It may come as a surprise to non-mathematicians, but the level of proof required for the acceptance of a mathematical theorem has a strong social element. Contributing factors include how surprising the result is, how it fits within the reigning beliefs of the field, the status of the author, as well as the general culture of rigour within the community. Mathematics, as a human activity, is as open to distortion and fraud as any area of science.

This uncomfortable fact is largely ignored by working mathematicians, although most have anecdotal evidence of incorrect results being waved through peer-review by “friendly” editors.

It is therefore a significant achievement for this paper, which raises questions about five big results in modern mathematics, to appear in print. The author has done the field a singular service by drawing attention to these gaps and opening up a debate about what constitutes a complete proof. It remains to be seen how the mathematical community responds to these considered and detailed critiques.

The gaps identified are of varying degrees of seriousness – all undermine the published proofs, although some of the gaps have been filled by other methods. As the author points out, even the fillable gaps must be acknowledged and corrected to ensure that the literature does not mislead future research using similar arguments in situations where they can not be filled. More worryingly, some of the gaps have not been filled and therefore, as they stand, the claimed results are not proven.

I will attempt to summarize the main arguments and conclusions regarding the five gaps.

Gap 1. Donaldson’s Theorem Donaldson, Simon K. "An application of gauge theory to four dimensional topology." Journal of Differential Geometry 18.2 (1983): 279-315. http://projecteuclid.org/euclid.jdg/1214437665

This paper revolutionized the study of smooth 4-manifolds and opened up new links between theoretical physics and mathematics. The key construction is that of the moduli space of self-dual Yang-Mills connections over a compact 4-manifold.

In the paper under review, the author identifies issue with the construction of slices, in particular, the manner in which the circle action at reducible connections interferes with the perturbation necessary for transversality.

These difficulties are explained in admirable detail. While the ultimate conclusion is that later work by Freed and Uhlenbeck [3] (involving a metric perturbation argument) fills in this gap, drawing attention to these details is important to ensure that the understanding of the construction is complete.

Gap 2. The Positive Mass and Yamabe Conjectures R.Schoen, Variational Theory for the Total Scalar Curvature Functional for Riemannian Metrics and Related Topics, 1365 (1987), Topics in Calculus of Variations, Lecture Notes in Mathematics, Montecatini Terme, 120-154.

The proofs of these two famous conjectures, and the relationship between them, were a milestone in mathematics and theoretical physics. There are various proofs in different dimensions attributed to different authors. The published work, according to Bahri, is incomplete in dimensions 4 to 7 for the Positive Mass Conjecture, and dimensions 4 and 5 for the Yamabe Conjecture in the non-conformally flat case.

The gap identified is the properties of the minimal hypersurfaces constructed in the proof: in particular that they are graphs with bounded gradient. According to Bahri, these properties are claimed without proof and there is no obvious way to extract them.

Following the posting of this critique on his website, it transpired that this gap could be filled by an (unpublished) dissertation in the University of Bonn [4]. In this paper, Bahri also goes through this argument and concludes that the gap has indeed been filled by this work. However, the gap in the published proof raises many interesting questions, which are explored in some length, whose answer would contribute to the understanding of the field.

Gap 3. The 3-dim. Poincare Conjecture J.Morgan and G.Tian, Ricci Flow and the Poincare Conjecture, vol. 3, Clay Mathematics Monograph, AMS and Clay Institute, 2007.

The sensational proof of this conjecture by Grigori Perelman [5] [6] remains one of the most fascinating and disturbing episodes in modern mathematics. Posting on the internet the proof of a century-old conjecture that was being aggressively pursued by dozens of mathematicians, the reclusive outsider, Perelman, triggered an enormous international effort to fill in the details/usurp the glory (take your pick).

The ensuing debacle, memorably documented in the New Yorker

involved unprecedented levels of conflict, publication fraud, and the rejection by Perelman of both a million dollar prize and the Fields Medal.

While it is generally deemed that Perelman’s papers contained all of the ideas for the proof, at some points his arguments are sketchy and a number of mathematicians set out to fill in these details.

According to this paper, a concluding argument involving the curve shortening flow, considered in detail in the paper above, contains a serious gap. According to Bahri, this gap is not fillable by any easy correction.

Rather the completion of the proof depends upon a second approach – also suggested by Perelman – which is completed by others [1] [2]. Once again, while the overall result appears to be correct, the dangers inherent in allowing the gap to remain unacknowledged far out-weigh any embarrassment its exposure may entail for the authors.

