isomorphismes

The historical impetus for using sheaves in algebraic geometry comes from the theory of several complex variables, and in that theory sheaves were introduced, along with cohomological techniques, because many important and non-trivial theorems can be stated as saying that certain sheaves are generated by their global sections, or have vanishing higher cohomology. (I am thinkin of Cartan’s Theorem A and B, which have as consequences many earlier theorems in complex analysis.)

If you read Zariski’s fantastic report on sheaves in algebraic geometry, from the 50s, you will see a discussion by a master geometer of how sheaves, and especially their cohomology, can be used as a tool to express, and generalize, earlier theorems in algebraic geometry. Again, the questions being addressed (e.g. the completeness of the linear systems of hyperplane sections) are about the existence of global sections, and/or vanishing of higher cohomology. (And these two usually go hand in hand; often one establishes existence results about global sections of one sheaf by showing that the higher cohomology of some related sheaf vanishes, and using a long exact cohomology sequence.)

These kinds of questions typically don’t arise in differential geometry. All the sheaves that might be under consideration (i.e. sheaves of sections of smooth bundles) have global sections in abundance, due to the existence of partions of unity and related constructions.

There are difficult existence problems in differential geometry, to be sure: but these are very often problems in ODE or PDE, and cohomological methods are not what is required to solve them (or so it seems, based on current mathematical pratice). One place where a sheaf theoretic perspective can be useful is in the consideration of flat (i.e. curvature zero) Riemannian manifolds; the fact that the horizontal sections of a bundle with flat connection form a local system, which in turn determines the bundle with connection, is a useful one, which is well-expressed in sheaf theoretic language. But there are also plenty of ways to discuss this result without sheaf-theoretic language, and in any case, it is a fairly small part of differential geometry, since typically the curvature of a metric doesn’t vanish, so that sheaf-theoretic methods don’t seem to have much to say.

If you like, sheaf-theoretic methods are potentially useful for dealing with problems, especially linear ones, in which local existence is clear, but the objects are suffiently rigid that there can be global obstructions to patching local solutions.

In differential geomtery, it is often the local questions that are hard: they become difficult non-linear PDEs. The difficulties are not of the “patching local solutions” kind. There are difficult global questions too, e.g. the one solved by the Nash embedding theorem, but again, these are typically global problems of a very different type to those that are typically solved by sheaf-theoretic methods.

The demands of university teaching, addressed to students … with [only] a modest (and frequently less than modest) mathematical baggage, led me to … start from an intuitive baggage common to everyone, independent of any technical language used to express it, and anterior to any such language—it turned out that the geometric and topological intuition of shapes, particularly two-dimensional shapes, formed such a common ground.

These themes can be grouped under the general names “topology of surfaces” or “geometry of surfaces”, … the main emphasis being on the … combinatorial aspects which form the most down-to-earth technical expression of them—and not on the differential, conformal, Riemannian, holomorphic, [Kähler, contact, symplectic, Moishezon] aspects—and from there on to ℂ algebraic curves.

our civilization continues to depend on activities that require large flows of energy and materials, and alternatives to these requirements can’t be commercialized at rates that double every couple of years. Our modern societies are underpinned by countless industrial processes that have not changed fundamentally in two or even three generations. These include the way we generate most of our electricity, the way we smelt primary iron and aluminum, the way we grow staple foods and feed crops, the way we raise and slaughter animals, the way we excavate sand and make cement, the way we fly, and the way we transport cargo.

In the process of mathematical research, dialectical conflicts are of fundamental significance: analytic/synthetic, axiomatic/constructive, … algebraic-geometric….

The development of analytic geometry is usually attributed to Descartes and Fermat. Ancient Greek geometers used relations between curve segments which, from a present-day standpoint, are like equations for the curves in cartesian coordinates. However, the analytic perspective was not fully developed.

Th basic idea of each analytic method is the reduction of a system to a few basic elements. The advantage of simplification gained in this way may possibly be opposed by the disadvantage of complexity in the reconstruction of the system from the basic elements.

In the case of analytic geometry the reduction consists in choosing two perpendicular lines in the plane [or any two intersecting lines!] and determining the position of points on a curve by their distances x,y from these lines. The simplicity of the reduction of the description to a position relative to only two lines is opposed by the complexity of the equation f(x,y)=0. The Greeks, whose thinking was perhaps more synthetic than analytic, preferred to muse many auxiliary lines, in order to attain simple relationships.

