On different reliability standards in current mathematical research
aa r X i v : . [ m a t h . HO ] J a n ON DIFFERENT RELIABILITY STANDARDSIN CURRENT MATHEMATICAL RESEARCH
A. SKOPENKOV
Abstract.
In this note I expose the reliability standards I share, and give examples ofdifferent reliability standards.
Contents
1. On different reliability standards 12. Reviewing peer review system 7References 91.
On different reliability standards
By ‘reliability’ I understand validity and novelty of results. This is closely related, butnot the same, as clarity. This is independent of value and importance of results.Reliability standards in mathematics change. E.g. a long time ago, oral proof was con-sidered as reliable. As proofs gradually became more complicated, there appeared the un-derstanding (and then the rule) that only written proof can be considered to be reliable ornot (and so could constitute a fair claim for a result). ‘All times are changing times, but ours is one of massive, rapid moral and mental trans-formation. Archetypes turn into millstones, large simplicities get complicated, chaos becomeselegant, and what everybody knows is true turns out to be what some people used to think.It’s unsettling. For all our delight in the impermanent, the entrancing flicker of electronics,we also long for the unalterable... Don Quixote sets out forever to kill a windmill.(U. K. Le Guin, Tales From Earthsea)
Different reliability standards coexist. So it would be nice if mathematicians and mathjournals publicly and explicitly reveal their reliability standards, by publishing their (poten-tially different) opinions on particular examples. This would allow more competent decisionsof the math community, sponsors and tax-payers to support this or other trend in mathe-matics.
It is not merely true that a creed unites men. Nay, a difference of creed unites men — solong as it is a clear difference.I am quite ready to respect another man’s faith; but it is too much to ask that I shouldrespect his doubt, his worldly hesitations and fictions, his political bargain and make-believe.(G. K. Chesterton, What’s Wrong With The World)
Moscow Institute of Physics and Technology, and Independent University of Moscow. Email: [email protected] . https://users.mccme.ru/skopenko/ .I am grateful to R. Karasev and M. Skopenkov for useful discussions. A reliable reference is a text whose results mathematicians can use without doing theauthor’s work: to check thoroughly the validity of the results and to describe properlyrelation to earlier publications. See also Remark 2.1.Journal publications practically rule the mathematical world. So peer review standardssignificantly affect standards of mathematical research. See § I omit ‘in my opinion’ for brevity.
A genius makes his own rules, but a ‘how to’ article is written by one ordinary mortalfor the benefit of another... Authors of articles such as this one know that, but in the firstapproximation they must ignore it, or nothing would ever get done. [Ha74]
Remark 1.1. (a) Among the first steps of checking the proof it is advisable to(i) write the statements of the results, together with all the definitions used, on the firstpages;(ii) structure the paper so as to separate motivations from statements and proofs, e.g.by having a separate subsection ‘motivations’ (see e.g. [Sk20o]);(iii) give formal proofs of all statements, either under a head ‘proof of such and suchnumbered statement’, or in phrases like ‘Such and such numbered statement follows from [alist of numbered statements from this or other papers]’, ‘The proof is obtained from suchand such proof [reference] by the following substitution: [replace this by that, etc.]’, ‘Theproof is analogous to such and such proof [reference]’ ;(iv) study proofs of most closely related results and send the paper to their authors forcomments (see more in Remark 1.4.a). We should at least try to understand arguments even from reliable references. However, to understandan argument omitting details on the assumption that they are already checked requires much less effort thanto check an argument including all details on the understanding that every detail might potentially be aproblem. Everybody who uses a computer program knows the big difference between a computer programand a computer program that works. Everybody who performs work for a user knows that quite some timeis required from the developer to test the result. These recommendations are based on traditions I learned, as well as on my own experience as an author,a reader, and a referee. I served as a referee for ‘Advances in Math.’, ‘Algebraic and Geometric Topology’,‘Archiv der Math.’, ‘Arnold Math. J.’, ‘Ars Combinatoria’, ‘Commentarii Math. Helvetici’, ‘Contempo-rary Math.’ AMS book series, ‘Discrete and Computational Geometry’, ‘European J. of Math.’, ‘IzvestiyaRossiyskoy Akad. Nauk’, ‘J. of Dynamics and Differential Equations’, ‘J. of Graph Theory’, ‘J. of Math.Physics’, ‘J. of Math. Analysis and Applications’, ‘J. of the Math. Society of Japan’, ‘Math. Proc., Cam-bridge Philosophical Society’ ‘Mat. Sbornik’, ‘Mat. Zametki’, ‘Proc. A of the Royal Society of Edinburgh’,‘Proc. of the American Math. Society’, ‘Revista Mat. Iberoamericana’, ‘SIAM J. on Discrete Math.’, ‘StPetersburg Math. J.’, ‘Topology and Its Application’, ‘Transformation Groups’. The latter phrase should be avoided because it might be not considered as a formal proof, when theanalogy requires too much time and efforts from a user.All this should be done instead of, or in addition to, informal phrases like ‘it is somewhat well-known to be’,or instead of some informal observations ended with ‘we obtain the following: [statement]’ (see e.g. [Sk20e,the third paragraph of footnote 4]).
