Mathematician unleashes 'a wave of new results' in geometric analysis

Mathematician unleashes 'a wave of new results' in geometric analysis: "If you drive into a parking garage and go up a level, that spiral ramp is part of a helicoid," he says. "And one of the things we were able to show was that every embedded minimal surface could be built out of these things. So the minimal surface either looks like a nice flat thing where the area is bounded, or it looks exactly like one of these double spiral staircases..."

They have been able to prove, he says, that of the infinite number of singularities that could possibly affect a surface through this curvature flow, only two types are stable enough to survive in reality. "If you were to wiggle your surface ever so slightly, in fact only two are stable of the infinitely many that are possible," Minicozzi says. "So if you are trying to understand [mean curvature flow] and you have to deal with all of these cases, it's much better to have to deal with two cases than an infinite number."

Watch a singularity form in a stretchy soap film - physics-math - 28 May 2014 - New Scientist

Watch a singularity form in a stretchy soap film - physics-math - 28 May 2014 - New Scientist

Magnetic charge crystals imaged in artificial spin ice

Magnetic charge crystals imaged in artificial spin ice: In the honeycomb pattern, where three magnetic poles intersect, a net charge of north or south is forced at each vertex. The magnetic "monopole charge" at each vertex influences the magnetic "charge" of the surrounding vertices. The team was able to image the crystalline structure of the magnetic charges using magnetic force microscopy...

The research team's new annealing protocol—heating the material to a high temperature where their magnetic polarity is suppressed (here, about 550 degrees Celsius)—allows the nanomagnets to flip their polarity and freely interact. As the material cools, the nanomagnets are ordered according to the interactions of their poles at the vertices...

"This work demonstrates a direction in condensed matter physics that is quite opposite to what has been done in the last sixty years or so," said Nisoli. "Instead of imagining an emergent theoretical description to model the behavior of a nature-given material and validating it indirectly, we engineer materials of desired emergent properties that can be visualized directly."

Twisted Magnetic Fields Tie Information in a Knot: Scientific American

Twisted Magnetic Fields Tie Information in a Knot: Scientific American: Writing in Science, von Bergmann and her collaborators describe how they created skyrmions on a thin magnetic film of palladium and iron on an iridium crystal. They began with a sample in which all the atomic bar magnets were aligned. The team then used the tip of a scanning tunnelling microscope to apply a small current made up of electrons that had their spins aligned, or polarized, in a particular way. The polarized current interacted with the atomic bar magnets to twist them into knot-like configurations of skyrmions, each a few nanometers, or about 300 atoms, in diameter, says von Bergmann. The scientists could also use the polarized current to erase the knot, deleting the skyrmion...

...this is the first time that scientists have created and deleted individual magnetic skyrmions...

First fluid knots created in the lab

First fluid knots created in the lab: To investigate, Dustin Kleckner and William Irvine of the University of Chicago, Illinois 3D-printed strips of plastic shaped into a trefoil knot and a Hopf link. Crucially, the strips had a cross section shaped like a wing, or hydrofoil (see picture).

Next, the researchers dragged the knots through water filled with microscopic bubbles. Just as a wing passing through air creates a trailing vortex, the acceleration of the hydrofoils created a knot-shaped vortex that sucked in the bubbles. The result was a knot-shaped flow of moving bubbles – the first fluid knot created in a lab – which the team imaged with lasers.

Fairly Simple Math Could Bridge Quantum Mechanics and General Relativity

Fairly Simple Math Could Bridge Quantum Mechanics and General Relativity: The analysis does not model gravity explicitly, and so is not an attempt to formulate a theory of ‘quantum gravity’ that brings general relativity and quantum mechanics under one umbrella. Instead... their work might provide a simplified framework for understanding the effects of gravity on quantum particles, as well as describing other situations in which the spaces that quantum particles move in can radically alter, such as in condensed-matter-physics experiments...
Wilczek and his co-authors set up a hypothetical system with a single quantum particle moving along a wire that abruptly splits into two. The stripped-down scenario is effectively the one-dimensional version of an encounter with ripped space-time, which occurs when the topology of a space changes radically. The theorists concentrate on what happens at the endpoints of the wire — setting the ‘boundary conditions’ for the before and after states of the quantum wave associated with the particle. They then show that the wave can evolve continuously without facing any disruptions as the boundary conditions shift from one geometry to the other, incompatible one.

Topology: The Secret Ingredient In The Latest Theory of Everything

Topology: The Secret Ingredient In The Latest Theory of Everything: Today, Wen combines topology, symmetry and quantum mechanics in a new theory that predicts the existence of new states of matter, unifies various puzzling phenomena in solid state physics and allows the creation artificial vacuums populated with artificial photons and electrons...

Xiao-Gang Wen's approach is to explore the properties of matter when the topological links between particles become much more general and complex. He generalises these links, thinking of them as strings that can connect many  particles together. In fact, he considers the way many strings can form net-like structures that have their own emergent properties...
That makes string nets a kind of "quantum ether" through which electromagnetic waves travel. That's a big claim.

A one-way street for spinning atoms

A one-way street for spinning atoms: In previous experiments, the RLE researchers created a superfluid — a completely frictionless gas — of lithium atoms. In their new experiment, the researchers used laser beams to trap a cloud of lithium atoms about 50 micrometers in diameter. The atoms were cooled to just a few billionths of a degree above absolute zero...

The researchers illuminated the gas with a pair of laser beams, sorting the atoms into two lanes, each of which consists of atoms with the same spin moving in the same direction. For the first time in an atomic system, this correlation of atoms’ spins with their velocities was directly measured.