Gap 4. Floer Homology and Seiberg-Witten Theory (and the 3-dim. Weinstein Conjecture) P.Kronheimer and T.Mrowka, Monopoles and Three-Manifolds (2007), Cambridge University Press, New Mathematical Monographs.

Since Donaldson’s spectacular use of gauge theory in smooth 4-manifold topology, the study of the moduli space of the Seiberg-Witten equations has become the dominant ingredient in many advances in smooth topology. The resulting Floer Homology has become an industry in itself, one which grows every year.

According to Bahri, the above monograph, a standard reference in this area, contains a serious gap in its construction of the moduli space. In particular, the theory is invariant under a circle action, which fixes reducible solutions and gives circles of irreducible solutions. This leads to problems in the construction of the moduli space, referred to as “point to circle” Morse relations.

After considering the analytic and dynamical aspects of these difficulties, the author concludes that these deficiencies have not been addressed and the theory, as expounded in the above monograph, is incomplete.

As a consequence, Taubes’ proof of the 3-dim. Weinstein Conjecture [7], which depends critically on the results in the monograph, is incomplete. Of the five gaps identified, this seems to be the most serious, as no alternative method of proof has been developed for this conjecture.

It is also worth pointing out that the common feature of many of these issues is the extension of finite dimensional arguments to infinite dimensional spaces and the non-compactness of associated gauge groups. These can easily lead to false conclusions, as identified in more recent work by Taubes:

Gap 5. Pseudoholomorpic Curves and Contact Structures

The final gap is only briefly discussed, as the author has published a number of papers on these difficulties. As in the previous case, it appears that a lack of understanding of “point to circle” Morse relations lies at the heart of these problems.

In summary, the paper does an exemplary job of identifying and elucidating gaps in the literature in a manner that is neither aggressive nor pejorative. Enough detail is given to high-light the concerns without overloading the reader.

Even if all of the five gaps can be filled by other methods – which does not seem to be the case at this juncture - the process of understanding the holes and completing the claimed results should be a valuable lesson in rigorous mathematics.

References [1] T.Colding and W.P.Minicozzi II, A Course in Minimal Surfaces, vol. 121, Graduate Studies in Mathematics, AMS, 2011. [2] T.Colding and W.P.Minicozzi II, Width and finite extinction time of Ricci flow, Geometry and Topology, 12 (2008), 2737-2786. [3] D.Freed and K.Uhlenbeck, Instantons and Four Manifolds, vol. 1, MSRI Publications, Springer-Verlag, New-York-Berlin-Heidelberg-Tokyo, 1984. [4] E.Kuwert, Der Minimalflächenbeweis des Positive Energy Theorem. Vorlesungsreihe des SFB 256 Nr. 14, Universität Bonn 1990. [5] G.Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, math.DG/0307245. [6] G.Perelman, Ricci flow with surgery on three-manifolds, math.DG/03031109. [7] C.H.Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117-2202.

The New Yorker article referred to has the following tidbit that is quite relevant to the recent discussion of the roles of editors in chief:

On April 13th of this year, the thirty-one mathematicians on the editorial board of the Asian Journal of Mathematics received a brief e-mail from Yau and the journal's co-editor informing them that they had three days to comment on a paper by Xi-Ping Zhu and Huai-Dong Cao titled "The Hamilton-Perelman Theory of Ricci Flow: The Poincare and Geometrization Conjectures," which Yau planned to publish in the journal. The e-mail did not include a copy of the paper, reports from referees, or an abstract. At least one board member asked to see the paper but was told that it was not available. On April 16th, Cao received a message from Yau telling him that the paper had been accepted by the A.J.M., and an abstract was posted on the journal's Web site.

...

By the end of the following week, the title of Zhu and Cao's paper on the A.J.M.'s Web site had changed, to "A Complete Proof of the Poincare and Geometrization Conjectures: Application of the Hamilton-Perelman Theory of the Ricci Flow." The abstract had also been revised. A new sentence explained, "This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow."

[The accent of Poincare does not survive editing.]

They graciously thank Bahri "for his persistent belief that these inequalities in the book were incorrect".

Perhaps the authors of the other four gaps identified by Bahri would follow suit and address the carefully considered and cogently explained issues that he has raised?

http://www.advancednonlinearstudies.com/ANLS_V15N2art.html

Preprint available on the author’s webpage:

http://www.math.rutgers.edu/~abahri/papers/five gaps.pdf

It may come as a surprise to non-mathematicians, but the level of proof required for the acceptance of a mathematical theorem has a strong social element. Contributing factors include how surprising the result is, how it fits within the reigning beliefs of the field, the status of the author, as well as the general culture of rigour within the community. Mathematics, as a human activity, is as open to distortion and fraud as any area of science.

