Many here will probably be interested to know about an interesting popsci article in the arXiv:

Moataz H. Emam,
"So what will you do if string theory is wrong?",
Americal Journal of Physics, July 2008
http://arxiv.org/abs/0805.0543

(A more accurate title would have been: What will Physicists do if String Theory Fails to Produce a Testable GUT?)

Note that Emam publishes research papers related to string theory. His point is pretty much the same one that I have tried to make in various public discussion forums: superstring theory was originally envisioned as a unique (or almost unique) theory of fundamental physical interactions (a GUT, in fact), but it has turned out to be better characterized as grand subject in pure mathematics which may one day contribute to viable and testable physical theories. Emam gives a nice short sketch of some of the reasons why superstring theory has proven so popular with mathematicians, but I'd like to flesh that out just a bit.

One interesting historical point which has been lost in all the popsci discussions of string theory I have seen is that in some sense superstring theory could be characterized as theoretical physics reciprocating the gift from pure mathematics of the notion of a symmetry group (especially Lie groups), plus vector bundles, Chern classes, and many other things, which have played such key roles in the development of theoretical physics during the last century. I dare say that not many non-mathematicians fully appreciate how closely the origins of symmetry groups and the current state of superstring theory lie to one of the most venerable topics in mathematics, algebraic geometry, the study of curves and surfaces defined by (systems of) polynomial equations in many variables; for example,
x0^5 + x1^5 + x2^5 + x3^5 + x4^5 - 5 psi x0 x1 x2 x3 x4 = 0
defines a family of degree 5 hypersurfaces in four dimensional complex projective space CP^4 (the family is parameterized by the variable psi). Here, we are using homogeneous coordinates for CP^4, in which nonzero complex scalar multiples of the nonzero vector (x0, x1, x2, x3, x4) are identified.

During the sixteenth century, Newton breathed new life into algebraic geometry (a subject initiated by the ancient Greeks) with his profound investigations into the theory of cubic curves (degree 3 curves) in the euclidean plane. One of the notable features of his work was that he gave a procedure for obtaining a qualitative description of an algebraic plane curve. To do this, he introduced the concept of the Newton polygon of a curve, which marks the beginning of convex geometry. (Late in the twentieth century, Newton polygons turned out to be an important concept in many applications seemingly quite unrelated to plane curves.)

In the next century, Stirling and McLaurin made further strides in the reinvigorated theory of algebraic curves (curves defined as the zero sets of suitable systems of polynomial equations in many variables), especially plane curves. Jumping ahead two centuries to a concept not yet invented, Stirling proved in effect that the space of degree d plane algebraic curves can be identified with a complex projective space of dimension d (d+3)/2. This ensures that the theory of plane algebraic curves--- which, as we will see, are best thought of as creatures "living" in CP^2, not C^2--- has a delightful "recursive" character.

McLaurin then proved that a degree m plane curve intersects a degree n plane curve in mn points (a result now known as Bezout's theorem, but Bezout neither discovered nor proved the theorem). But, to his alarm, while mulling over his result, McLaurin noticed the following discrepancy:

• Stirling's theorem says that a set of 9 points determines a unique cubic curve (degree 3 curve).
• The McLaurin-Bezout theorem says that a pair of cubic curves intersects in 9 points--- yet only one cubic should pass through 9 given points!

The nature of the problem becomes even clearer if we consider the next example:

• Stirling's theorem says that 14 points determine a unique quartic curve (degree 4 curve).
• The McLaurin-Bezout theorem says that a pair of distinct quartic curves intersects in 16 points---yet only one quartic should pass through 14 given points!
• Indeed, it should not be possible to pass any quartic curve through 16 given points, much less two of them, just as (generally speaking) it is not possible to pass a line through 3 given points.

McLaurin admitted that he couldn't explain the discrepancy, and for another three decades, neither could anyone else.

Euler introduced the notion of generating functions, in which (typically) a rational function is used to neatly "encode" an infinite series of facts. That is, the McLaurin series of this function has integer coefficients which answer some enumeration problem; for example, the question "how many rooted binary trees with n vertices are there?" might be answered by the n-th coefficient. Keep that in the back of your mind; we'll take quite a while to get back to it! And, oh yes, Euler also solved the McLaurin paradox: he observed that we must add the phrase points in general position to the statement of Stirling's theorem and we must add the phrase a generic pair of distinct curves to the statement of the McLaurin-Bezout theorem. This kind of qualification has become an enduring preoccupation of the theory of algebraic curves!

During the course of the nineteenth century, German, French, and English mathematicians vied with each other in discovering and investigating a great variety of novel geometries and many new notions which had never before been investigated. Most important was the introduction of projective geometry and the discovery of a fundamental principle of duality. For example, in complex projective 3-space, CP^3, any statement involving points, lines, and planes remains true if the roles of lines and planes are interchanged. We say that "point" and "plane" are dual notions, while "line" is selfdual, in CP^3. Various geometers proved decisive results showing that many theorems which require listing many special cases over the real number field and in affine real or complex spaces C^n become much simpler when restated in the context of the appropriate complex projective space CP^n.

