söndag 12 juni 2011

Demetrios Christodoulou & Richard Hamilton.

Demetrios Christodoulou one of the Shaw Prize winners has one single article from 2008 on arXive, a whole book on 'The Formation of Black Holes in General Relativity', a book where a result which complements the stability result is proved. Namely, that a sufficiently strong flux of incoming gravitational waves leads to the formation of a black hole. Prologue 34 p, sensible for everyone. Read it!

The Formation of Black Holes in General Relativity, (monograph, 589 pp.),
EMS Monographs in Mathematics, EMS Publishing House (ISBN 978-3-03719-068-5), 2009.

Research field: Partial di fferential equations, geometric analysis, general relativity, fluid mechanics. His publications started from1970 with black holes, which also was his theme for the thesis 1971. Investigations in Gravitational Collapse and the Physics of Black Holes. So I guess he is qualified enough. Mathematical Problems of General Relativity I, 2008, The formation of shocks in 3-dimensional fluids, The Euler equations of compressible fluid flow, 2007, Recent developments in nonlinear hyperbolic PDE, 2001 etc.

On wikipedia: well known in the field of general relativity for his proof, together with Sergiu Klainerman, of the nonlinear stability of the Minkowski spacetime of special relativity in the framework of general relativity. The extraordinarily difficult proof of the stability result is laid out in detail.
  • Christodoulou, Demetrios & Klainerman, Sergiu (1993). The global nonlinear stability of the Minkowski space. Princeton: Princeton University Press. ISBN 0-691-08777-6.
  • Christodoulou, Demetrios (2000). The action principle and partial differential equations. Princeton: Princeton University Press. ISBN 0-691-04957-2.

Richard Hamilton's mathematical contributions are primarily in the field of differential geometry and more specifically geometric analysis. He is best known for having discovered the Ricci flow and suggesting the research program that ultimately led to the proof, by Grigori Perelman, of the Thurston geometrization conjecture and the solution of the Poincaré conjecture. Research field: Partial differential equations, differential geometry. (Peter Woit is also at the same institution. Algebraic Geometry, Mathematical Physics and Number Theory are other fields.) Woits blog. Not many words for the 1mill prize! The second Nobel?

Several stages of Ricci flow on a 2D manifold. Wikipedia. Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

Colombia Universitys release. Analytic number theory is the study of the distribution of prime numbers. One of the most important unsolved problems in mathematics is the Riemann hypothesis about the zeros of the Riemann zeta function, which gives a square root type error term for the number of primes in a large interval. One of the greatest applications of Grothendieck's theory of schemes is Deligne's proof of the Riemann hypothesis for L-functions for varieties over finite fields (which was first formulated by Weil). Thanks to the profound insight of Langlands, now embodied in the Langlands program: there is a sweeping vision of connections between automorphic L-functions on the one hand, and motivic L-functions, on the other. This vision encompasses the Artin and Shimura-Taniyama conjectures, both of which played a key role in Wiles' proof of Fermat last theorem. The main technique of Wiles, the deformation of Galois representations, is a new direction, now quite extensively developed

Wiles' proof of his modularity lifting theorems is a perfect illustration of p-adic techniques in number theory where the basic objects are deformation of Galois representations, congruences between modular forms, and their deep connections with special values of L-functions. Another spectacular illustration of the p-adic techniques for automorphic forms attached to higher rank reductive groups is the recent proof of the Sato-Tate conjecture. Mazur's theory of deformations of Galois representations used in Wiles' proof has been inspired by the theory of p-adic families of automorphic forms developed originally by Hida. This theory has been developing for reductive groups of higher rank and has many powerful applications for the understanding of the connections between L-functions (or p-adic L-functions) and Galois representations which are at the heart of modern research in algebraic number theory and arithmetic geometry. The theory of p-adic families has also inspired some of the new developments of p-adic Hodge theory and the so-called p-adic Langlands program which establishes a conjectural connection between p-adic Galois representations of a local field of residual characteristic p and certain p-adic representations of p-adic reductive groups. These subjects where the notion of p-adic variation is involved are advancing very quickly and a substantial breakthrough is expected in the near future.

