Posts filed under ‘geometry’
I had to give a talk to high school students about some mathematical notion: I decided to tell them something about elliptic curves, but not the usual speech about cryptography, finite fields and the group law on a cubic curve…
Instead, I talked about the perhaps less known appearances of elliptic functions as solutions of classical ODEs (even if I don’t really know much about these myself). The simplest mechanical system whose motion is governed by an elliptic curve is the pendulum: the reason for this is that the ODE which classically describes the time evolution of the angle of the pendulum is best rewritten in terms of the altitude of the pendulum: the law of energy conservation is then written as
where 0 and 2l are the extremal values of the altitude , is the highest altitude which can be reached with a given energy (even if , which corresponds to the pendulum make full rotations around its axis), and is the vertical momentum of the pendulum.
In this setting, there are classical Hamilton relations , , so the differential form turns out to be the canonical non-vanishing abelian differential on the elliptic curve. This explains why the period of the pendulum is an elliptic integral, which can be calculed by an arithmetic-geometric mean, and why the position of the pendulum at can be deduced from its position at times and by the classical secant-tangent law.
The notes for the talk (in French) are available here.
I made a cardboard dodecahedron for the needs of a talk.
If you draw 5-coloured stars on all facets, by choosing smartly the colours, you can get five coloured cubes whose vertices are vertices of the dodecahedron. This trick can be used to show that the symmetry group of the dodecahedron is the alternate symmetric group : it replaces a star by a star with a different arrangement of colours.
Since there are 12 facets and 5 ways of rotating each of them, 60 colourings can be seen by rotating the dodecahedron by direct isometries. However, it is NOT true that you can see the 120 possible colourings by allowing also reflections (the full isometry group of the dodecahedron is the symmetric group on five colours). An easy reason for this is that the colouring is invariant under symmetry through the central point (which is a determinant -1 transformation). You can also argue that reflections act as double transpositions of the colours of a star.
People also talk about five tetrahedra in a isocahedron, which can also be obtained in the dodecahedron by choosing a tetrahedron in each cube in a consistent way. The tetrahedra have faithful action of the isometry group: there are two sets of five tetrahedron, which are exchanged under signature -1 transformations, and even permutations of the tetrahedra correspond to direct isometries.
I wrote a small package to compute Gröbner bases in Haskell with the Buchberger algorithm (with applications to variable elimination). Performance is quite bad compared to specialised software like Macaulay, but it seems to work ! I put a Cabal package here. Maybe I’ll add several functions afterwards.
A testcase :
type R = Polynom QQ VarXYZ
[x,y,z,t,u,v] = map returnp [X,Y,Z,T,U,V] :: [R]
-- projection from a point on the intersection of quadrics
main = do
print $ step_eliminate [T] $ MakeIdeal
[x^2 - 3*y*z + z*t + 2*x*t,
z^2 + 5*y^2 + z*x - 2*t*z]
The output should be :
André Weil was among the first ones to point out the importance of having a local description of varieties, especially projective spaces, which can always locally be described as an affine space with completion by a hyperplane at infinity, and projective varieties, which similarly look like varieties in affine space. The use of sheaves in local description of spaces was magnified by Cartan and Serre, in the context of complex analytic spaces, and generalised to the algebraic setting by Serre in Faisceaux algébriques cohérents.
The projective space is the simplest example of an algebro-geometric object which cannot be described by the prime spectrum or the functor of points of a ring. For example, there is no obvious ring whose ideals describe varieties in projective space, which come from homogeneous equations. We would like to give a correct definition of gluing affine lines (with coordinates and ) to define the projective line as the gluing of with given by . For functors of points, the latest article by Alain Connes and Caterina Consani, gives a definition. For prime spectra, one has to be aware that gluing only topological spaces do not give meaningful information on algebraic properties. This is illustrated by the case of differentiable manifolds, which are not the same as topological manifolds: gluing differentiable manifolds has to induce a correspondance between differentiable functions (this is equivalent to the requirement that gluing maps between charts be differentiable).
