It can do this without tearing itself and without ever leaving the surface. Any object with a simply connected surface can be smoothed out to look like a sphere. Imagine, by contrast, an elastic band that passes through the hole in a doughnut. If this band is slowly shrunk, it becomes necessary to cut the doughnut or break the band. In this example, the surface is not simply connected and any smoothed-out object looks like a torus with at least one hole. On the next rung up the ladder of difficult mathematics comes doing the same thing in four-dimensional space.
His conjecture, made in , was that in this four-dimensional world, all closed three-dimensional surfaces that are simply connected could be transformed to look like a three-dimensional sphere. The equation smoothes out the irregularities of an object, transforming it mathematically into something that looks like a uniform three-dimensional sphere. This does not change the object's essential properties: the transformed shape is equivalent to the starting shape. Richard Hamilton of Columbia University had previously realised the usefulness of this heat-flow model but became unstuck when he found that, under this transformation, the object may stretch out to form singularities—spikes that could not be easily manipulated into a sphere.
Dr Perelman overcame this difficulty by cutting off the singularities, continuing with the Ricci-flow application and then rejoining the transformed objects later. Fields medals are only awarded once every four years but the organisers give four medals at each ceremony. These winners accepted their medals but Dr Perelman stayed at home, reportedly because he has no desire to be a figurehead of the mathematics community. For Dr Perelman, transforming a conjecture into a theorem appears to have been prize enough.
Join them. Subscribe to The Economist today. Media Audio edition Economist Films Podcasts. New to The Economist? Sign up now Activate your digital subscription Manage your subscription Renew your subscription. Topics up icon. Blogs up icon. Then, the addition of the handle is equivalent to a handle on a sphere and, of course, that's equivalent to a donut with an aneurysm, which is equivalent to a donut.
View all New York Times newsletters. This, of course, is not formal Mathematics; I'm just running animations in my imagination, which behavior is what interested me in math in the first place in when I was in tenth grade. I went on to get a mathematics BA but, ironically, never went anywhere with the degree having been repelled by, you guessed it, topology or, rather, a haughty, dismissive Topology professor in mysenior year.
Yes, you need a handle on the mug for it to be transformable into a sphere.
Your topology professor dismissed you too soon. Your example of a rubber band stretched around an apple or a doughnut is unclear -- at least to a scientific ignoramus like me. When I picture a band stretched around an apple shrinking, the band starts to cut into the apple and bisect it. I don't know what you mean when you say the band can be "shrunk without limit," or when you say that the band will be "stopped" by a doughnut hole. Can you illustrate this some other way? You could imagine a little lasso lying on the surface of whatever. As I think about this, and obviously I do not have the tools of any of these mathematicians and I have the fear of being thought crazy even to think I really understand this proof let alone the implications.
Still, I wonder if this mathematical proof is saying in effect that space of more than three dimensions must be 'spherical,' means that a Worm hole tunneling between two points in our universe cannot exist because the shape that space-time must assume can not exist. Darn it sounds crazy.
In addition, space time can not be torn because the shape it would need to form can not exist. If I am on the wrong track and off base for trying to understand the physics through the math please humor me with some response. You raise a challenging question. Wormholes would catch them up. Before you go on reporting on the conjecture, you should certainly read A. Garciadiego's book shows that the supposed paradoxes were not paradoxes at all. This immediately brings into question the goal of the conjecture. I think you will find that there are assumptions about the nature of a set, built into the conjecture, which are highly questionable.
Certainly you will find Cantor's set notions highly questionable, after you read Garciadiego. This is something which has not been considered by the mathematicians you name. Thanks for bringing it to my attention.
source link You have made a very difficult concept somewhat accessible to many of us who are not mathematicians. Topology has been central to the field of chemistry for as long as we know. Since ancient Greek description of materials atoms as spheres with hooks on them , those of us interested in the properties of matter have found it necessary to rely on an abstract symbolic representation of the size and shapes of atoms and molecules in order to effectively understand and describe the properties that we observe.
Our views today are slightly more sophisticated than those of the Greeks. Molecular structures are represented by connected partial spheres representing the constituent atoms, and lead to shapes of molecules.
Reasonably impenetrable surface electron clouds provide that provide the shapes of molecules as we have come to perceive them. Together with differences in electron density, these shapes provide the foundation to our description of how molecular structures interact with each other. This understanding is central to modern chemical reaction studies, and is essential to the rapidly developing field of molecular biology and genetics.
In your interviews and research while preparing the article, did you encounter any topologists who have begun to speculate about the possibility that current work that might lead to an improvement in our understanding of the geometry, shapes and interaction of molecular structures? You can appreciate that a significantly improved understanding of the topology of molecules could provide profound improvement in our understanding of chemical reactions.
I would be interested in learning if your preparation of this article provided you any information on researchers who are actively pursuing this point. A quick Google search yielded many results relating to topology and chemistry, though none of the people I talked to are involved. Yau and his colleagues have applied some of these techniques to analyzing images of brains from M. The idea is that knots or loops in a fluid of cold electrons could form the basis of a quantum computer. Tell us what you think. Please upgrade your browser.
In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in. Poincaré Conjecture. If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and.
See next articles. Katz A.
See this for an example. He shows that the singularities can be dealt with, since the pieces of the space where they occur can be cut out from the original space. The book is bracketed by two modern stories. Vita, New Haven, Connecticut. He had many brilliant students who further developed his theories, not least by producing powerful computer programs that could test any given space to try to find its geometric structure. For those not already familiar with the history of mathematics and its Dramatis Personae there is a Glossary of Terms, a Glossary of Names and a Timeline. It asserts that if any loop in a closed three-dimensional space without boundary can be shrunk to a point without tearing either the loop or the space, then the space is equivalent to a three-dimensional sphere.
Horwitz A. Morgan Q.
Vita, New Haven, Connecticut Q.