N DIFFERENT RELIABILITY STANDARDS IN CURRENT MATHEMATICAL RESEARCH 3
Using unsophisticated language and sticking to statements and proofs of the main resultshelps to find (and correct) mistakes.(b) Authors sometimes first invent a complicated proof. Sometimes this happens becausethey are not familiar with simple expositions of the subject. However, a reader wants a shortproof not made artificially inaccessible by making it dependent on more complicated ideasunnecessary for the short proof (even if those ideas did appear in the authors’ approach).Freeing the proofs from complications appeared in the invention of those proofs (but notnecessary for the proofs) is a way of checking the proofs, not only of exposing them. (E.g.compare longer proofs in [KS20, § § § §
1] citing [BZ16] to [BZ16, §
1] not citing [Sk16], and [Sh18] citing [Sk16] to[BS17], [FS20] not citing [Sk16].)
Remark 1.2. (a)
Listeners [AS: and readers] are prepared to accept unstated (but hinted)generalizations much more than they are able ... to decode a precisely stated abstraction andto re-invent the special cases that motivated it in the first place. [Ha74](b)
The modern world is full of theories which are proliferating at a wrong level of gener-ality, we’re so good at theorizing, and one theory spawns another, there’s a whole industryof abstract activity which people mistake for thinking. (I. Murdoch, The Good Apprentice) (c) See [GKP, Preface] for discussion of similar issues.(d) It is advisable to postpone technical results and definitions to later (sub)sections andbring bright results to earlier (sub)sections. In my opinion, bright results and their proofs arepotentially more useful than technical versions of known constructions which so far did notyield any bright results. Here by bright results I mean non-trivial results whose statementsare accessible to mathematicians specialized in this area of mathematics, but not necessarilyin the subject of the paper.(e) When one uses a specific theory, it is advisable to explicitly state results to be provedwith the help of this theory but in terms not involving the theory (see e.g. [Sk16, Sk19,Sk20e]). This makes the application of the result accessible to mathematicians who havenot specialized in the theory. So one is motivated to study the theory and sees explicitstatements which could guide this study.Instead of the above way, some papers start exposition with details of a specific theorywhich are matter-of-fact to specialists but are much less accessible to mathematicians fromother (even close) areas. (E.g. compare [BZ16, § § B from known A and known A ⇒ B is not a research achievement.(Even it is hidden by sophisticated exposition that the novel part is deduction of B from A and A ⇒ B .) Same holds for other ‘Aristotle’ deductions (like C from A , A ⇒ B and A. SKOPENKOV B ⇒ C ). However, such deductions could be expository achievements, even important onesif B is an important statement while groups of mathematicians knowing A and knowing A ⇒ B are distant. (See e.g. the last section of arxiv version of [Sk16], [Skw, § Remark 1.3. (a) A published paper is for a much wider audience of mathematicians thanjust referees and Editors. So motivations for main results, details of the proof etc. has to bewritten in the paper, not in letters to referees and Editors.(b) A published paper is for users, not for developers. Working on details could be aninteresting task for a developer but is usually not within intents of a user.The following are good lower estimates of how hard work on details is: • the amount of time required for authors (or for other mathematicians) to make thedetails publicly available upon request of a reviewer. • the amount of text written by the authors to justify that the details need not be madepublicly available, and to reveal real or imaginary flaws in correspondence asking for details.(c) Updating the arxiv version is not considered as an indication that the published versionhas any serious gaps. Some authors previously updated arxiv versions upon my suggestions asa math reviewer, and we didn’t have any discussion about that. See e.g. arXiv:1609.06573v3,arXiv:1209.1170v4 and a forthcoming paper by D. Gugnin.(d) Discussions of a text involving a user of this text should refer (at least upon requestof the user) to the text, not to any non-existent text obtained from the discussed text bysome change (see e.g. [Sk20e, the third paragraph of footnote 4]). Of course, this need notapply to discussions of a text between developers (e.g. coauthors) of this text. Remark 1.4 (important steps to prepare a quality submission) . (a) It is advisable beforeputting a paper to arxiv to discuss it among specialists in its area. Such a pre-submissiondiscussion usually allows the author to check whether his/her results are clearly stated, new,and completely proved. This allows to improve quality of the paper.Improving quality may involve a disillusion and dissatisfaction (as a part of learning). ‘He who practises the Way (Tao) suffers daily loss of its false shine’. A dissatisfaction which might appear during such work is a natural part of improvingquality of the paper and qualifications of the author. Such work is interesting if authors(=developers) recognize the importance of learning and fulfilling the wishes of their readers(=users). Such work is annoying only if authors write under the assumption (howeverunconscious) that