Grassmann invented what is now called exterior algebra and defined the space of k-flats in n-dimensional projective space. Plucker then showed that every such Grassmannian is in fact an algebraic surface in some higher dimensional projective space. For example, the space of lines in CP^3, which is a Grassmannian, turns out to be a degree 2 hypersurface in CP^5. Furthermore, Plucker showed that the subspace of all lines in CP^3 which meet a given line (for example) is degree 2 codimenion 2 surface in CP^5. This means that in many cases, the problem of identifying all lines meeting a list of conditions reduces to finding the intersection of algebraic surfaces in some projective space! So the McLaurin-Bezout theorem is the prototype of many important results concerning the space of all algebraic surfaces of some kind which satisfy certain conditions.

Plucker also shed new light on the McLaurin paradox when he proved that if you delete any one point from a set of d (d+3)/2 points in GP, then a one-parameter family or pencil of degree d curves passes through the remaining points, and any pair of these will intersect in d^2 points. In other words, the remaining d (d+3)/2 - 1 points determine (d-1) (d-2)/2 additional points, all lying on the original degree d curve, which form a non-generic set of d^2 pinch points common to a pencil of degree d curves.

Thus, for example, a pair of quartic curves in GP will intersect in 16 points, but these points are not in GP! A set of 14 given points in GP determines a unique quartic curve, but omitting any one point allows us to pass a pencil of quartic curves through the remaining 13 points. These 13 points uniquely determine 3 additional points, forming a set of 16 pinch points in special position which is common to the entire pencil, since any pair of curves from the pencil intersect in precisely these 16 points. These 16 pinch points are in fact multiply-special, in the sense that no subset of 14 is in GP.

But Plucker was not done: he revolutionized curve theory with his introduction of inflection points.

Cayley reacted to Plucker's work by introducing projective metrics, which he used to achieve a kind of mathematical unification, clearly showing that spherical geometry and hyperbolic geometry both arise by adding additional structure to the geometry of the real projective plane RP^2.

Independently, Lie and Klein became intrigued by these developments. Fortuitously, they met in Berlin and soon discovered that they shared a common interest in geometry. They then traveled together to Paris where they learned much from such French luminaries as Darboux and Jacobi. Beating a hasty retreat to Berlin with the outbreak of war in 1870, Klein and Lie began to work closely together. Within a few years, Klein formulated a program for the unified study of geometry, the Erlangen program , while Lie forumlated a program a unified systematic attack on (systems of) differential equations, including the difficult nonlinear systems of PDEs.

In the Erlangen program, Klein introduced the notion of symmetry groups and explained how essentially all the dozens of "geometries" then known (all but Riemannian geometry). For example, Klein showed that both spherical metric geometry and hyperbolic metric geometry are specializations of affine geometry, which is in turn a specialization of projective geometry. In spherical metric geometry, we have a notion of distance and angle; in spherical conformal geometry, we lose the former. In affine geometry, we have a notion of geometric averaging, but no notion of angle. In projective geometry, we lose the notion of geometric averaging, but still have other notions such as cross-ratio. Unfortunately, BAUT may not allow inclusion of a Hasse diagram here, but interested readers should try to sketch one in which we have vertices for spherical metric geometry and hyperbolic metric geometry, connected by edges to lower vertices for spherical conformal geometry and hyperbolic metric geometry, and both of these connected to a still lower vertex for affine geometry, which is connected to a still lower vertex for projective geometry. Here, the various "notions" I mentioned are invariants under the action of the appropriate symmetry group on some projective space, and this is how Klein defines the "additional structure" needed to organize all these geometries (and dozens more) into a hierarchical structure.

Klein's unification greatly impressed his contemporaries. Cayley, Sylvester and Gordan, among others, were thrilled by invariants and rapidly developed that subject. Cayley developed a symbolic method which efficiently gave results but was also deeply troubling because it didn't entirely make sense.

Lie was stimulated by his knowlege, acquired during his stay in Paris (where he roomed in the same house as Klein) of the achievements of Galois, who had used group theory to precisely identify which polynomials have roots expressible in terms of radicals plus elementary arithmetical operations. To oversimplify shamelessly, Galois showed how to define a kind of symmetry group for a given polynomial, and he showed how the algebraic structure of that group determines whether the roots can expressed in terms of radicals, and if so, the form of the expression. This solved a problem which had been open since the days of the ancient Greeks.

Galois's great achievement led Lie to formulate his dream of doing for differential equations what Galois had done for polynomials. In the years after returning from Paris, while corresponding often with Klein, he began to work out a theory. One of his greatest achievements here was the discovery that even a thorny system of nonlinear differential equations has a point symmetry group defined by a linear system of DEs, which means it can be computed (although the work isn't much fun without computer assistance!). Furthermore, he showed how the resulting algebraic structure can be analyzed to obtain solutions by exploiting extra symmetries, when they exist, and he showed that his method gave a unified approach to the bewildering variety of "special tricks" which had been developed in the eighteenth and nineteenth centuries for solving various kinds of ODEs.

