how many of  you know about donaldson's theorem and classification of three and four manifolds?
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According to a celebrated theorem of Donaldson, if the intersection form of a smooth closed four-manifold is definite then it is diagonalizable. Later proofs of this result use Elkies' characterization of the diagonal definite uni-modular form. I will describe a generalization of Elkies' theorem to forms of arbitrary determinant. Combined with a theorem of Ozsváth and Szabó this gives a generalization of Donaldson's theorem to four-manifolds with boundary. This is joint work with Brendan Owens. \nReally simple isn't?
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Calculus on Manifolds \nMichael Spivak's Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus (1965) is a text treating analysis in several variables in Euclidean spaces and on differentiable manifolds. Notable features include a long problem entitled "A First Course in Complex Analysis" and the book's cover, a reproduction of Lord Kelvin's letter to Stokes in which Stokes's theorem is first stated.
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Think what you are looking for is Euclidian Space:\n\nAround 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "Euclidean geometry", which is the study of the relationships between angles and distances in space. Euclid first developed "plane geometry" which dealt with the geometry of two-dimensional objects on a flat surface. He then went on to develop "solid geometry" which analyzed the geometry of three-dimensional objects. All of the axioms of Euclid have been encoded into an abstract mathematical space known as a two- or three-dimensional Euclidean Space. These mathematical spaces may be extended to apply to any dimensionality, and such a space is called an n-dimensional Euclidean space or an n-space. This article is concerned with such mathematical spaces.\n\nIn order to develop these higher dimensional Euclidean spaces, the properties of the familiar Euclidean spaces must be very carefully expressed and then extended to an arbitrary dimension. Although the resulting mathematics is rather abstract, it nevertheless captures the essential nature of the Euclidean spaces we are all familiar with.\n\nAn essential property of a Euclidean space is its flatness. Other spaces exist that are not Euclidean. For example, the surface of a sphere is not a Euclidean space, nor is the four-dimensional spacetime described by the theory of relativity when gravity is present.
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This is the old four color problem of topology.  Mathemeticians showed years ago that no more than four colors can satisfy a map where each country is a single unit.
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This might help --> http://www.uwm.edu/~ericskey/TANOTES/Geometry/node15.html
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I'm going to assume that you want a proof of Lagranges theorem in group theory which states that the number of elements of a subgroup divides the number of elements of the whole group. \n\nFirst, if G is a group and H is a subgroup, a collection gH={gh:h in H} is called a coset. It turns out the everything in the group is in exactly one coset and that every coset has the same number of elements as H does. If we write [G:H] for the number of cosets, we then get that the number of elements of G is [G:H] times the number of elements of H, hence the result.
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Fermat's Last Theorem has been solved ashas the Four Color Theorem. The Poincare conjecture has recently been solved. Trisecting a general angle with unmarked straightedge and compass has been shown to be unsolvable (although it is possible with marked straightedge and compass).\n\nHigh on the list of open problems:\n\nRiemann Hypothesis.\nP=NP?\nTwin prime conjecture.\nAre there any odd perfect numbers?\nProof of existence of solutions to Navier-Stokes equations.
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Differential geometry is the study of smooth manifolds and differential equations on such manifolds. As such, it mostly gives examples for these other areas of study. For example, the DeRham complex is a relatively easy to understand example of a chain complex. The homology of this chain complex turns out to be the same as for singular homology. The collection of continuous functions (or various levels of differentiable functions) gives an example of a Banach space, which is the fundamental concept in functional analysis. Also, a good part of the Hodge theory uses function analytic techniques. Finally, the differential equations that come up in differential geometry are often of practical use and need to be solved, at least approxiamtely, so numerical analysis comes into play.
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http://mathworld.wolfram.com/PoincareConjecture.html
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Spiral, lenticular, elliptical and irregular\n\nSee the link
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Think of the projective plane as the disk with opposite boundary points identified. Let the base point for the fundamental group be the center of the disk. The loop that goes directly across the disk and comes back through the other side is the generator of the fundamental group.\n\nAnother way to see this is to notice that the 2-sphere is the universal cover and is a double cover. Thus the fundamental group has two elements. There is only one group of order two up to isomorphism.