From 1996 Oswald Veblen Prize motivations.
The Ricci flow equations were introduced to geometers by Hamilton in 1982 (“Three manifolds with positive Ricci curvature”, J. Differential Geometry 17 (1982), 255–306). These equations form a very nonlinear system of differential equations (of essentially parabolic type) for the time evolution of a Riemannian metric on a smooth manifold. The equations assert simply that the time derivative of the metric is equal to minus twice the Ricci curvature tensor. (The Ricci curvature tensor is a symmetric, rank two tensor which is obtained by a natural average of the sectional curvatures.) This flow equation can be thought of as a nonlinear heat equation for the Riemannian metric. After an appropriate, time-dependent rescaling, the static solutions are simply the Einstein metrics. In introducing the Ricci flow equations, Hamilton proved that compact, three-dimensional manifolds with positive definite Ricci curvature are diffeomorphic to spherical space forms. (These are quotients of the three-dimensional sphere by free, finite
group actions.)
... understand the nature of the singularities which arise under the flow. (Hamilton proved that singularities do not arise in three dimensions when the Ricci curvature starts out positive.)
Hamilton has come to understand the geometric constraints on the singularities which arise under the Ricci flow on a compact, threedimensional Riemannian manifold and under a related flow equation (for the “isotropic curvature tensor”) on a compact, four-dimensional manifold. This understanding has allowed him, in many cases, to classify all possible singularities of the flow. In the four-dimensional case, Hamilton was recently able to give a topological characterization of the possible singularities which arise from the isotropic curvature tensor flow if the starting metric has positive isotropic curvature tensor. The conclusion is as follows: If a singularity arises, then it can be described as a lengthening neck in the manifold whose cross-section is an embedded spherical space form with injective fundamental group. Hamilton deduced from this fact that simply connected manifolds with positive isotropic curvature are diffeomorphic to the four-dimensional sphere.
For the compact 3-manifold case, Hamilton, in a recent paper, analyzed the development of singularities in the Ricci flow by studying the evolution of stable, closed geodesics and stable, minimal surfaces under their own, compatible, geometric flows. This analysis of the flows of stable geodesics and minimal surfaces leads to a characterization of the developing singularities in terms of Ricci soliton solutions to the flow equations along degenerating, geometric subsets of the original manifold. (A Ricci soliton is a solution whose motion in time is generated by a 1-parameter group of diffeomorphisms of the underlying manifold.)
He shared the prize with Gang Tian: The basic Kähler-Einstein problem is to find necessary and sufficient conditions for the existence of a Kähler metric on a given complex manifold whose Ricci curvature is a constant multiple of the metric itself. The sign of the constant is determined by the degree of the manifold’s first Chern class. The case where the sign is negative was solved independently by Aubin and Yau, while the sign zero case (where the first Chern class vanishes) was solved by Yau in his celebrated solution to the Calabi Conjecture.

This is still today very actual.

From the Columbia University research pages:
Topology is concerned with the intrinsic properties of shapes of spaces. One class of spaces which plays a central role in mathematics, and whose topology is extensively studied, are the n dimensional manifolds. These are spaces which locally look like Euclidean n-dimensional space.

Historically, topology has been a nexus point where algebraic geometry, differential geometry and partial differential equations meet and influence each other, influence topology, and are influencedby topology. More recently, topology and differential geometry have provided the language in which to formulate much of modern theoretical high energy physics. This interaction has brought topology, and mathematics more generally, a whole host of new questions and ideas. Because of its central place in a broad spectrum of mathematics there has always been a great deal of interaction between work in topology and work in these neighboring disciplines.

Ironically, in topology, the case of manifolds of dimensions 3 and 4, the physical dimensions in which we live, has eluded undestanding for the longest time. The case of manifolds of dimension n=1 is straightforward, and the case where n=2 was understood thoroughly in the 19th century. Moreover, intense activity in the 1960's (including the pioneering work of Browder, Milnor, Novikov, and Smale) expresses the topology of manifolds of dimension n>4 in terms of an elaborate but purely algebraic description.

The study of manifolds of dimension n=3 and 4 is quite different from the higher-dimensional cases; and, though both cases n=3 and 4 are quite different in their overall character, both are generally referred to as low-dimensional topology.

