Subject: The deadend of Ashtekar's quantization program
Date: Thu, 24 Mar 2005 23:13:54 +0200 From: Dimi Chakalov <dchakalov@surfeu.at> To: Karel Kuchar <kuchar@physics.utah.edu> CC: Abhay Ashtekar <ashtekar@gravity.phys.psu.edu>, Chris Isham <c.isham@imperial.ac.uk> Dear Professor Kuchar, On Mon, 27 Jan 2003 15:13:51 0700 (MST), you wrote: > If you reread my paper which you quote, you will see
that I
I believe have asked you to defend your insulting statement, by proving that I have been doing "superficial reading" of what you have actually said. Instead, you chose to reply in a very vague way. Let me quote from your email from Tue, 28 Jan 2003 13:09:32 0700 (MST): "The sentence which you quote "the PROPOSAL AS IT STANDS is selfcontradictory" refers to one specific preceding statement on the same page marked by a bullet. It is followed by a suggestion how `the proposal as it stands' can possibly be amended to remove the contradiction and what are the difficulties involved. What you claim in your letter to Professor Ashtekar is that I "stated that YOUR QUANTIZATION PROGRAM is selfcontradictory". Different words mean different things and fortunately litera scripta manet." My immediate reply (Wed, 29 Jan 2003 02:30:36 +0200): "I take this as an introduction to your explanation of how the proposal referred to you as [39] can possibly be amended to remove the contradiction, and what are the difficulties involved. "NB: Please send me your rigorous mathematical proof that this task can indeed be achieved." Please note that I have not yet received any proof from you. To avoid misunderstandings, let me write three points below. 1. Fact: The phrase 'the proposal as it stands' refers to Ashtekar's quantization program, Ch. 10 in his 1991 book, http://www.Goddoesnotplaydice.net/Ashtekar.html#Kuchar_1993 2. Fact: Your have not yet elaborated on your suggestions "how 'the proposal as it stands' can possibly be amended to remove the contradiction", firstly, and secondly  "what are the difficulties involved". 3. Opinion: It is my understanding that your two suggestions above were clearly showing the DEADEND of Ashtekar's quantization program. It is indeed selfcontradictory, as you *hinted* in your grqc/9304012. Again, I am respectfully urging you to defend your insulting statement by showing that I have indeed been doing "superficial reading" of what you and A. Ashtekar have actually said. Again, please elaborate on your two suggestions (cf. (2) above) and demonstrate that my reading were indeed "superficial" (cf. (3) above). Once I receive your rigorous explanation (cf. (2) above), I will do my very best to show you that "the difficulties involved" are totally insurmountable. Again, please use math, not just words. As you said, different words mean different things, but if you use math to elaborate on your two suggestions (cf. (2) above), I am definitely positively sure that you will expose the deadend of Ashtekar's quantization program. Again, please be gentlemen and defend your insulting statement from Mon, 27 Jan 2003 15:13:51 0700 (MST). This time with math, please. Looking forward to hearing from you, Yours faithfully, D. Chakalov
Note: It goes without saying that in 1993 (see grqc/9304012 mentioned above) Prof. Karel Kuchar knew very well that "the difficulties involved" (see above) are indeed insurmountable. Otherwise he himself would have fixed the contradictions in Ashtekar's quantization program. This hasn't happened in the past twelve years, for reasons explained thirteen years ago: "classical geometrodynamics does not seem to possess a natural time variable, while standard quantum theory relies quite heavily on a preferred time." The crux of the matter is solving the Hamiltonian constraint problem and the "reconstruction problem", namely, the problem of recovering a classical limit "with a nearly vanishing cosmological constant" (S. Carlip, Quantum gravity: a progress report, grqc/0108040 v1, 14 August 2001, p. 48). To quote again from S. Carlip's report (p. 33): "An original rationale for Ashtekar variables was that they simplified the Hamiltonian constraint. But this simplification requires the choice y = i, and thus a complex connection and problematic reality conditions." But the problem is of fundamental nature: there is a fundamental difference between gravity in 2+1D and 3+1D spacetime, and A. Ashtekar hasn't been able to reach the latter. More from S. Carlip (p. 24): "The Hamiltonian constraint does not merely constrain fields; it also generates transformations that leave physical states invariant, and these must somehow (Sic!  