their work need not be useful. Improved quality of the paper publiclyavailable on arxiv improves the author’s reputation, while low quality damages it.It is advisable to put a paper on arxiv before submitting it to a journal. Without thissimple procedure there remains a possibility that the results of a paper are already (partly)known. There are many mathematicians, so what is unknown to one group can be knownto another. Putting a paper on arxiv allows including into pre-submission discussion (seethe previous paragraphs) people who work in related areas but are not in contact with theauthor. Improved quality of a paper published in a journal improves the author’s reputation,while low quality damages it even more significantly. By user I mean user (reader, reviewer, etc) of the paper, who could be developer (author, advisor, etc.)of another paper. Often users read papers for the purpose of developing other papers. This is an English translation of a citation from the Russian translation [Ch, Chapter 22] of
Chuang tzu .I am grateful to Yan Pan for informing me that the following translation (said to be by Y. Lin) seems moreacceptable to Chinese scholars: ‘He who practises the Tˆ a o, daily diminishes his doing’. In particular, thewords ‘of its false shine’ are not present either in Chuang tzu or in English or Russian translations differentfrom [Ch] and available to me. Still, because of the preceding text in [Ch, Chapter 22] I find ‘of its falseshine’ a proper commentary.
N DIFFERENT RELIABILITY STANDARDS IN CURRENT MATHEMATICAL RESEARCH 5
In a less formal pre-submission discussion it is easier to help the author, to share ideaswith him/her, and to minimize the critical part of such help. Publication of a paper onarxiv without prior discussion with a colleague means that the author expects a public, nota private approval or criticism of this colleague. The colleague might still prefer privatecriticism, but could be compelled to criticize publicly if the arxiv text contain flaws whichobstruct progress of mathematics.Journal publications practically rule the mathematical world. So writing a referee reporton a paper is a responsible task involving double-checking. In this time-consuming form itis much harder to help the author than via informal discussions.The above steps do not absolutely protect against significant flaws, see e.g. [Sk08p].However, with the above steps done, the responsibility is shared with math community.(b) During pre-submission discussion specialists in the area might send their specific sug-gestions/criticism which they consider important (below this is shortened to just ‘sugges-tions’). Then it is advisable to put on arxiv (or submit to a journal) a revised versionapproved by specialists. Of course this might not be easy to do. E.g. the authors can receiveno comments from somebody; they can disagree with some suggestions (and they cannot besure that they would not receive another stupid or essential suggestions the day after theyfinalized their work, based on previous suggestions). Hence the authors can decide to submittheir paper to arxiv/journal even if they did not(1) receive a feedback from some persons,(2) take into account some suggestions, or(3) give a chance to a person who sent feedback to learn his/her opinion whether his/hercritical remarks are properly taken into account.There is no need to mention (1), but it is fair to mention (2) or (3) in the text. E.g. • because of the way my name and work is mentioned in [MW16], I find it misleadingthat [MW16] does not mention that a pre-arxiv version of [MW16] was sent by the authorsto me, I liked the idea of proof and had important specific criticism on its realization, but[MW16] was put to arxiv without (3); cf. [Sk17o]. • because of the way M. Skopenkov’s work is mentioned in [FK17], I find it misleadingthat [FK17] does not mention that pre-arxiv versions of [FK17] were sent by the authors toM. Skopenkov, who liked the idea of proof and sent the authors important specific criticismon its realization, but [FK17] was put to arxiv without (3); the authors kindly sent someupdates to M. Skopenkov, he answered with more critical remarks and encouragement, andthe arxiv updates of [FK17] still do not mention that M. Skopenkov did not confirm thathis criticism is reasonably resolved. • see the Zentralblatt review on [Cu20].On the other hand, mathematicians would not find it misleading that [Ad18] does notmention that I had some specific criticism on the argument (their opinion might or mightnot change if they have learned of the criticism, see Remark 1.6); it is sufficient that Karimremoved upon my request my name from acknowledgements in version 3 or later. Remark 1.5 (claims and responsibility) . Here I present some motivations for the recom-mendations of Remark 1.4. A natural reaction to Remark 1.4 is as follows:
The opinions ofpeople vary on how to use arXiv. It is good if there is some diversity. Some people would puton arXiv only a finished paper which appeared in a journal, some people would put there very When I saw the arxiv papers [MW16, FK17] and journal publication [Cu20], I informed the authors ofmy opinion expressed in the bullet points and suggested to update the papers. When the correspondingupdates will appear, I would be glad to remove these examples. I am grateful to Rado Fulek for many fruitful discussions in spite of our differences on this point.