I just explained how the affine plane could be described by the ring . A point M of the affine plane whose coordinate ring is R is a morphism defined by the assignment , where are the coordinates of M. In the case of points corresponding to morphisms , there is a natural way of recovering the point from the ring morphism by looking at his equations, which are elements of the kernel of the morphism. If M satisfies the equations and , then M has the form . This motivates the abstract definition of point of the affine plane as a morphism to some ring.
Conversely, the set of equations of M defines a canonically associated point , which is the morphism , where is the ideal generated by the equations. But this morphism has no reason to totally recover M if it wasn’t a point with integral coordinates. For example, the point is a special point, satisfying a lot of equations, which characterize it. But do not satisfy any polynomial equations with integral coefficients, so the set of its equations is empty, and cannot be used to recover it. Moreover, the point does not satisfy equations either: their algebraic properties are exactly the same. These points are called generic.
The prime spectrum of a ring is a convenient way of describing equivalence classes of points of a given ring.
Definition. The prime spectrum of is the set of points for prime ideals . If is any point of the affine plane with coordinates in an integral domain , then Mis canonically associated to some , where is the kernel of the map .
I think most people blogging around algebraic geometry eventually write about schemes, (as in Rigorous trivialities or algebraic stacks (in the Secret Blogging Seminar), which are traditionnally seen as the main reason (not) to study algebraic geometry today. My turn now. I recommend Igor Dolgacev’s lectures which is one of my favorite ways of speaking of schemes.
Interesting mathematics come up when algebraic varieties (things defined by several polynomial equations) are no longer defined as mere sets (sets of tuples of numbers satisfying the equation) but mope complex mathematical objects. Differential geometry, for example, gives the structure of a complex manifold to algebraic varieties in , which is still an efficient way of proving theorems. However, during the 20th century, a lot of mathematicians tried to develop a new structure which would avoid the use of analysis to concentrate on the algebraic aspects (I don’t know exaclty who, but expect Hilbert, Zariski, Chevalley, Grothendieck to have played a role). Grothendieck approach using category theory and functors of points is now widely used and is a very impressive way of tying together intuition, commutative algebra, and geometry.
There are many ways of reverse engineering Grothendieck’s definition of a scheme (see EGA1 if you want to know how this is related to Chevalley’s definition of a scheme). The first thing to say is probably what properties and notions are needed for schemes:
When talking about “the” cohomology of mathematical objects, we do not always explicitly mention which cohomology is used, because it is obvious (in cases there is only one possible definition), or because we really don’t care (since as we will see, it is frequent that different definitions lead to equivalent results). The case of differentiable manifolds or algebraic varieties is particularly impressive, since there were a lot of equivalent cohomology theories defined during last century in order to simplify proofs or allow generalisations. Most cohomology theories, if not all, are defined as the cohomology of a complex : i.e. a sequence of vector spaces or modules with a boundary map . The kernel of is called the set of cycles, while the images by are called boundaries : the cohomology is then the quotient of cycles in by the subspace of boundaries.
Classical topologists, for example, will use preferably (see MacLane, Homology or the book of Allen Hatcher) :
- simplicial (co)homology : it is defined for a triangulated space, i.e. the manifold is cut by curves, surfaces, etc. which make it isomorphic to a sort of polyhedron (a complex); simplicial homology describe non-triviality (holes) in the combinatorial structure of this polyhedron; here the boundary map is really the boundary map.
- singular (co)homology : a more abstract version of simplicial homology; now we consider the set of all possible simplexes (curves, polygons, polyhedra, and their generalisations…) drawn on the manifold; this definition was given by Eilenberg; I don’t know who first defined simplicial homology, but MacLane mentions that Poincaré and Noether gaves important contributions to this theory.
Then come sheaves, which were explicitly defined by Leray (tales for young mathematicians help remember that Leray made considerable efforts as a prisoner in concentration camps to focus his work on especially “useless” subjects to avoid helping Nazis). (more…)