D.C.) be factored out when one forms an inner product." I believe Karel Kuchar holds an opinion on this issue (see the footnote on p. 24), and can elucidate the "observables"  if any  of Ashtekar's quantum gravity. Note that Kuchar's Perennials are timeindependent and unobservable, and they cannot facilitate any dynamical evolution that can promote Ashtekar's purely kinematical model into a viable physical theory. To sum up, the chain of tasks for A. Ashtekar is this: solve the Hamiltonian constraint, build a true physical Hilbert space, define an inner product, write down the observables, and check out if you have recovered a smooth 3D space: look around, and see as far as you can. I believe Karel Kuchar is uniquely qualified to make a 'reverse engineering' of Ashtekar's quantization program: start from a smooth 3D space, get down to the requirements for solving the Hamiltonian constraint, and show A. Ashtekar that these requirements cannot be met in principle. Karel Kuchar is optimist, as he stated above. I'm not. Recall that Ashtekar's hypothesis is explicitly backgroundindependent, and thus he cannot recover the 3+1D spacetime in principle. He would need to 'hold onto' Kuchar's Perennials to solve the "reconstruction problem" (cf. S. Carlip above), but there is no such option in his hypothesis, hence it is doomed to fail. This is my opinion, but maybe I'm wrong and Karel Kuchar has reasons to be optimistic, as he stated above. Is this "superficial reading", Professor Kuchar? Just show me your math, please. Then I will do my best to elaborate on Ashtekar's "natural Hermitian inner product", from his 'Background independent quantum gravity: A status report', grqc/0404018 v2. Meanwhile, see Ashtekar's recent
effort to define "semiclassical" quantum state which is peaked at its
classical
state, by the socalled group averaging technique [Ref.
1]. Surely Karel Kuchar can comment on these issues, which he has studied
back in 1971.
D.C.
[Ref. 1] Abhay Ashtekar, Luca Bombelli, and Alejandro Corichi, Semiclassical States for Constrained Systems, grqc/0504052 v1, 12 April 2005. "In the standard Hamiltonian descriptions
of fundamental interactions, the canonical variables are subject to constraints.
Notable examples of systems of this type are gauge theories, general relativity
and supergravity. In the gravitational case, a key question faced by any
backgroundindependent approach, such as loop quantum gravity, is whether
specific constructions used to impose constraints in the quantization procedure
lead to a theory with ‘a sufficient number of semiclassical states’. To
analyze this issue one needs a framework which spells out the precise meaning
of the term ‘semiclassical states’, introduces strategies to construct
them and provides tools to analyze their properties. The purpose of this
paper is to propose such a framework and illustrate its use with simple
examples.
"However, our restriction excludes
several important cases, most notably the Hamiltonian
constraint of general relativity.
"First, the group averaging procedure
[3] need not result in a state which has finite and positive norm. Second,
even if it does, and therefore defines a state [psi]_phy in the physical
Hilbert space H_phy, this state may not at all be semiclassical.
"Secondly, our goal here is not to
provide a complete and exhaustive analysis of the application of the group
averaging technique to construct candidate semiclassical physical states.
We only wish to present a few simple examples in the hope that the explicit
and rather intriguing results in these cases may stimulate further research."
The group averaging procedure, p.
5: "For a large class of physically interesting systems, the group averaging
method provides a technique to extract physical states starting from kinematical
ones. For a discussion of more general cases and a treatment of subtleties
p. 6: "Our task is to spell out what
we mean by a semiclassical quantum state which is ‘peaked at this classical
state’.
p. 7: "The intended meaning is intuitively
clear and, (...). However, such semiclassical states simply don’t exist
unless the class of observables is greatly restricted.
"However, there
is in fact an interesting situation in which the analogous observable is
of direct physical interest. This is provided by the quantum theory of
the EinsteinRosen waves in 4dimensional general relativity [6].
NB: see p. 7, Footnote 2 (C. Rovelli).
p. 24: "However, in the general case,
this canonical chart would not be adapted to the linearity of the phase
space, whence it would be difficult to identify coherent states in this
representation. Nonetheless, it should be possible to introduce some
notion of kinematical semiclassical states [psi]_kin (q)."