Thus, both Klein and Lie achieved far reaching generalizations of vast bodies of mathematics, and revealed with unprecedented clarity how geometries and methods of solving differential equations "work".

(to be continued..)

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

At about the same time that Klein and Lie were working out their programs, Schubert (the mathematican Hermann Schubert, not the composer Franz Schubert!) became intrigued by the fact that certain problems in algebraic geometry have finite answers. For example: in complex projective three-space CP^3, "how many lines meet four given lines (lacking any particular geometric relationship to each other, such as lying in a common plane; we say they are in general position)?" The answer is two, and Schubert found many other problems with finite answers and computed their solutions. Here are some examples of the kinds of statements he (or others) came up with, in what we now call enumerative geometry:

• In CP^2, a degree 4 curve has 28 bitangents (lines tangent to the curve in two points), and a degree 5 curve has 120 bitangents. (The symmetry groups of these configurations are interesting examples if the theory of finite groups!)
• In CP^2, 3264 conic (degree 2) curves are tangent to 5 given conics in GP. There is 1 conic meeting 5 points in GP, 2 conics meeting 4 points and tangent to 1 line in GP, 4 conics meeting 3 points and tangent to 2 lines in GP, 4 conics meeting 2 points and tangent to 3 lines in GP, 2 conics meeting 1 point and tangent to 4 lines in GP, and 1 conic tangent to 5 lines in GP. (This statement exhibits another duality principle.)
• In CP^3, there are 27 lines on a generic degree 3 surface (and 15, 7, or 3 lines on various special cases of degree 3 surfaces). There are 666,841,048 degree 4 surfaces tangent to 9 given degree 4 surfaces. There are 5,819,539,783,680 twisted cubic curves (i.e. nonplanar cubic curves) tangent to 9 given degree 4 surfaces.
• In CP^4, 5 lines meet 6 given 2-flats in GP.
• In CP^4, there are 2875 lines and 609,250 conic curves on a generic degree 5 hypersurface.
• In CP^5, 11,010,048 2-flats meet 9 Veronese surfaces in GP.
• In CP^6, 42 lines meet 10 given 4-flats in GP, and 462 2-flats meet 12 given 3-flats in GP.

These are all very striking statements! The trouble was that Schubert's highly original computations, while very ingenious, rested upon an idea which didn't really make sense, which he called the principle of conservation of number.

(Lest anyone doubt the urgent importance of this topic for all mankind, I note that the correct question whose answer explains the meaning of life, the universe, and everything, is: in six-dimensional complex projective space, how many lines meet ten given four-dimensional flats in general position?)

What Schubert did was basically to devise a kind of symbolic calculus in which expressions like a^3 b^2 stood for something like this: a might stand for "meets a line, with no special relations to any objects so far mentioned", while b might mean "tangent to a generic conic curve, with no special relations to any objects so far mentioned", and then a^3 b^2 would mean that we want to find all configurations of "points" in the space of lines (or degree 4 surfaces, etc.) which satisfy the combined conditions: "meets 3 lines and tangent to 2 generic conics, all in general position".

For example, consider the question: "In CP^3, how many lines meet 4 given lines in GP?" Schubert's computation of the answer rested upon the equation s1^2 = s2 + s1.1, where s1 means "meets a given line, with no special relation to any objects so far mentioned", s2 means "contains a given point, with no special relation to any objects so far mentioned", and s1.1 means "contained in a given plane, with no special relation to any objects so far mentioned". Thus s1^2 means "meets two given lines, with no special relation to each other or any other objects so far mentioned". To prove it, Schubert said: consider two coplanar lines (in CP^3, any two coplanar lines intersect in a point which is of course also in the common plane). Then a line can meet them both in two ways: it can be a third line meeting the intersection point, or it can lie in the common plane but miss that point. Thus, s1^2 = s2 + s1.1 holds when the two given lines are coplanar, which means they are not in GP. Next, and this is the key, Schubert contended that by the (unproven) "principle of conservation of number", this equation also holds when we perturb the two given lines so that they are no longer coplanar.

Similarly, according to Schubert, s2^2 = s2.2, which says the condition that a line contain two specific points (in GP) is equivalent to the condition that a line is a specific line--- namely, the unique line passing through the two given points. And s1.1^2 = s2.2, which says that the condition that a line be contained in two specific planes (in GP) is also equivalent to the condition that a line is a specific line--- namely, the unique line which is the intersection of the two given planes. And s2 s1.1. = 0, which says that the condition that a line contain a specific point (in GP) and the condition that a line be contained in a specific plane (in GP) has no solutions, since the given point won't lie in the given plane, unless these elements are not in GP at all. Thus,
s1^4 = (s2 + s1.1)^2 = s2^2 + 2 s2 s1.1 + s1.1^2 = s2.2 + 0 + s2.2 = 2 s2.2
which gives the desired conclusion: in CP^3, two lines meet four given lines in GP.

