impact on cosmogony. His famous “Conjecture” held that if any loop in a given three-dimensional space can be shrunk to a point, the space is equivalent to a sphere; it remained unsolved until Grigori Perelman completed a proof in 2003.
And we might also send amusingly-phrased birthday greetings to Ludwig Josef Johann Wittgenstein; the philospher of logic, math, language, and the mind was born on this date in 1889.
“All you really need to know for the moment is that the universe is a lot more complicated than you might think, even if you start from a position of thinking it’s pretty damn complicated in the first place”*…
When you gaze out at the night sky, space seems to extend forever in all directions. That’s our mental model for the universe, but it’s not necessarily correct. There was a time, after all, when everyone thought the Earth was flat, because our planet’s curvature was too subtle to detect and a spherical Earth was unfathomable.
Today, we know the Earth is shaped like a sphere. But most of us give little thought to the shape of the universe. Just as the sphere offered an alternative to a flat Earth, other three-dimensional shapes offer alternatives to “ordinary” infinite space.
We can ask two separate but interrelated questions about the shape of the universe. One is about its geometry: the fine-grained local measurements of things like angles and areas. The other is about its topology: how these local pieces are stitched together into an overarching shape.
Cosmological evidence suggests that the part of the universe we can see is smooth and homogeneous, at least approximately. The local fabric of space looks much the same at every point and in every direction. Only three geometries fit this description: flat, spherical and hyperbolic…
Alternatives to “ordinary” infinite space: “What Is the Geometry of the Universe?”
* Douglas Adams, The Hitchhiker’s Guide to the Galaxy
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As we tinker with topology, we might recall that it was on this date in 1811 that Percy Bysshe Shelley was expelled from the University of Oxford for publishing the pamphlet The Necessity of Atheism. Shelley, of course, went on to become a celebrated lyric poet and one of the leaders of the English Romantic movement… one who had a confident (if not to say exalted) sense of his role in society:
Poets are the hierophants of an unapprehended inspiration; the mirrors of the gigantic shadows which futurity casts upon the present; the words which express what they understand not; the trumpets which sing to battle, and feel not what they inspire; the influence which is moved not, but moves. Poets are the unacknowledged legislators of the world.
“Reality is merely an illusion, albeit a very persistent one”*…
To look for the strange wave-like properties of quantum particles, physicists hurtle them through a long tunnel-like instrument known as an interferometer
Magnify a speck of dirt a thousand times, and suddenly it no longer seems to play by the same rules. Its outline, for example, won’t look well-defined most of the time and will resemble a diffuse, sprawling cloud. That’s the bizarre realm of quantum mechanics. “In some books, you’ll find they say a particle is in various places at once,” says physicist Markus Arndt of the University of Vienna in Austria. “Whether that really happens is a matter of interpretation.”
Another way of putting it: Quantum particles sometimes act like waves, spread out in space. They can slosh into each other and even back onto themselves. But if you poke at this wave-like object with certain instruments, or if the object interacts in specific ways with nearby particles, it loses its wavelike properties and starts acting like a discrete point—a particle. Physicists have observed atoms, electrons, and other minutiae transitioning between wave-like and particle-like states for decades.
But at what size do quantum effects no longer apply? How big can something be and still behave like both a particle and a wave? Physicists have struggled to answer that question because the experiments have been nearly impossible to design.
Now, Arndt and his team have circumvented those challenges and observed quantum wave-like properties in the largest objects to date—molecules composed of 2,000 atoms, the size of some proteins. The size of these molecules beats the previous record by two and a half times. To see this, they injected the molecules into a 5-meter-long tube. When the particles hit a target at the end, they didn’t just land as randomly scattered points. Instead, they formed an interference pattern, a striped pattern of dark and light stripes that suggests waves colliding and combining with each other…
One possibility physicists are exploring is that quantum mechanics might in fact apply at all scales. “You and I, while we sit and talk, do not feel quantum,” says Arndt. We seem to have distinct outlines and do not crash and combine with each other like waves in a pond. “The question is, why does the world look so normal when quantum mechanics is so weird?”…
A record-breaking experiment shows an enormous molecule is also both a particle and a wave—and that quantum effects don’t only apply at tiny scales: “Even Huge Molecules Follow the Quantum World’s Bizarre Rules.”
Read the paper published in Nature Physics by Arndt and his team here.
* Albert Einstein
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As we dwell on duality, we might spare a thought for August Ferdinand Möbius; he died on this date in 1868. A German mathematician and theoretical astronomer, he is best remembered as a topologist, more specifically for his discovery of the Möbius strip (a two-dimensional surface with only one side… or more precisely, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space).
“The cyclical rebirth of caste in America is a recurring racial nightmare”*…
Dorothy and Lillian Gish and D.W. Griffith at the White House, 1922. Library of Congress
The year 1915 marked the fiftieth anniversary of the end of the Civil War. Monuments to Confederate and Union heroes were being dedicated all over the country. Woodrow Wilson, a fan of Jim Crow laws, was president. He had allowed federal workplaces to segregate again.
Enter Thomas Dixon Jr., Wilson’s classmate from Johns Hopkins. A film had just been made of Dixon’s second novel, “the true story” of the South under Reconstruction. Would the president, he wondered, be interested in viewing it? (He would.)
“History written with lightning,” Wilson declared of The Clansman, the second film ever to be screened in the White House. It was an endorsement guaranteed to head off resistance from town censor boards charged with shutting down entertainment deemed unsuitable or incendiary to the public…
The Clansman was a silent movie with title cards. It depicted whites as victims and blacks as villains. Benevolent former masters were denied votes and subjugated by newly freed blacks taking over the country. In an early scene, black legislators sit at desks, shoeless and drunk, too busy stuffing their faces with fried chicken to work. The title card read: “An historical facsimile of the State House of Representatives of South Carolina in 1870.” South Carolina had been the first state to elect a majority-black legislature and that the card implied that the apish behavior depicted was historically accurate, too.
In a later scene, the white heroine (played by Lillian Gish) is threatened by a black man unable to contain his urge to “mongrelize” the white race. Before she is ravaged, a savior army rides in: The Ku Klux Klan. The title-card copy comes straight from the president’s five-volume History of the American People, published in 1902:
More of this sad story, and its aftermath, at “Hatred Endorsed by a President.”
The New Jim Crow: Mass Incarceration in the Age of Colorblindness
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As we ruminate on recurrence, we might send never-ending birthday greetings to August Ferdinand Möbius; he was born on this date in 1790. A German mathematician and theoretical astronomer, he is best remembered as a topologist, more specifically for his discovery of the Möbius strip (a two-dimensional surface with only one side… or more precisely, a non-orientable two-dimensional surface with only one side when embedded in three-dimensional Euclidean space). See ““It might help to think of the universe as a rubber sheet, or perhaps not.”
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