This uncomfortable fact is largely ignored by working mathematicians, although most have anecdotal evidence of incorrect results being waved through peer-review by “friendly” editors.

It is therefore a significant achievement for this paper, which raises questions about five big results in modern mathematics, to appear in print. The author has done the field a singular service by drawing attention to these gaps and opening up a debate about what constitutes a complete proof. It remains to be seen how the mathematical community responds to these considered and detailed critiques.

The gaps identified are of varying degrees of seriousness – all undermine the published proofs, although some of the gaps have been filled by other methods. As the author points out, even the fillable gaps must be acknowledged and corrected to ensure that the literature does not mislead future research using similar arguments in situations where they can not be filled. More worryingly, some of the gaps have not been filled and therefore, as they stand, the claimed results are not proven.

I will attempt to summarize the main arguments and conclusions regarding the five gaps.

Gap 1. Donaldson’s Theorem

Donaldson, Simon K. "An application of gauge theory to four dimensional topology." Journal of Differential Geometry 18.2 (1983): 279-315.

http://projecteuclid.org/euclid.jdg/1214437665

This paper revolutionized the study of smooth 4-manifolds and opened up new links between theoretical physics and mathematics. The key construction is that of the moduli space of self-dual Yang-Mills connections over a compact 4-manifold.

In the paper under review, the author identifies issue with the construction of slices, in particular, the manner in which the circle action at reducible connections interferes with the perturbation necessary for transversality.

These difficulties are explained in admirable detail. While the ultimate conclusion is that later work by Freed and Uhlenbeck [3] (involving a metric perturbation argument) fills in this gap, drawing attention to these details is important to ensure that the understanding of the construction is complete.

Gap 2. The Positive Mass and Yamabe Conjectures

R.Schoen, Variational Theory for the Total Scalar Curvature Functional for Riemannian Metrics and Related Topics, 1365 (1987), Topics in Calculus of Variations, Lecture Notes in Mathematics, Montecatini Terme, 120-154.

The proofs of these two famous conjectures, and the relationship between them, were a milestone in mathematics and theoretical physics. There are various proofs in different dimensions attributed to different authors. The published work, according to Bahri, is incomplete in dimensions 4 to 7 for the Positive Mass Conjecture, and dimensions 4 and 5 for the Yamabe Conjecture in the non-conformally flat case.

The gap identified is the properties of the minimal hypersurfaces constructed in the proof: in particular that they are graphs with bounded gradient. According to Bahri, these properties are claimed without proof and there is no obvious way to extract them.

Following the posting of this critique on his website, it transpired that this gap could be filled by an (unpublished) dissertation in the University of Bonn [4]. In this paper, Bahri also goes through this argument and concludes that the gap has indeed been filled by this work. However, the gap in the published proof raises many interesting questions, which are explored in some length, whose answer would contribute to the understanding of the field.

Gap 3. The 3-dim. Poincare Conjecture

J.Morgan and G.Tian, Ricci Flow and the Poincare Conjecture, vol. 3, Clay Mathematics Monograph, AMS and Clay Institute, 2007.

The sensational proof of this conjecture by Grigori Perelman [5] [6] remains one of the most fascinating and disturbing episodes in modern mathematics. Posting on the internet the proof of a century-old conjecture that was being aggressively pursued by dozens of mathematicians, the reclusive outsider, Perelman, triggered an enormous international effort to fill in the details/usurp the glory (take your pick).

The ensuing debacle, memorably documented in the New Yorker

http://www.newyorker.com/magazine/2006/08/28/manifold-destiny

involved unprecedented levels of conflict, publication fraud, and the rejection by Perelman of both a million dollar prize and the Fields Medal.

While it is generally deemed that Perelman’s papers contained all of the ideas for the proof, at some points his arguments are sketchy and a number of mathematicians set out to fill in these details.

According to this paper, a concluding argument involving the curve shortening flow, considered in detail in the paper above, contains a serious gap. According to Bahri, this gap is not fillable by any easy correction.

Rather the completion of the proof depends upon a second approach – also suggested by Perelman – which is completed by others [1] [2]. Once again, while the overall result appears to be correct, the dangers inherent in allowing the gap to remain unacknowledged far out-weigh any embarrassment its exposure may entail for the authors.