As you might expect, Schubert's leap of faith in appealing to a murky "principle" which was not only not proven by him but which-- as everyone knew, including Schubert himself-- did not always hold true was sharply criticized by other leading mathematicians, such as Study. Many decades later, in his history of geometry, Coolidge (the English mathematician, not the American president) commented that nothing in the history of geometry had proven so contentious as the principle of conservation of number.

Backing up a bit, I should point out that Hamilton's work on optics led to a very beautiful approach to classical mechanics, Hamiltonian mechanics, which is related to a landmark achievement from the dawn of the nineteenth century, Lagrangian mechanics by a magical tool called the Legendre transformation, which has a simple and beautiful geometrical interpretation which invokes projective duality in CP^2.

Meanwhile, both Lie and Klein soon realized that both the Erlangen program and Lie's theory of the symmetry of (systems of) differential equations required the development of a theory of groups which are also smooth manifolds (such that the group operations are "smooth"). This led Lie to spend many decades developing what we now call Lie groups and Lie algebras; in the latter subject, Lie was able to replicate in great generality his feat of replacing the study of "nonlinear" objects (Lie groups) with "linear" ones (Lie algebras).

At the same time, Poincare discovered beautiful relationships between hyperbolic geometry and function theory, and introduced the twin pillars of algebraic geometry; for us, the important idea is the notion of homology groups. One key point which emerged from the work of Poincare, building upon earlier work of Riemann, is that curves in CP^2 have a natural topology. Perhaps the most important topological invariant of such a curve is its genus (a concept which had been invented long ago by Euler!). The genus of a degree d plane algebraic curve is given by the formula
g= (d-1) (d-2)/2
where you might recall from Plucker's theorem bearing on the McLaurin paradox!

Around 1890, Hilbert proved several landmark theorems in the theory of invariants, and ascended to the first rank of mathematicians. One of these established a perfect correspondence between ideals of certain rings and zero sets of systems of polynomials in many variables. At the dawn of the twentieth century, he gave an address in which he gave a list of problems which predicted (with astounding accuracy) the development of twentieth century mathematics. One of these was the problem of putting Schubert's calculus on a sound footing. His student Noether applied Lie's ideas to establish a useful symmetry principle in the theory of differential equations, which later proved invaluable to physicists.

A few decades into the twentieth century, Cartan provided a beautiful unification of Kleinian geometry with Riemannian geometry, using what we now call fiber bundles. Reacting to gtr, Weyl proposed the first GUT, which was immediately shot down by Einstein as a physical theory, but which is important because it introduced the notion of a gauge theory.

Soon thereafter, Poincare's homology and Hilbert's contributions to invariant theory gave rise to cohomology and then an abstract theory, homological algebra. DeRham found a beautiful cohomology theory formulated using differential forms, and various mathematicians developed the theory of vector bundles and K theory (which related operator theory to some of the geometrical ideas we are discussing). These later found employment in physics and dynamical systems respectively.

If I might interject a global comment here: one might humorously characterize homological algebra as the subject which seeks to reduce mathematics to computations in which one adds up terms obtained by multiplying many signs (i.e. multiplying strings of +/-1). And one might humorously characterize all mathematical attacks on nonlinear objects as attempts to emulate Lie's trick of reducing the problem to a linear one. These are caricatures, but they both have a kernel of truth.

By about 1930, several leaders in the development of algebraic topology realized that cohomology provided a way to finally make sense of Schubert's notion of "principle of conservation of number": equations like s1^2 = s2 + s1.1 turned out to make perfect sense in the appropriate cohomology ring. This takes advantage of the fact that cohomology has extra structure than it's "dual", homology, namely the multiplication in the ring. Even better, this multiplication has, in the context of Schubert's work, an interpretation in terms of intersection: if J, K are submanifolds of a manifold such as a Grassmannian, we can find corresponding classes (elements of the cohomology ring) such that
[J] [K] = [J cap K]
That is, the product in the cohomology ring corresponds to intersection of (the cohomology classes of) submanifolds, and thus to combining various geometric conditions.

The following program then emerged for making sense of Schubert's computations:

• determine the compact complex topological manifold which serves as the appropriate parameter spaceof geometric configurations in the problem (e.g. the space of lines in CP^3, which happens to already be compact),
• compute its cohomology ring,
• find submanifolds in the parameter space which model the various conditions to be imposed (e.g., "meets a given line", "tangent to a given degree 4 hypersurface"),
• identify the corresponding classes in the cohomology ring,
• calculate their product in the ring,
• evaluate the integral over the parameter space to obtain the desired positive integer.