=== Subject: Re: The deadend of Ashtekar's
quantization program
Dear Karel, I don't know why you're keeping quiet. You were born in a communist country, and I believe you should really hate to keep quiet in a situation like Ashtekar's quantization program, also known as "Der Herr Der Quanten", http://cgpg.gravity.psu.edu/people/Ashtekar/articles.html I wonder what is your contribution to "100 Years of Relativity

Paddy and Alan Rendall were invited, T. Padmanabhan, Understanding Our Universe
Alan D. Rendall, The nature
of spacetime singularities
Recall Boetius: Si tacuisses, philosophus mansisses (wenn du geschwiegen hättest, so wärest du ein Philosoph geblieben). Kindest regards, Dimi On Thu, 24 Mar 2005 23:21:49 +0200, Dimi Chakalov wrote:
==== Note added on 28 March 2005: I just receive an email from The Eberly Professor of Physics and Director of the Institute for Gravitational Physics and Geometry at Penn State University A. Ashtekar! I've been trying to contact him since September 1998, and this is the first email from him ... well, not quite. See below. D.C.
Subject: away from my mail
From 3/27/05 to 4/10/05 I will be away. From Monday 28th March to Sunday
3rd April, in Pavia, Italy.
From April 3rd to 9th, at the AEI
conference:
During this time I may not be able
to access my email.
====
Dear Professor Kuchar, I sincerely hope that you are well and in good health. One whole year I've been patiently waiting to hear from you regarding the quantization program of your colleague Prof. A. Ashtekar, http://members.aon.at/chakalov/Ashtekar.html#Kuchar Please recall that I asked you to explain how the Hilbert space problem  the problem of fixing the inner product in the Hilbert space of physical states by requiring that it is both invariant under Diff(M) and conserved in *time*  can be resolved in Ashtekar's quantization program. Ten years ago, your colleague was only able to say that this is a fascinating issue, http://members.aon.at/chakalov/Shimony.html#Ashtekar You were far less poetic, of course, and claimed that his quantization program is selfcontradictory (Canonical Quantum Gravity, grqc/9304012, p. 25), http://members.aon.at/chakalov/intro.html#Ashtekar_program But one year ago, you suddenly decided to back off (please see the first URL above). Let's be honest, okay? Ashtekar's quantization program is indeed selfcontradictory. There is no background whatsoever, no metric, no fields, no nothing. Just a manifold, http://members.aon.at/chakalov/Ashtekar.html#1 NB: Neither you nor Ashtekar can fix any instant of *time* w.r.t.w. one can solve the Hilbert space problem, http://members.aon.at/chakalov/Schwarz.html#2 No way. There is a severe logical contradiction in Ashtekar's program: recall the Barber Paradox, http://members.aon.at/chakalov/ReflexiveParadoxes.html The Barber (Ashtekar) has to introduce some sort of *time* to his bare manifold, because the manifold itself cannot do this miracle (=barber's clients do not shave themselves). Do you believe he can succeed? To sum up, if you really and honestly believe that Ashtekar's quantization program is *not* selfcontradictory, please provide the proof. With math, and with scrupulous intellectual honesty, as Carlo Rovelli used to say. This was my request sent to you one year ago. Alternatively, if you cannot defend Ashtekar's quantization program and stick to your previous opinion that this program is selfcontradictory, please say it in plain flat English, and with scrupulous intellectual honesty, as Carlo Rovelli used to say. Only please don't take all this personally. It's not about you and Ashtekar. It's all about Einstein, the nature of gravity, and the 3D space, http://members.aon.at/chakalov/Thiemann.html#note I do hope to hear from you before 2005. Perhaps you can write a paper dedicated to Einstein's Annus Mirabilis, and elaborate on Ashtekar's quantization program and the Barber Paradox. Just a suggestion. Yours sincerely, Dimi Chakalov P.S. Ceterum censeo Carthaginem delenda esse.
Note: I just read the latest article by Martin Bojowald, in which he again talked about "work in progress" on the "fascinating issue" of his mentor Abby Ashtekar. Martin Bojowald knows Claus Kiefer, and is certainly aware of the problems encountered by Lee Smolin. How is he going to solve the task of choosing one out of infinitely many "surplus solutions"? Fuhgeddaboudit, Martin. You're
a young guy, enjoy your life. Try to understand the nature of
continuum.
Example: the circumference of a circle, after Archimedes.
