Month: January 2011

Terry Loring, distinguished professor of mathematics and statistics, published and co-authored a new research piece involving his research on K-theory with the major advances in applications to critical problems in physics.

The study titled, “Revealing topology in metals using experimental protocols inspired by K-theory,” was published in Nature Communications. Loring used mathematical properties of K-theory to help advance the understanding of topological materials in the physics world.

The main focus of the study was to discover how electricity, sound, or light can be trapped in a portion of a material. “This experiment was done in what is called a meta-material, built from individual sound resonators coupled in a fashion that mimics how atoms can come together to form a crystal. Three-dimensional printing allows us to make customized resonators that we join in a precise way to make the physics match the mathematics,” explained Loring. The study was part of a larger project that covered many areas of physics.

According to Loring, there are different forms of K-theory that arise in many different mathematical fields, however, the form of K-theory that he used in this study was focused on being best suited for studying matrix models of physical systems.

Loring explains that matrices are simply square tables of numbers, with a peculiar rule for how two matrices are multiplied. This rule has an asymmetry in it that leads to having AB and BA sometimes being very different, meaning that the commutative law for multiplication is violated.

“Physicists like Heisenberg realized that matrices are terrific at modeling uncertainty in molecular- and atomic-scale physics. K-theory can tell us when certain matrices can be connected except by a path that goes through what we call a singular matrix. This guaranteed singularity turns out to have an important meaning when the matrices come from models of physical systems,” Loring said.

The researchers were mainly looking at topological materials which include topological insulators. A topological insulator can have an index associated to it, which is a number computed using K-theory. If a device is built from two topological insulators that each have a different index, there is guaranteed to be a conducting region where the two materials come together.

“This conducting region exactly corresponds to where a certain matrix goes singular. To demonstrate this fact we use results about determinants one learns in linear algebratogether with the intermediate value theorem that people learn in their first calculus class,” said Loring.

This research is attempting to advance the theory of topological metals. Topological metals mix up conducting and insulating properties in very confusing ways. Loring and team built an acoustic crystal that had a specific pattern, they then deliberately broke the pattern in the middle thus inserting a defect in the system.

“During the experiment, and computer simulations, we were able to show how sound can get trapped at the defect. The hope is that it teaches us how to better trap light in small-scale photonic devices, and more generally start to manipulate light in a similar way to how electronic circuits manipulate electricity. There are advantages to moving information with light, as this can sometimes eliminate/reduce the energy wasted by the heat associated with electronics,” Loring stated.

Another part of the experiment which was more delicate included modifying the acoustic resonators by a formula from K-theory. The modified system removed the metallic properties in many parts of the crystal, isolating the binding metallic nature of the defect.

“Of course our acoustics system is not a metal, but shares mathematical properties with metals that harbor topology in their electronic structure. The hope is we will be able to devise experimental probes of photonic and electronic systems that bring the K-theory off the blackboard and into the lab,” explained Loring.

Mathematics was central to the design of this experiment. The project began with a discussion of formulas in K-theory that might lead to a matrix that can describe the energy in an acoustic system.

“We started with the analysis we would use to explain the system and then built a system that could be analysed this way. This backwards flow is somewhat common in the field of ‘topological physics’ where clean formulas in math suggest the search for physical systems that match that formula,” Loring stated.

In finding, Loring and his team discovered that new mathematics can classify very local patches of material as insulating or conducting. Loring points out that initially it was not clear if this classification had any meaning that a physicist would care about.

“This experiment showed that we can manufacture materials where this local classification is physically meaningful. While this material has no practical application, it is expected that materials and devices will be discovered or manufactured that have these local variations and that these local variations will give us even more control over light and electricity than we now enjoy.”

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Credit of the article given to Dani Rae Wascher, University of New Mexico

Two weeks ago, a modest-looking paper was uploaded to the arXiv preprint server with the unassuming title “On the invariant subspace problem in Hilbert spaces.” The paper is just 13 pages long and its list of references contains only a single entry.

The paper purports to contain the final piece of a jigsaw puzzle that mathematicians have been picking away at for more than half a century: the invariant subspace problem.

Famous open problems often attract ambitious attempts at solutions by interesting characters out to make their name. But such efforts are usually quickly shot down by experts.