But as Schubert himself knew, this is not sufficient! As the nineteenth century geometers appreciated, the relation between algebraic and geometric concepts in algebraic geometry is muddled by phenomena of multiplicity and reality (as in real number field rather than complex number field). This means that even after doing all the work just sketched, one has much more to do:

• eliminate possible overcounting in the above computation,
• eliminate degenerate objects which might occur in the configuration sought,
• (possibly) show that all objects can be taken to be real rather than complex.

By about 1960, it was clear what needed to be done, and finally, by about 1990, most of Schubert's computations had been rigorously verified.

During this period, in a seemingly unrelated development, category theory, an even more abstract theory which had arisen, in part, from homological algebra and which serves as a kind of grand unifying theory in pure mathematics, gave rise to the theory of structors (also called "combinatorial species"), which unified the kind of enumerative counting with generating functions introduced by Euler with Polya enumeration, another approach to solving enumeration problems which arose in chemistry and which involves symmetry groups and (finite) group actions. The symmetry groups in question are oligomorphic groups, which also turn out to be intimately involved in a part of model theory concerning general relations and first order logic.

(to be continued...)

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

(Sorry, due to editing limitations the parts now appear out of order...)

And now for the sociological dimension:

If Emam (and I) are correct, many or most departments of physics are now dominated by researchers working in a subject in pure mathematics, namely superstring theory. Furthermore, many professors of mathematics are also working on that subject. This raises a sociological issue in that not since Newton's say, perhaps, have the "cultural" lines between mathematics and physics been so blurred in terms of what labels researchers wear and in which department they teach. (In another sense, commentators like Woit argue--- and I agree--- it's easy to tell the difference between pure mathematics and mathematical physics, because the last concerns, above all, testable predictions.)

As one indication of the truth of this, I note that the highest award in Mathematics (four times as rare as a Nobel Prize, and thus arguably four times more prestigious) is the Fields Medal. Witten has not and may never win the Nobel Prize in Physics, but he has been awarded a Fields Medal for his contributions to pure mathematics (which are not limited to superstring theory, incidently!). So has Kontsevich, and some other recent Fields Medals resulted from related work.

This raises the question of whether or not this anomaly will have a long term impact on academic research in math and physics, since many of these "improperly placed" researchers are young and have long researcher careers ahead of them (but in math, not in physics, unless they switch fields).

Will string theorists working in physics departments be encouraged to change fields to a subject in physics? Will they move to mathematics departments? Or, as Emam suggests, will string theorists in math and physics departments form new Departments of String Theory? I find such a prospect horrifying, since it's bad enough that so many math departments have splintered into Math and Applied Math, following five decades after a split between Math and Statistics (in many universities). To my mind, applied math, statistics, computer science, and superstring theory are all part of mathematics, and splitting math departments into four creates artificial and harmful barriers to the recognition of close connections between all these topics.

For example, category theory has given rise to topos theory, an extremely abstract subject which nonetheless is used in CS departments to study things like (I think) compiler design.

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

Ok; I thought I might be hopelessly lost on this subject, but I didn't expect to screech to a halt on the first sentenct.

A popsci article? Surely you don't mean the magazine formerly known as Popular Science, or do you? That's the only reason I looked. It sounded rather anachronistic.

I see your article catagorized as Popular Physics... Is that what you meant?

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Numbers are not case sensitive. (me)

Yes, pop sci = popular science, the genre of writing about science for general audiences.

Where you thinking of Popular Mechanics?

The popsci paper I cited is by Emam, not by me. The book from which most of Part II is drawn is by Katz, not by me.

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

First, String theorists are a small subset of physicists. The school I am at, University of Houston, has exactly zero string theorists in the few score faculty.

Not only that, but most researchers are now more in solid state or nanophysics.

This also leads to an answer to the question posed in the OP.

Most physicists will do nothing if string theory is wrong. It wont affect their lives or careers. There will be alot of discussion on the new theories, but no earth shaking changes.

Or they can simply go back to a normal life of drinking beer and doing illicit drugs. Isn't that where these ideas originated in the first place?
(just kiddin').

YCSM

One of the greatest twentieth century mathematicians was Andre Weil (not to be confused with Hermann Weyl) who among many many other things helped use cohomology to put Schubert calculus on a sound footing. I seem to recall that he quipped to his friend Auden (referring to Coleridge's method of composing Kubla Khan) that Schubert calculus had been conceived in an opium den.

But seriously: one of the points I am trying to make is that Schubert (like Heaviside) introduced a brilliant mathematical method (the utility of their methods was quickly recognized in both cases)--- these innovations just took some time to fully justify. Until that happened, their critics were quite right to continue to stress that the work was not yet solid. Shannon's great 1947 paper founding information theory (another topic which can be examined from Klein's point of view, incidently) was also not entirely rigorous, and was roundly criticized by Doob for that reason, but in this case the gaps were quickly filled, using ergodic theory, which had been developed by Birkhoff, Kolmogoroff and others in order to put certain topics in dynamical systems on a firm footing.