D. Chakalov
Martin Bojowald, Loop Quantum Cosmology: Recent Progress, grqc/0402053, Thu, 12 Feb 2004 10:13:47 GMT "Work in progress includes a systematic
derivation of perturbative corrections as well as approaches to find the
physical inner product and observables.
"This representation has the same
properties as observed before for loop quantum gravity in general [20]:
There is a nonseparable Hilbert space, only holonomies but not
c
itself are promoted to welldefined operators, and the flux p
has a discrete spectrum. It is not possible to derive a coperator because
the holonomy operators are not continuous in µ at µ = 0 such
that the derivative does not exist. The flux has a discrete spectrum in
the sense that it has normalizable eigenstates µi, even though
the range of eigenvalues is continuous, the real line.
"One of the most important open issues
conceptually is that of the physical inner product and quantum observables.
(...) The meaning of the infinitely many surplus
solutions, however, and the possible role of superselection can only be
understood with the help of observables and the physical inner product."
===== Subject: The Unmoved Mover
Dear Professor Kuchar, In your 1999 article "The Problem of Time In Quantum Geometrodynamics" [Ref. 1], you wrote: "The profound message of general relativity is that spacetime does not have any fixed structure which is not dynamical but governs dynamics from outside as an unmoved mover. The problem of time is only one facet of the missing unmoved mover, and canonical quantization only a handy tool for laying bare the consequences. Unless the deep structure of the world reveals a hidden unmoved mover and thus overturns all expectations raised by general relativity, the quantum theory of the deep level is likely to encounter a problem similar to the problem of time, though in a different guise. If so, the problem of time in quantum geometrodynamics may be only a wind that precedes the storm. "Saying that much as a physicist, I have missed my last chance to pretend to be a philosopher: Si tacuisses, philosophus mansisses." To the best of my knowledge, the last sentence belongs to Boetius (Anicius Manlius Severinus Boethius), and could be translated roughly as "Should you have kept quiet, you would have stayed a philosopher". If so, why are you keeping quiet like a philosopher? I believe have sent you three very polite email messages on Fri, 27 Dec 2002 05:22:47 +0200, Fri, 27 Dec 2002 15:08:25 +0200, and Tue, 21 Jan 2003 04:34:32 +0200. The subject was "Request for opinion". I haven't heard from you regarding my request, so please see my explanatory note of Thursday, January 29, 2003 at http://members.aon.at/chakalov/Ashtekar.html#note The Unmoved Mover which you mentioned three times in your article [Ref. 1] could be the Holon of Arthur Koestler, http://members.aon.at/chakalov/faq.html It seems to me that the task we need to solve is the nature of 3D space. Please see again my note at http://members.aon.at/chakalov/Ashtekar.html#note and my guess about the nature of 3D space at http://members.aon.at/chakalov/Zurek.html Please correct me if I'm wrong. And, most importantly, please do not keep quiet, like a philosopher. I'm sure you can say much more on the Unmoved Mover of Aristotle and your Perennials. [Ref. 2] I very much hope to hear from you as a physicist. The
opinion of your colleagues will be highly appreciated, too.
With kindest regards, Dimi Chakalov
References [Ref. 1] Karel Kuchar, "The Problem of
Time In Quantum Geometrodynamics", in "The Arguments of Time", ed. by Jeremy
Butterfield, Oxford University Press, Oxford, 1999, p. 193.
[Ref. 2] Karel Kuchar (1993). Canonical
Quantum Gravity.
"Our space is threedimensional and a theory which makes
an effective use of this fact is not to be blamed.
"No perennial ever
changes along a dynamical trajectory."
=========== Subject: It's about the 3D space,
not H[Psi]=0
Dear Colleagues, I think the "mysterious time" introduced by Bill Unruh in 1988 strikes back as the mystery of 3D space, http://members.aon.at/chakalov/Kuchar.html I will appreciate your critical comments. Will keep them private and confidential. Regards, Dimi Chakalov
=========== Subject: Re: Addendum
P.S. I heard that you've been working on the consistent histories approach to quantum gravity, and some thoughts on your "Time and interpretations of quantum gravity" emerged, in some coarsegrained way, at http://Goddoesnotplaydice.net/Giachetta.html#addendum D.C. On Sat, 23 Oct 2004 07:54:34 +0300, Dimi Chakalov wrote:
Note: If you follow
the two links above, in my email to K. Kuchar from Tuesday, 26 October
2004, you might be lost in the jungle of presentday research in quantum
gravity. Let me try to keep you focused on the main issues, by referring
to Karel Kuchar's article "The Problem of Time In Quantum Geometrodynamics"
[Ref. 1].