Low-dimensional topology is currently a very active part of mathematics, benefiting greatly from its interactions with the fields of partial differential equations, differential geometry, algebraic geometry, modern physics, representation theory, number theory, and algebra.

The case of manifolds of dimension n=4 remains the most elusive. In view of the foundational results of Freedman, understanding manifolds up to their topological equivalence is a theory which is similar in character to the higher-dimensional manifold theory. However, the theory of differentiable four-manifolds is quite different. The subject was fundamentally transformed by the pioneering work of Simon Donaldson, who was studying moduli spaces of solutions to certain partial differential equations which came from mathematical physics. Studying algebro-topological properties of these moduli spaces, Donaldson came up with very interesting smooth invariants for four-manifolds which demonstrated the unique and elusive character of smooth four-manifold topology. In the case where the underlying manifold is Kähler, these moduli spaces also admit an interpretation in terms of stable bundles, and hence shed light on the differential topology of smooth algebraic surfaces. Since Donaldson's work, the physicists Seiberg and Witten introduced another smooth invariant of four-manifolds. Since then, the study of four-manifolds and their invariants has undergone several further exciting developments, tying them deeply with ideas from symplectic geometry and pseudo-holomorphic curves, and hence forming further bridges with algebraic and symplectic geometry, but also connecting them more closely with knot theory and three-manifold topology.

Geometry and analysis are vast fields, with myriad facets reflected differently in the leading mathematics departments worldwide. At Columbia, they are closely intertwined, with partial differential equations as the common unifying thread, and fundamental questions from several complex variables, algebraic geometry, topology, theoretical physics, probability, and applied mathematics as guiding goals.

The theory of partial differential equations, PDE, at Columbia is practically indistinguishable from its analytic, geometric, or physical contexts: the d-bar-equation from several complex variables and complex geometry, real and complex Monge-Ampère equations from differential geometry and applied mathematics, Schrodinger and Landau-Ginzburg equations from mathematical physics, and especially the powerful theory of geometric evolution equations from topology, algebraic geometry, general relativity, and gauge theories of elementary particle physics. Of particular interest are manifestations of non-linearity and curvature, long-time behavior and inherently non-perturbative aspects, formation of singularities, generalized and viscosity solutions, and global obstructions to the existence and regularity of solutions. Although real and complex differential geometry can be quite different in orientation - the latter having closer ties with algebraic geometry and number theory - both are strongly represented at Columbia.

Other less analytic aspects of the theory of partial differential equations also thrive at Columbia. Of particular importance is the theory of solitons and integrable models, with their hidden symmetries and deep geometric structures, and stochastic differential equations, with the ever growing manifestations of random phenomena.

From its PDE and differential geometry core, the group branches out for strong interactions with other groups in the department and the university, notably the groups in algebraic geometry, topology, number theory, string theory, and applied mathematics.
In arXive is so many articles, but look at this one from 2009!
arXiv:0905.4215 [ps, pdf, other]
Quaternionic Soliton Equations from Hamiltonian Curve Flows in HP^n

15 kommentarer:

  1. Ulla,

    interesting but somewhat dated yet a little evolved.
    "(Hamilton proved that singularities do not arise in three dimensions when the Ricci curvature starts out positive.)"

    "The study of manifolds of dimension n=3 and 4 is quite different from the higher-dimensional cases; and, though both cases n=3 and 4 are quite different in their overall character, both are generally referred to as low-dimensional topology."

    Surely by now you understand very much of this if only to highlight and quote intelligently.

    People make money from this? amazing!

    I still think Kea (I see no problem there and do not understand why you asked) and Pitkanen with different approaches are way beyond this despite the standard methods and terminology. Based on that alone the methods may appear limited to that terminology.

    I see no problem with Lubos stance either (I mean the terms do not show what is in TGD by sounding the same- so is string theory needed in the mix after all?) Lubos boldly claims today that quantum theory is all of physics! While that can be seen true in some logical results to absolute theory- (an Otto-Motle statistics of sorts, more like Cantor dust) there is still more to the quantum theory as we do not have all of it.

    What would his post today say about how we should see the life processes if it the only fact and model?