Having said that, I can't resist adding that one of the loveliest applications of these ideas shows how geodesics in the hyperbolic plane are related to simple continued fractions. This is only one of many connections between number theory and dynamical systems. A possibly even more striking example: prime geodesics on discrete (compact) quotients of the hyperbolic plane H^2 possess asymptotics governed by a dynamical zeta function which is similar to the asymptotic distribution of prime numbers. (This connection uses ergodic theory, which arose in the context of studying the long-term behavior of dynamical systems; this topic subsumes the most abstract subfield of dynamical systems, symbolic dynamics, which includes the topic of my own diss, on Sturmian tilings, a kind of generalized Penrose tiling.)

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

Most will carry on with what they were doing before because, as has already been mentioned, GUTs are a very small part of physics as a whole.

As for the few that are on this particular "bleeding edge" of physics, they will do what scientists have always done: Say "Oh well" and go on looking for something that works!

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Any day you wake up on "the right side of the dirt" is a good day.

T. Anderson

The OP is a nice way to put several advances in what are thought of as theoretical physics into the broader context of pure mathematics. I don't understand much of what those words really mean, but many I see come up over and over in areas of physics theory (especially relativity), so it is useful to see how they fit into a mathematical landscape (pun intended).

You might have found this thread interesting in this light:
Mathematics! Truth or Tool?

Personally, I think that the old Greek way of combining math and physics into a study of what "has to be" was a bad approach for physics, and physics really took off with Galileo and the idea that observations should be accepted as the path to truth (theory was more like a way of organizing what you see in the most unified way possible). I am seeing a bit of a return to the Greek approach, starting when Einstein told gravity how it "should" work (with amazing success, yet some rough edges around unification), and then taking that success and going way overboard into concepts like multiverses and many-worlds interpretations. I'd say it remains to be seen on which side of that divide string theory lives, but I think Chris' view that it belongs on neither side, but rather in pure mathematics, is a very interesting one indeed. That makes a lot of sense to me-- if we see it as pure mathematics, we in a sense already know how to use it, and how not to use it, in physics.

Thanks, Ken!

Elaborating: superstring theory has already proven useful for a number of things--- in pure math.

What might be the very general lines of the most likely future development? I'd like to suggest an analogy which probably won't sustain being taken too far: harmonic analysis first arose in the work of Fourier on temperature distributions, and it was immediately effective there. But suppose it hadn't. Then mathematicians would have noticed that Fourier's ideas suggested how to develop a fundemental method of analyzing some purely mathematical problems, and after another century of development of things like operator theory, with the rise of some new branch of physics (quantum physics played this role in the history of harmonic analysis), it would have turned out that harmonic analysis provided the perfect tools to solve problems involving this new kind of physics.

In other words: simply because superstring theory is universally acclaimed (by those mathematicians who have studied it) as beautiful and compelling, and because it has already proven useful in solving hard mathematical problems, I have no doubt that superstring theory will be developed even further, in ways we can hardly guess at right now, and eventually, like any great theory in mathematics, it will be applied succesfully to the sciences. Just how and when I cannot guess, but I am confident it will be applied (barring a mass die-off of Apis mellifera, or something else causing the collapse of the global economy, which would presumably result in a mass die-off of Homo sapiens, to the detriment of science, unless nonhuman scientists have been allowed to flourish beforehand).

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

y'all do realize that the bulk of mathmatics was invented to explain the physics, dont you?

I think that depends on how one defines the "bulk" of mathematics. If you simply bean count the mathematical papers, I'm pretty sure that contention would not be correct. If you weight the math by its "significance in reality", then you are pretty much talking about physics anyway, so you are unfairly selecting the math that dovetails with physics and may have been stimulated by physics. But even if you do that, then when the OP talks about "gifts from pure math to physics" (symmetry groups, gauge theories, complex analysis, etc.), it is really talking about things that were developed primarily as pure math endeavors, not motivated by physics. Of course, it kind of depends on how one defines "physics"-- our brains are conditioned by our experience in the real world, so unless we are genetically "hard-wired" to prove certain math theorems, one can never completely divorce math from physics. Similarly, physics divorced from math is purely descriptive and ambiguous. I think it's hard to identify a preferred direction of flow of information between those pursuits, but it is important to identify their different goals and standards of proof.

As Ken G said, that's going much too far.

But if you change the emphasis our positions might not be so far apart. Some points I often try to make included these:

• There has been a close interaction between "theoretical" and "applied" mathematics from the beginning,
• Some mathematical milestones resulted when analyzing a completely new phenomenon engendered a new field of "pure" mathematics; equally often, they resulted from when an unexpected application was found for a pre-existing theory from "pure" mathematics,
• At various times the relationship between "pure mathematics" and "mathematical physics" has been particularly close; examples include Newton's time and our own time,
• The relationship between number theory and application has probably never been as close as it is today,
• Ditto, for algebraic geometry and homological algebra,
• Parts of mathematics continue to become ever more abstract; but even the most abstract portions find powerful applications, and I expect this trend will only increase as science as a whole (as well as economics and public policy) becomes ever more mathematized,
• Those societies which jump most enthusiastically on the mathematical bandwagon have always been those which flourish most.