However, the author of this short note, Swedish mathematician Per Enflo, is no ambitious up-and-comer. He is almost 80, has made a name for himself solving open problems, and has quite a history with the problem at hand.

Per Enflo: Mathematics, music, and a live goose

Born in 1944 and now an emeritus professor at Kent State University, Ohio, Enflo has had a remarkable career, not only in mathematics but also in music.

He is a renowned concert pianist who has performed and recorded numerous piano concertos, and has performed solo and with orchestras across the world.

Enflo is also one of the great problem-solvers in a field called functional analysis. Aside from his work on the invariant subspace problem, Enflo solved two other major problems—the basis problem and the approximation problem—both of which had remained open for more than 40 years.

By solving the approximation problem, Enflo cracked an equivalent puzzle called Mazur’s goose problem. Polish mathematician Stanisław Mazur had in 1936 promised a live goose to anyone who solved his problem—and in 1972 he kept his word, presenting the goose to Enflo.

What’s an invariant subspace?

Now we know the main character. But what about the invariant subspace problem itself?

If you’ve ever taken a first-year university course in linear algebra, you will have come across things called vectors, matrices and eigenvectors. If you haven’t, we can think of a vector as an arrow with a length and a direction, living in a particular vector space. (There are lots of different vector spaces with different numbers of dimensions and various rules.)

A matrix is something that can transform a vector, by changing the direction and/or length of the line. If a particular matrix only transforms the length of a particular vector (meaning the direction is either the same or flipped in the opposite direction), we call the vector an eigenvector of the matrix.

Another way to think about this is to say that the matrix transforms the eigenvectors (and any lines parallel to them) back onto themselves: these lines are invariant for this matrix. Taken together, we call these lines invariant subspaces of the matrix.

Eigenvectors and invariant subspaces are also of interest beyond just mathematics—to take one example, it has been said that Google owes its success to “the $25 billion eigenvector.”

What about spaces with an infinite number of dimensions?

So that’s an invariant subspace. The invariant subspace problem is a little more complicated: it is about spaces with an infinite number of dimensions, and it asks whether every linear operator (the equivalent of a matrix) in those spaces must have an invariant subspace.

More precisely (hold onto your hat): the invariant subspace problem asks whether every bounded linear operator T on a complex Banach space X admits a non-trivial invariant subspace M of X, in the sense that there is a subspace M ≠ {0}, X of X such that T(M) is contained back in M.

Stated in this way, the invariant subspace problem was posed during the middle of last century, and eluded all attempts at a solution.

But as is often the case when mathematicians can’t solve a problem, we move the goalposts. Mathematicians working on this problem narrowed their focus by restricting the problem to particular classes of spaces and operators.

The first breakthrough was made by Enflo in the 1970s (although his result was not published until 1987). He answered the problem in the negative, by constructing an operator on a Banach space without a non-trivial invariant subspace.

What’s new about this new proposed solution?

So what is the current status of the invariant subspace problem? If Enflo solved it in 1987, why has he solved it again?

Well, Enflo settled the problem for Banach spaces in general. However, there is a particularly important kind of Banach space called a Hilbert space, which has a strong sense of geometry and is widely used in physics, economics and applied mathematics.

Resolving the invariant subspace problem for operators on Hilbert spaces has been stubbornly difficult, and it is this which Enflo claims to have achieved.

This time Enflo answers in the affirmative: his paper argues that every bounded linear operator on a Hilbert space does have an invariant subspace.

Expert review is still to come

I have not worked through Enflo’s preprint line by line. Enflo himself is reportedly cautiousabout the solution, as it has not yet been reviewed by experts.

Peer review of Enflo’s earlier proof, for Banach spaces in general, took several years. However, that paper ran to more than 100 pages, so a review of the 13 pages of the new paper should be much speedier.

If correct, it will be a remarkable achievement, especially for someone who has already produced so many remarkable achievements over such a large span of time. Enflo’s many contributions to mathematics, and his answers to many open problems, have made a big impact on the field, generating new techniques and ideas.

I’m looking forward to finding out whether Enflo’s work now closes the book on the invariant subspace problem, and to seeing the new mathematics that may emerge out of its conclusion.

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Credit of the article given to Nathan Brownlowe, The Conversation