Gap 4. Floer Homology and Seiberg-Witten Theory (and the 3-dim. Weinstein Conjecture)

P.Kronheimer and T.Mrowka, Monopoles and Three-Manifolds (2007), Cambridge University Press, New Mathematical Monographs.

Since Donaldson’s spectacular use of gauge theory in smooth 4-manifold topology, the study of the moduli space of the Seiberg-Witten equations has become the dominant ingredient in many advances in smooth topology. The resulting Floer Homology has become an industry in itself, one which grows every year.

According to Bahri, the above monograph, a standard reference in this area, contains a serious gap in its construction of the moduli space. In particular, the theory is invariant under a circle action, which fixes reducible solutions and gives circles of irreducible solutions. This leads to problems in the construction of the moduli space, referred to as “point to circle” Morse relations.

After considering the analytic and dynamical aspects of these difficulties, the author concludes that these deficiencies have not been addressed and the theory, as expounded in the above monograph, is incomplete.

As a consequence, Taubes’ proof of the 3-dim. Weinstein Conjecture [7], which depends critically on the results in the monograph, is incomplete. Of the five gaps identified, this seems to be the most serious, as no alternative method of proof has been developed for this conjecture.

It is also worth pointing out that the common feature of many of these issues is the extension of finite dimensional arguments to infinite dimensional spaces and the non-compactness of associated gauge groups. These can easily lead to false conclusions, as identified in more recent work by Taubes:

https://pubpeer.com/publications/E5F634A91EFD3351B2D56A792045D9

Gap 5. Pseudoholomorpic Curves and Contact Structures

The final gap is only briefly discussed, as the author has published a number of papers on these difficulties. As in the previous case, it appears that a lack of understanding of “point to circle” Morse relations lies at the heart of these problems.

In summary, the paper does an exemplary job of identifying and elucidating gaps in the literature in a manner that is neither aggressive nor pejorative. Enough detail is given to high-light the concerns without overloading the reader.

Even if all of the five gaps can be filled by other methods – which does not seem to be the case at this juncture - the process of understanding the holes and completing the claimed results should be a valuable lesson in rigorous mathematics.

References

[1] T.Colding and W.P.Minicozzi II, A Course in Minimal Surfaces, vol. 121, Graduate Studies in Mathematics, AMS, 2011.

[2] T.Colding and W.P.Minicozzi II, Width and finite extinction time of Ricci flow, Geometry and Topology, 12 (2008), 2737-2786.

[3] D.Freed and K.Uhlenbeck, Instantons and Four Manifolds, vol. 1, MSRI Publications, Springer-Verlag, New-York-Berlin-Heidelberg-Tokyo, 1984.

[4] E.Kuwert, Der Minimalflächenbeweis des Positive Energy Theorem. Vorlesungsreihe des SFB 256 Nr. 14, Universität Bonn 1990.

[5] G.Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, math.DG/0307245.

[6] G.Perelman, Ricci flow with surgery on three-manifolds, math.DG/03031109.

[7] C.H.Taubes, The Seiberg-Witten equations and the Weinstein conjecture, Geom. Topol. 11 (2007), 2117-2202.

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The New Yorker article referred to has the following tidbit that is quite relevant to the recent discussion of the roles of editors in chief:

On April 13th of this year, the thirty-one mathematicians on the editorial board of the Asian Journal of Mathematics received a brief e-mail from Yau and the journal's co-editor informing them that they had three days to comment on a paper by Xi-Ping Zhu and Huai-Dong Cao titled "The Hamilton-Perelman Theory of Ricci Flow: The Poincare and Geometrization Conjectures," which Yau planned to publish in the journal. The e-mail did not include a copy of the paper, reports from referees, or an abstract. At least one board member asked to see the paper but was told that it was not available. On April 16th, Cao received a message from Yau telling him that the paper had been accepted by the A.J.M., and an abstract was posted on the journal's Web site.

...

By the end of the following week, the title of Zhu and Cao's paper on the A.J.M.'s Web site had changed, to "A Complete Proof of the Poincare and Geometrization Conjectures: Application of the Hamilton-Perelman Theory of the Ricci Flow." The abstract had also been revised. A new sentence explained, "This proof should be considered as the crowning achievement of the Hamilton-Perelman theory of Ricci flow."

[The accent of Poincare does not survive editing.]

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http://arxiv.org/abs/1512.00699

They graciously thank Bahri "for his persistent belief that these inequalities in the book were incorrect".

Perhaps the authors of the other four gaps identified by Bahri would follow suit and address the carefully considered and cogently explained issues that he has raised?

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How To)## Are you sure you want to delete your feedback?