Elaborating on the next to last, one can already see the most progressive and advanced societies moving toward an economy in which all the important decisions (economical, legislative, criminal justice) are made by statistical methodology. Government by R; this is one reason why I often try to make another point: by far the most important and most neglected philosophical problem of our time is: what is probability, that we are governed by it?

Further reading:

• Some general audience attempts to survey great chunks of mathematics
  • Mark Kac and Stanislaw Ulam, Mathematics and Logic, Dover 1992 (reprint of 1968 classic); short and sweet; the authors were both leading mathematicians,
  • Ian Stewart, The Problems of Mathematics, Oxford University Press, 1987; the best recent short survey,
  • Hans Rademacher and Otto Toeplitz, The Enjoyment of Mathematics, Princeton University Press, 1957 (translation of 1933 classic; I believe there is now a very new paperback reprint); my own favorite book for conveying the flavor of mathematics; the authors were both leading mathematicians,
  • Felix Klein, Elementary Mathematics from an Advanced Standpoint, Dover, 1932 (abridged translation from volume 1 of the three volume 1932 classic); the author is the same mathematician featured above!,
  • D. Hilbert and S. Cohn-Vossen, Geometry and the Imagination, Chelsea, 1990 (translation of 1932 classic); probably the best general audience invitation to the study of geometry; Hilbert is the same mathematician featured above!,
  • E. T. Bell, Men of Mathematics, Touchstone, 1986 (reprint of 1937 classic); the author was a distinguished number theorist; in addition, he illustrates the existence problem!,
  • James R. Newman (editor), The World of Mathematics, Five Volumes, Tempus, 1988 (reprint of 1956 classic); samples popular mathematical writing by Poincare, Hardy, and many other masters,
  • E. Atlee Jackston, Perspectives on Nonlinear Dynamics, Cambridge University Press, 1991; two volumes; a delightful pictorial romp through this vast subject, full of useful (and valid!) insights,
  • David Salsburg, The Lady Tasting Tea, Freeman, 2001; a delightful and thought-provoking survey of twentieth century mathematical statistics,
• Perhaps my all-time favorite example of book with lots of beautiful and fascinating mathematics which has almost no prerequisites is Bollobas, Modern Graph Theory, GTM 184, Springer, 1988.
• A current BAUT thread on the greatest mathematical theory of all time, which is also to my mind the perfect theory.
• The R Project, for the future of all decision-making in an open-source society. (Ah, Wikipedia, what you could have been! ... sigh...)

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

Backing up a bit, in possibly the greatest paper of all time, Shannon introduced his stunningly original information theory, which posed and solved in an elegant and completely unexpected way the problem of efficient communication in the presence of noise. In this same paper, Shannon employed an earlier suggestion by Fano involving a finite projective geometry to also found a separate but closely related theory, algebraic coding theory. His ideas explained the fundamental issues in cryptanalysis, and information theory was soon shown to provide close links between statistics, statistical mechanics, and decision theory. The central quantity in Shannon's work, entropy, written H(.), was mathematically identical to an expression which had arisen decades earlier as an approximation to the "entropy" introduced in physics by Boltzmann.

In Shannon's paper, perhaps the key idea is that of choosing one of many alternatives. In other words, if we have some space of alternatives, we need some quantity which measures the information required to make a unique choice from those alternatives; this quantity is Shannon's entropy. A central role is played by the quotient law:
H(A) + H(B/A) = H(A v B)
This says that the entropy of A, the information required to identify A, plus the conditional entropy of B given A, [I]the information required to identify B, if the identity of A is known[I], equals the joint entropy of A and B, [I]the informaton required to identify A,B jointly[I]. This might remind you of Plucker's law:
d (d+3)/2 + (g-1) = d^2

Even better: suppose a group G acts on some space. Given a subset A let <A denote the pointwise stabilizer subgroup of A. Then as Galois showed
<(A u B) = <A cap <B
(stabilizer of union is intersection of stabilizers). Furthermore:

• the complexion G/<A = G.A is a homogeneous space which measures the variety of motions of A under an unknown element of G.
• the conditional complexion <A.B measures the variety of motions of B under an unknown element of G if we know the motion of A under that element.
• the joint complexion of A and B is a fiber bundle over the complexion of A, with fibers isomorphic to the conditional complexion of B given A.