“The combined system of ultracold atoms and the light we shine on them forms a material with unique properties,” says Lawrence Cheuk, lead author of the paper and a graduate student in MIT’s physics department. “The gas acts as a quantum diode, a device that regulates the flow of spin currents.”

Mathematicians develop new method for describing extremely complicated shapes

Mathematicians develop new method for describing extremely complicated shapes:Bridging the topology and fractals, as described in the American Institute of Physics' Journal of Mathematical Physics (JMP), relies upon a recently developed mathematical theory, known as "persistent homology," which takes into account the sizes and number of holes in a geometric shape. The work described in JMP is a proof of concept based on fractals that have already been studied by other methods – such as the shapes assumed by large polymer molecules as they twist or bend under random thermal fluctuation.

Could a planet have only one pole?

Could a planet have only one pole?: You are on Boy's Surface, discovered by Werner Boy in 1901, and represented as a strange planet by John Pierre Petit in his book, Le Topologicon, in 1902. Each of the 'wings' can be traced by a Mobius Strip. The planet is a four dimensional object that penetrates itself without causing any holes or edges.

Silica microspheres in liquid crystals offer the possibility of creating every knot conceivable

Silica microspheres in liquid crystals offer the possibility of creating every knot conceivable: "The glass plates were treated in such a manner as to force the liquid crystalline molecules to align parallel to the surface", explains Tkalec. A single silica microsphere entering the layer changes the surrounding alignment substantially: around the sphere a ring-shaped region forms in which no preferred direction can be discerned... "It looks as if every microsphere were surrounded by its own ring – similar to the planet Saturn..."
In an essential step, the researchers discovered a way of manipulating the regions between the spheres by joining and separating neighbouring rings. First, they heated the region between the spheres with a laser. This destroys the characteristic alignment of the molecules. After switching off the laser, the alignment is re-established – but often in a different way than before.

Miniature 'knot lab' could help untangle DNA mystery - New Scientist - New Scientist

Miniature 'knot lab' could help untangle DNA mysteryEach silica particle was coated with a surfactant, making its surface hydrophobic. This disrupted the crystal's highly ordered structure – any liquid crystal molecule adjacent to a silica particle aligned itself perpendicular to the curved surface of the particle and these "disordered" molecules formed a three-dimensional Saturn's ring around the surface. "It's visible like a black ring around the particle," says Tkalec.

When the team trapped the loops with a laser and brought them close together, they immediately joined up to form a bigger, twisted loop around both the particles. A similar thing happened with three particles. By bringing just the right combination of twisted loops into contact, these arrays could be made to unknot and then re-knot to form loops that aren't just twisted, but are intertwined.

Make: Online | The Knot Zoo

Make: Online | The Knot Zoo: Old but very interesting page from Canadian Robert Scharein, programmer of the visualization software KnotPlot. The knots are organized by the number of crossings, and can be clicked on to display rotatable 3D models.

Exotic sphere discoverer wins mathematical 'Nobel' - physics-math - 23 March 2011 - New Scientist

Exotic sphere discoverer wins mathematical 'Nobel': Imagine splitting an ordinary sphere into two halves along the middle, so that each half has a copy of every point on the equator. Now rejoin the two halves so that the southern copy of a point doesn't join its northern counterpoint. In two dimensions, there's only one way to do this: by twisting the sphere. But in seven dimensions the points can be mixed up with respect to each other in multiple different ways...
It turns out there are a total of 28 exotic spheres in seven dimensions, and they also exist in other dimensions. Dimension 15 has as many as 16,256, while others like dimensions five and six only have the ordinary sphere. Mathematicians don't yet know whether exotic spheres exist in four dimensions – a problem known as the smooth Poincaré conjecture, and related to the generalised Poincaré conjecture, which was solved in 2003.

Make way for mathematical matter - physics-math - 05 January 2011 - New Scientist

Make way for mathematical matter:But Baas's more complicated hyperstructures have radically different topologies from anything yet seen in nature. If groups of particles can be made to bond in this way, they would create matter with previously unseen properties, Baas reckons. "When you go to a higher level, something completely new happens mathematically - and I would suspect it does in the real world too," he says.
Baas has teamed up with Ned Seeman of New York University in New York City to figure out how to build the hyperstructures.

Topologist Predicts New Form of Matter� - Technology Review

Topologist Predicts New Form of Matter� - Technology Review: "And topology is about to extend its influence, if Baas has his way. He points out that Borromean rings are just the simplest example of an entire periodic table of topological structures. And if it's possible to make Efimov states that are equivalent to Borromean rings, then it ought to be possible to make the others too.

This family of stuff will be a new state of matter that is governed by news rules, a kind of 'Efimov physics'.

How might this stuff behave? That isn't yet clear but Baas raises an interesting possibility. The deep and unworldly link between particles in Efimov states is remarkably similar to quantum entanglement."

Hypersphere Exotica: Kervaire Invariant Problem Has a Solution!

http://www.scientificamerican.com/article.cfm?id=hypersphere-exotica

Fundamental secrets are tied up in knots

Knot theory and quantum mechanics

http://www.newscientist.com/article/mg20026781.500-fundamental-secrets-are-tied-up-in-knots.html

Tied Up in Knots

By tumbling a string of rope inside a box, biophysicists Dorian Raymer and Douglas Smith have discovered that knots—even complex knots—form surprisingly fast and often.

http://www.sciencenews.org/view/feature/id/9226/title/Tied_Up_in_Knots

Minimal Surfaces, Maximal Bricks

Topology with legos

http://www.wired.com/wired/archive/14.10/start.html?pg=3