Point #1. The difference between the intrinsic and extrinsic properties of a curved surface is based on the presumption that neither the ribs nor the fabric of the umbrella are being "appreciably stretched"; see Fig. 7.1 in [Ref. 1, p. 171]. Perhaps this seemingly innocent expression "appreciably stretched" contains too much poetry. Why can we ignore the obvious
stretching? This whole web site is about replacing the simple "flat Euclidean
space" in which the umbrella can bend on a windy day [Ref.
1, p. 170] with a new 'reference fluid' called 'global mode of spacetime'.
Yes, we can ignore the stretching of the umbrella, because the ordering
of "points" on the continuum
of the local mode of spacetime is facilitated by the global
mode of spacetime, as explained here.
The "stretching" occurs only in the global mode.
Point #2. The intrinsic metric and extrinsic curvature are not localized at a single spot, but "distributed along a threedimensional hypersurface" [Ref. 1, p. 179]. If we toss a ball, we can imagine that at some "point" from its trajectory the ball will have a single time value, instantaneous velocity, welldefined energy, etc. In canonical gravity, we substitute this instantaneous snapshot with a whole threedimensional hypersurface, and the laws of this instant are drastically different: we run into the famous manyfingered time. The energy conjugated to this brand new manyfingered time is the "energy density and its flux at all points" of the threedimensional hypersurface [Ref. 1, p. 179]. However, to "draw such a hypersurface" we must specify where each of these infinitely many points lies in a fourdimensional spacetime [ibid.], which in turns requires fixing the error with applying double standards in treating space and time in GR, as explained here. Otherwise we cannot even begin to "draw" the picture of manyfingered time. We must not do "grave injustice to spacetime covariance that underlies general relativity", as did A. Ashtekar. The intrinsic properties of 3D space are 'inside' and 'outside', but they are excluded from the basic principle of general covariance, hence presentday geometrodynamics incorporates a Newtonian notion of 3D space but operates with a manyfingered time. Hence we have 'the problem of time' but not a 'problem of 3D space', as if the intrinsic properties of 3D space could be immune to the basic rules of reparametrization invariance of GR. To fix this crucial error, I believe we need to extend the rules of active diffeomorphism invariance to include a new symmetry group called 'space invariance'. It should eliminate the notion of some absolute 3D space with foreverfixed spatial relations 'inside and outside' from Newtonian mechanics, just as we have eliminated the notion of absolute time and have replaced it with the manyfingered time. This is a whole new ball game, however. The fundamental object that can encode the dynamics in GR  the elementary increment of the universal Heraclitian time called instant  has to be placed in the global mode of spacetime, where it can not only "wiggle up and down" along its Tinvariant manyfingered time, but can also "wiggle inside and outside" along a fourth spacelike "direction", as explained here. Note that this forth spacelike "direction" corresponds to the socalled imaginary time. All this is needed for the next two steps, Point #3 and Point #4 below, because we need to expand the phase space of potential states to include all possible "unphysical" states from which one physical 4D instant will be actualized, along the universal Heraclitian time. Oneatatime. Going back to the question of where
each of these infinitely many points lies in a fourdimensional
spacetime [Ref. 1, p. 179], I believe we could answer
it by providing a new "holder" of these potential states in the
manyfingered time: the global mode of spacetime.
Point #3. Karel Kuchar doesn't like the socalled negative energy densities and negative probabilities, because the latter would imply that a particle "would not be found anywhere at all" [Ref. 1, p. 184]. Bingo! This is exactly what we need
to make all states in the the manyfingered time (global mode of spacetime)
potential.
I believe we should explore again the hints from the
complex KleinGordon
state and discover the 'alter ego' of positive mass
solutions, hence recovering the whole set of potential states.
Their dynamics has been truncated by eliminating the additional field degrees
of freedom called 'polarization' [Ref. 1, p. 186]. We
have to recover these additional filed degrees of freedom that pertain
exclusively to 4D spacetime; see S. Carlip here.
This is yet another grave injustice to Einstein's GR, which we must undo
as soon as possible. We know from STR that luxons (photons are the only
known example of this specie) do not have an alter ego [Ref.