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Well there are 3 spatial dimensions and one temporal dimension, so there's 4 to begin with. On a macroscopic scale the universe behaves like a minkowski space. However on a microscopic scale even 4 dimensions isn't enough, the maths suggests possibly 11 or more dimensions.\n\nHowever I have a feeling you're not talking in the strictly mathematical sense. If you meant "are there other universes 'beyond' ours", then I have no idea. There are theories that suggest there are, but as far as I know there are no known observations that could answer that yet. We'll just have to wait and see what is discovered.
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There is a delightful article by the late Dr. John Kenny on that, back a few years ago, looking at the formulae for surface area and volumes of spheres; I'll see if I can find it for you--it's in my files SOMEWHERE--and give you the reference, and the formulae he generated. It turns out that they're not precisely the same as what you've got--but not terribly far from it.\n\nAs I said, I'll see if I can find it, and I'll get you the reference and the formulae.\n\nAbout 9pm CST...\n\nOK, here is the formulae, done the best I can on this thing...\n\n\nFor a sphere of dimension 2*k, radius R\n\nVolume: (pi^k)*(R^(2*k))/k!\n\nFor a sphere of dimension 2*k + 1 and radius R\n\nVolume:\n(2^(2*k+1))*k!*(pi^k)* (R^(2*k+1))/(2*k+1)!\n\n\nTo get the surface areas for each of these, all you do is differentiate with respect to R...\n\nAs I said, it was a little more complex than you thought--the surface areas come out a little bit differently for odd and even numbered dimensions.  I do not recall why, frankly.
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None.  That is the point.  If they exist is not proved, and hypothetical tests to see if they exist are flawed.  \n\nOn your other question of visualizing multiple dimensions, I have a few hints.  Have you read Flatland yet?  Start there, its good, and not overly mathematical.  Multiple dimensions can mean different things to different scientists.  We use them all the time for their mathematical properties.  But mathematics is just a description, the real world is something observable.  In terms of observation, you would not be able to detect a separate dimension by any means pedantic.  We may be able to prove that physical space has more than three dimensions, but that does not mean that those other dimensions are physical space.  Just like time.  Some particle interactions appear to go backward in time briefly.  What if there is more than one time dimension?  \n\nSo the classic three dimensional physical picture of multiple dimensions will always be flawed by the observers expectation of the properties of the unseen dimension.  Which is why we describe those dimensions using 'imaginary' constructions.  Hope this helps a bit.
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The answer is the free group on two letters. In fact, this space is homotopy equivalent to the space consisting of two circles joined at one point. To see this, expand the hole until you get one circle going one way around the torus and another going the other way. Alternatively, think of the torus as a square with opposite sides identified and remove the center point. It's easy to see the homotopy to the boundary sides of the square, which correspond to the two circles under the identification map.
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Differential geometry/manifolds, Optimization word problems in 3D, topology proofs.
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You must think about the 3d visualization of 4d objects as partials projections or perspectives (as well as when you represent a cube drawing it on a paper sheet, you only describe some of its properties by analogy): so you cannot "see" 4d objects: you can only describe them.\n \n\nSee this link for a rough idea of an hypercube (there would be much more to say about it ...)\nhttp://answers.yahoo.com/question/index;_ylt=Apvwu5u3VyjMZfW_mDnnJUnsy6IX?qid=1006012001214\n\nSee this other one (expecially the links in all answers) for the so-called "Klein bottle"\n\nhttp://answers.yahoo.com/question/index;_ylt=As6vB06qMWLWdUaS3dhCfW_sy6IX?qid=1006013107146\n\n\nfour dimensions is interesting in maths because (they say: I'm starting now to study it) that there are many different "regular solids" which you can hypotize counting sides, vertexes and faces. From 5th dimension on, there is only one kind of regular solid, so it is not considered an interesting field of study.