In terms of dimension, this quotient law gives the formula
dim G.A + dim <A.B = dim G.(A u B)
Here:

• the dimension of the complexion G.A is analogous to the entropy H(A); it measures the information required to identify a pointwise motion of A,
• the dimension of the conditional complexion <A.B is analogous to the conditional entropy H(B/A); it measures the information required to identify a pointwise motion of B, if the simultaneous pointwise motion of A is known,
• the dimension of the joint complexion G.(A u B) is analogous to the joint entropy H(AvB); it measures the information required to identify a pointwise simultaneous motion of A and B.

As this suggests, Shannon's theory has an analog (generalization, even) formulated using group actions, in which entropy is considered to merely a numerical invariant (the dimension) underlying an algebraico-geometric object (the complexion) which measures asymmetry.

Incidentally, although G and <A are groups, G.A is usually not a group. However, if F is an equivariant map (one which respects the action of G), then the conditional complexion of F^(-1)(A) given A is always a group, in fact, a generalized Galois group. So we see once again that the ideas of Galois, Klein, Lie, and Shannon are closely related.

Near the close of the twentieth century, Witten first suggested that the emerging grand theory of superstrings might be related to enumerative geometry. In 1991, the introduction of mirror symmetry (motivated by physics, but from our perspective, pure mathematics in the mathematical subject known as superstring theory) led to revolutionary reorganization and reappreciation of the subject, and shortly thereafter, Kontsevich and Manin showed that Witten was correct. Again using reasoning suggested by physical motivations, but from our perspective as part of the purely mathematical subject of superstring theory, they found a generating function which enabled them to solve a problem in enumerative geometry which had never before been solved, by giving a kind of recursion relation.

I hope I have succeeded in sketching

• how superstring theory fits into the organic development of pure mathematics, specifically enumerative geometry,
• how it can be said to reciprocate the gift of Lie groups (from math to physics) with the gift of superstring theory (from physics to math),
• how the enduring theme of symmetry in mathematics, together with Lie theory (and many other topics essential to modern theoretical physics) and enumerative geometry arose in the context of algebraic geometry,
• how topology plays a central role in many interesting phenomena.

Last but not least, I have tried to hint how the notion of information is related to symmetry (and topology).

I have oversimplified some of the above discussion in the interests of space.

Further reading:

C. G. Gibson,
Elementary Geometry of Algebraic Curves
Cambridge University Press, 1998.

Undergraduate level; very readable.

David Cox, John Little, and Donal O'Shea,
Ideals, Varieties, and Algorithms
Springer, 1992.

Undergraduate level, one of the best books ever published.

Miles Reid,
Undergraduate Algebraic Geometry
LMS student texts 12.
Cambridge University Press, 1990.

A nice survey, with a good introduction to the Nullstellensatz, one of the fundamental results of Hilbert mentioned above.

Hal Schenck,
Computational Algebraic Geometry
LMS student texts 58.
Cambridge University Press, 2003.

Delightful introduction to homological algebra and some applications of algebraic geometry; see in particular discussion of the Syzygy theorem, another of the fundamental results of Hilbert which was mentioned above.

France Kirwan,
Complex Algebraic Curves
LMS student texts 23.
Cambridge University Press, 1992.

Moves beyond the McLaurin-Bezout theorem to Riemann surfaces, holomorphic differentials, and the Riemann-Roch theorem, a landmark result I didn't mention above for lack of space.

Joe Harris,
Algebraic Geometry.
Springer, 1992.

Graduate level; readable; of all the many textbooks on this vast subject, this is the one closest to Klein's viewpoint.

Sheldon Katz,
Enumerative Geometry and String Theory
American Mathematical Society, 2006

Lectures given to talented undergraduate students in Park City; the sketch of Schubert calculus in this post is closely modeled on Katz's book.

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

The ST (String Theory) is not a established theory, so i dont think there will be a great impact...

I do not know all the details of the ST, but this theory can be tested? any attempt to do it?, the ST seems to be the best efford to unite quantum physics with gravity and relativity, if the ST fails, it research, development and experience could be used partly on the new theories, so i think the impact will be even lower

This thread is not actually about the merits or demerits of superstring theory as a possible framework for developing theories of fundamental physical interactions, but about how it fits into the framework of pure mathematics, organized by the running theme of enumerative geometry.

Or were you simply trying to express agreement with my assertion that superstring theory, which everyone agrees comprises a body of beautiful mathematics, will one day find applications outside mathematics?

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!

An unified (whichever) physic theory give you new engineering fields, with that you can have the tools to start to playing with energy/force transformations

Im just saying the ST is not a established theory, is an unified theory, but then there is not a established unified theory, so i doubt it failure will have a big impact...with something you must begin, the ST is that "something"

Anyways, i really want to know how that theory can be tested...im not an expert, but maybe you know

It sounds like you want to criticize superstring theory as a possible framework for developing GUTs. That's not what this thread is about. Can you start your own thread , please? Thanks.

__________________
Chris Hillman

Read these PF posts. Avoid Wikipedia--- except for these versions. Read this and this suggested sticky. When asked for advice, I always say: never take advice!