The advantage of the analytic method, the reduction of geometric relations to complex quantitative relations — only came to light after algebra had been developed in the East and imported into the West.

Fermat had the idea of analytic geometry in 1629.

Descartes published his geometry in 1637, but had worked on it earlier, perhaps since 1619. Descartes’ Geometry brought classical geometry within the scope of algebra. The book was originally published as an appendix to Discours de la Method. Descartes searched for a general method of thinking. Since the only [decent system] was mechanics, mathematics became his means for understanding the universe.

His Geometry actually contains little analytic geometry in the modern sense, no “cartesian” axes and no derivation of the equations of conic sections as in Fermat.

Ancient Greek geometers had considered not only lengths x, y of segments but also their products, such as x², x³, xy, etc. But they were not regarded as numbers of the same type.

Descartes abolished this distinction: An algebraic equation became a relation between numbers, an advancing abstraction, which one can regard as the final adoption of the algorithmic tradition of the East by the West.

…

Thus it was that the geometry of plane curves became analytic geometry, the investigation of the equations defining curves by algebraic and analytic methods. This was the starting point for two hundred years of development, after which the tendency to synthesis again came to the fore.

Plato and Kepler were in the right ball-park, but not really right.  Both the solar system and atoms are described pretty well by similar laws—the inverse-square force laws for gravity and electrostatics.

And solving this problem (in either the classical or quantum case) does indeed require a deep understanding of rotations in 3-dimensional space.  …[T]he Platonic solids have as their symmetries finite subgroups of the rotation group in 3 dimensions….

[T]he real reason the inverse square force law problem is exactly solvable is a surprising symmetry in four-dimensional rotations.

Financial intermediation is premised on acquiring pools of cheap funding and a lending-to-credit loss/reward profile that creates profits.

In the United States, this funding used to come from deposits…. Things have changed. Now funding comes from money markets.

When long-term deposits funded assets held to maturity, banking was a safe business with reasonable margins. As margins compressed, banks sustained profits with leverage—requiring additional sources of financial capital.

Wholesale funding has now eclipsed deposits as a funding source.

Wholesale funding comes from money-market funds and from corporations or banks with excess short-term cash which they want a bit of interest on. Instruments include: repos, commercial paper, CD’s, fed fund borrowings.

the [solar] industry could readily eliminate many of the damaging side effects that do exist. …Although the overall track record for the industry is good, the countries that produce the most photovoltaics today typically do the worst job of protecting the environment and their workers.

To understand exactly what the problems are, and how they might be addressed, it’s helpful to know a little something about how photovoltaic panels are made. While solar energy can be generated using a variety of technologies, the vast majority of solar cells today start as quartz, the most common form of silica (silicon dioxide), which is refined into elemental silicon. …The quartz is extracted from mines, putting the miners at risk of the lung disease silicosis.

Refining turns quartz into metallurgical-grade silicon, a substance used mostly to harden steel and other metals. That happens in giant furnaces, and keeping them hot takes a lot of energy, a subject we’ll return to later. Fortunately, the levels of the resulting emissions—mostly carbon dioxide and sulfur dioxide—can’t do much harm to the people working at silicon refineries or to the immediate environment.

The next step, however—turning metallurgical-grade silicon into a purer form called polysilicon—creates the very toxic compound silicon tetrachloride. The refinement process involves combining hydrochloric acid with metallurgical-grade silicon to turn it into what are called trichlorosilanes. The trichlorosilanes then react with added hydrogen, producing polysilicon along with liquid silicon tetrachloride—three or four tons of silicon tetrachloride for every ton of polysilicon.

Most manufacturers recycle this waste to make more polysilicon. Capturing silicon from silicon tetrachloride requires less energy than obtaining it from raw silica, so recycling this waste can save manufacturers money. But the reprocessing equipment can cost tens of millions of dollars. So some operations have just thrown away the by-product. If exposed to water—and that’s hard to prevent if it’s casually dumped—the silicon tetrachloride releases hydrochloric acid, acidifying the soil and emitting harAd