1, p. 184], and only luxons can safely "separate"
the material and tachyonic worlds, and the "proper time" of this luxonic
world would be zero, if read by a physical clock. Hence the global mode
of spacetime could be physically manifested on a nullplane
only. It is indeed an "instant". But "before" it becomes an instant, all
potential states in the global mode of spacetime should be generated by
the proposed extended rules of active diffeomorphism invariance, which
should include a new symmetry group called 'space invariance'. Namely,
in the global mode of spacetime, the material and tachyonic worlds should
freely "wiggle" by conflating three spatial dimensions with one temporal.
I believe something similar happens in the KruskalSzekeres diagram. Here
I can only offer my hunch that, by making a maximum extended space of potential
states in the global mode of spacetime, we might see some possibility
for 'sumoverwhatever', which would eliminate all ambiguities in the presentday
approaches to quantum gravity. I see these ambiguities as 'richness', and
my gut feeling is that we should make our global mode of spacetime
'as rich as possible', and then this additional richness would hopefully
help us nail down an "instant" of fixed, alreadyactualized 4D spacetime
in the local mode of spacetime, oneatatime.
Point #4. We should
keep in mind the issue of actualization of the potential states,
which is also known as 'the emergence of time and space' in quantum gravity;
see C. Isham and J. Butterfield here. I believe
we should upgrade Einstein's GR with suitable "dynamics", in the global
mode of spacetime, of two worlds with inverted spacetime basis, and look
for some analog of the Born rule, as outlined here
and here. Instead of some "projection postulate",
we need a brand new 'cancellation mechanism' that would cancel all
unneeded potential states and allow only one set of states to be actualized
in an "instant", thus creating a 4D spacetime "instant", oneata
time. The rest are kept safe in the global mode of spacetime (resembling
the memory of our brain), ready to be
actualized if such need occurs. The "need" for particular "instant"
comes from our relational reality. It is
the ultimate "chooser", and is a genuine Perennial, since it "governs the
dynamics from outside as an unmoved mover" [Ref. 1, p.
193].
Point #5. Finally, let me quote again Karel Kuchar [Ref. 1, p. 178]: "I hope that any reader who has got this far feels cheated. (...) So the people who tried to quantize geometry surely made some slip, and they are misleading laymen and possibly themselves by the combined trickery of general relativity and quantum theory. I will try to explain here where the slip may be and what remedies have been proposed to avoid it. The problem is that none of them quite works (Kuchar, 1992)." I fully
agree. The first slip may be in our understanding of the "dynamics"
in quantum theory. If so, the remedy
has been proposed
here. I tried to
follow strictly the path of Karel Kuchar: "classical geometrodynamics does
not seem to possess a natural time variable, while standard quantum theory
relies quite heavily on a preferred time" (Kuchar,
1992). It's a bundle, and I believe its solution will require new physics.
Mind you,
96 per cent from the stuff around
us is made of some "dark" mass and "dark" energy,
so we better get used to new ideas.
It is a great pleasure to read Karel Kuchar. I think he and Chris Isham have developed schools in general relativity and quantum gravity. All we have to do is to study their schools and zoom on some subtle "poetic" details, such as the phrase "appreciably stretched" above. Regrettably, I cannot produce anything but poetryand cannot offer anything substantial. Just a bunch of speculations and gut feelings about some 'virtual geodesic path' formulation of Einstein's GR. Besides, I'm not a physicist but psychologist, and am old and biased. If you, my dear reader, are psychologist or neuroscientist, you may be baffled by the current situation in quantum gravity research. 'Die gegenwärtige Situation' looks quite murky, and you may ask a very simple question: Why would a fish need a bicycle? Well, I believe we need quantum gravity to understand how the brain works, which in turns requires understanding quantum mechanics. I haven't been able to understand quantum mechanics nor Einstein's GR, and my only hope is that we can model the universe as a human brain and develop a 'virtual geodesic path' formulation of Einstein's GR, the ultimate theory of He Who Does Not Play Dice. Only don't expect any help from the established theoretical physics community. There is too much money and politics involved with the current projects of "gravitational wave astronomy" and "the God particle", and nobody really cares about Einstein's legacy. As Max Planck stated, the future belongs to youth. If you want to win, you have to fight. Your main "weapon" is differential geometry and topology. Praise The Lord and pass the ammunition!
D. Chakalov