    Would it not follow also that in an infinite honeycomb flat space after all the resolved manifolds of whatever dimensions that no singularities would arise either? And if the hyperbolic geometry in such field resolutions can be seen in a view positive are they important on their own?

    Maybe you and the reductionist see philosophy as now irrelevant but I still need it and think they do.

    But what does it matter when we are trying ignorance again in the gold silver and bronze of Plato so failing the Teachers, Farmers, and Business Middle Class... the healing arts? Will we ever go back to the Golden Age without the cross of the Gold standards?

    The PeSla

  2. There is so much in your comment that I prefer to just say something, not directly as an answer.

    One big reason TGD has not been accepted is the 4-D and the p-adics and primes. Matti use 2x4D. Lowdimensionality has no place in M-theory. And the loops and spheres are problematic. That's why he dislike Smolin, that use loops differently from Matti. Nor is Nima Arkani-Hamed's loops as Mattis'. The essence is that sympletic algebra can be used for them all. I would really not be surprised if Matti ends up with a kind of M-theory :)

    This is just what they themselves write. Of course it is not TGD, but TGD can use this, which maybe M-theory cannot directly. It is linked to the Yang-Mills problem however.

    Why I highlighted "(Hamilton proved that singularities do not arise in three dimensions when the Ricci curvature starts out positive.)"? Of course because it then starts NEGATIVE, and think of the ZEO as negative! This is the 'nothingness'.

  3. And I seldom read what Lubos write. Not this time either, it doesn't pay back.

  4. Ulla,

    That certainty was an informative reply, thanks.

    I must admit my position on "nothingness" is more radical than such theories, Smolins baby universes is highly speculative if we ask for some mechanism not merely a default topology.

    I still cannot read Pitkanen in detail nor would I want to discourage him. Surely Lubos objectively is a control or test case if not a reality check on speculations?

    I imagine the preoccupation with compactification and sign changes and so on of six (but not 8 and above) is a rather narrow view of things.

    Beyond this we still need useful theories that do the hard work of not only counting the universal numbers on a turtle's shell but
    in the patterns read the cracks that the East divined when fire breaks it up- just as the bark of logs leads to binary or I Ching like theories.

    But this is more or less issues as physics. What do you think of how I applied this to the gene code since 74? Can you grasp that and sympathize when those in the profession see such theories as crackpot even as the evidence mounts?


  5. But you could say Pitkänens theory was simple? I really don't think so, maybe more that it is too complex. What theory Keas use is more than I can tell. I have tried to figure out, and also asked her personally. What I know is there is no biology, what is a severe drawback. I think a quantum gravity theory has to explain consciousness, and sadly Kea will never be able to do that.

    East divined in a turtle-shell is very poetic, but philosophy and poetry is another side of the story I like very much, but maybe it is better to keep away from here. Philosophy is needed for those chaotic jumps, and poetry for the breath of your skin:)

    I cannot understand your genecode, sadly so. In the code there must be also the surroundings. A sole code is nonsense, I think.

  6. BTW if as you describe it this is what they mean by LOW dimensions involved, from my view this is a Bogus and limiting distinction bases on faulty assumptions about how the continua are structured. Maybe it is better than assuming 3 and 4 are the only dimensions substantial but in a way that is a tenable position too.

    The PeSla

  7. Well, math is about limiting distinctions and assumptions, but it is almost the only tool, sadly so.

    It cannot be just Bogus, with so many excellent prizes. And they actually use p-adics too. Can you think a while why it would be necessary?

  8. Ulla, Ulla,

    Both Pitkanen's and Kea's theories are above your head in its philosphic depths as I suppose it is above most of their detractors here. Some of it has a lot in common as when we find deep connections surprising in the math and numbers. Lubos and the string model can also be seen as a tenable and viable position but in a different area. I see no reason you should say the greatly advanced Peter Rowlands is wrong but if you cannot understand his genetics mine is hopeless for you. Not that we all do not do important specialized work- as you say to include consciousness as part of the theory of everything.

    Here is a trick question really, for you and Pitkanen: How do you get around the fact that your general theory violates the third law of thermodynamics?

    The PeSla

  9. I know pretty well what TGD is up to, but I cannot understand Keas thinking, and she is bad in explaining. Non-linear braids look interesting, and her tetrychs.
    TGD is exactly the interactions of numbers and geometry AND consciousness. The trinity.

    My question was there first. "Can you think a while why it (p-adics) would be necessary?"

  10. And I have told you I have not read Rowland, just a little. The first glimse was fascinating, but I think he is not so much more than Tony Smith. He is not on my top priority list, but maybe sometimes I will read him. When I try to figure out the Platonic solids :) You know it is difficult to discuss something I have not read. That's why it is unfair of you to always refer to Rowlands.

    Although he is an university professor it doesn't mean he is right. There are big doubts on him. But I should read him first before I say something.

    I have great trust in Mattis thinking, because I have in many ways tried to 'violate' it, and he has almost always 'won the test', but not always :) His thinking is much more deep than yours or mine, or Keas. I have a glimpse of how deep it is. Can't you understand that? There is no way to simply explain the many layers, so...

    I try to show on the missing, obvious parts leading to his thinking. There are huge implicit things in TGD. This is one of these. Can you think of an Asia NOBEL given for the same thinking as is the basement for TGD? Does this tell you something?

  11. Ulla,

    OK, perhaps intuitively you dismiss something you have not read- not to lie but not offer the truth is a good policy, keeping quite so as to appear a philosopher, and I speaking the fool.

    p-adics is one of many approaches that asks philosophically why 2 and 2 is 4 and it applies to any such theory of various forms provided our enquiry is not so rigid. But it is not necessary in a world that makes more sense where there are no necessary realities.

    Are you asking me why it works? Why the world is intelligible? There is no conflict I see between the area Kea and Pitkanen is applied, if anything a convergence to possible higher ideas and they should be acknowledged.

    One theory is not verified at the expense of another- nor does propaganda make the truth- yes we should see something new in our own theories light and point it out- but the constant spamming of ones work like TGD reeks of Goerbels principles- not the lie big enough so to be believed, but the story told over and over again that it overwhelms us and our thinking.

    But I did not mean to have a conflict with you save perhaps to allow no condescending speaking as if to children. I would like a forum perhaps that presented at least once these alternative physics for consideration and response- how to design such a thing beyond the arbitrary and explosive cross linking of dialog and comments.

    But I love you little Finland, the second most connection of readers to this blog (USA the first) and feel a connection again to the rumbles in Christchurch, may no one get hurt.

    The PeSla

  12. Has I said there is a conflict in theories between Kea and Matti? No, I said they should collaborate.

    I like Lubos view that every theory has its place. It is as studying the famous elephant, each one in the own corner. "One theory is not verified at the expense of another" is also what Matti talk for. You have simply misunderstood.

    The truth takes care of itself, yes, but sometimes old dogmas are big obstacles. As is different languages.

    I think everyone would want that forum, except the leading scientists with fundings.

    To talk of lying like you do (not to lie but not offer the truth) would need some clarifying. What is a TRUTH? If we knew that this thinking and writing was completely unnecessary,just as the Catholic Church thought once. And (but the constant spamming of ones work like TGD reeks of Goerbels principles- not the lie big enough so to be believed,) What the heck is that? I don't like insinuations like that. I am a straight person.

    And I asked why p-adics was needed and necessary.

  13. http://marcofrasca.wordpress.com/2009/01/22/ricci-solitons/
    posted on Thursday, January 22nd, 2009

    I think that all of you will recognize these equations that for a Lorentzian metric are just Einstein equations in vacuum with a cosmological constant! Ricci solitons are resembling a kind of behavior of the metric under the flow that can be expanding, collapsing or static depending on the cosmological constant.

    As time goes by we learn something deeper about Einstein equations. Their very nature seems rooted in quite recent concepts coming from differential geometry and it is my personal view that whatever quantum gravity theory we will formulate, these are the questions we have to cope with.

  14. Intro to dimensions

  15. Geometry of Ricci solitons http://wwwth.mpp.mpg.de/members/strings/Ricci/ricci_files/Talks/Cao.pdf
    Tension as a matrix between wavelength and particle.

    Ricci Solitons and Einstein-Scalar Field Theory
    solutions of Euclidean-signature Einstein gravity coupled to a free massless scalar field with nonzero cosmological constant are associated to shrinking or expanding Ricci solitons.