A. SKOPENKOV early preprint. Overall, a paper on arXiv may be both: already a solid work or somethingjust very early. Many mathematicians, if they find some problems in an early preprint onarXiv, would either ignore this, or send their remarks to the authors.
A research paper (on arXiv or elsewhere) can have both positive and negative impacton development of mathematics. A usual example of a negative impact is as follows. Apaper claims an interesting result but the proof is not complete enough to provide a reliablereference (see beginning of § Mathematicians often have interesting ideas and no time to develop them to providecomplete proofs. Sharing preliminary ideas in the form of claiming results has a negativeimpact described above, but could also have positive impact. Sharing preliminary ideas aspreliminary ideas is usually useful and never harmful.A clear diversity in arXiv papers, when a paper in its first lines explains whether theauthors think it is a finished paper or a very early preprint, certainly stimulates progress ofmathematics. Lack of clear diversity allows attempt, however unintentional, to both makea claim and not have responsibility for making it. With some administrative support, thisattempt can well be successful in terms of having paper published, getting a grant or a job,etc. So without clarity (provided either by authors or by their critics) the style of not caringfor users has a significant advantage of development comparative to the style of caring forusers. This has negative impact on the development of mathematics.
Remark 1.6.
Different standards concerning Remarks 1.1.a.(iii) and 1.3.b are illustratedby the following discussion of whether the paper [Ad18] is a reliable reference (see explana-tion before Remark 1.1) for a proof of the Gr¨unbaum-Kalai-Sarkaria (GKS) conjecture (cf.[Fa20]). (AS to KA, 16.01.2019)
Dear Karim,Thank you for writing an interesting paper https://arxiv.org/abs/1812.10454 (
Addedlater: this is v1 of [Ad18] . ). I am interested in learning your proof of the Gr¨unbaum-Kalai-Sarkaria conjecture, but I could not find it in § §
4. If I am missing something, couldyou let me know in which page(s) the proof is written? If not, could you update your paperadding a head ‘proof of the Gr¨unbaum-Kalai-Sarkaria conjecture’ and such a proof underthis head?Best, Arkadiy. (KA to AS, 22.01.2019; I am grateful to Karim for allowing me to publish this letter)
Hi Arkadiy, In particular, a premature claim may involve unfair competition. This said, I have to clarify that[Sk17, Sk17o] are not results of a research competitive to [MW16], but are results of several-years attemptsto help the authors recovering their gap. See the following letter of A. Skopenkov to S. Avvakumov, U.Wagner, I. Mabillard, from January 15, 2017.
When I wrote a remark to part III [added in 2020: to [AMS+] ], I realized that the idea of recovering the gapin part II [added in 2020: to [MW16] ] by smoothing, which I suggested early in 2015, works:The smooth version of the Local Disjunction Theorem 1.16 [added in 2020: from version 2 of [AMS+] ] iscorrect. Moreover, the smooth version implies (by approximation) the piecewise-smooth and then the PLversion. The same holds for the non-injective version of the metastable Local Disjunction Lemma [MW16,Lemma 10] . Thus by proving the smooth non-injective version of the metastable Local Disjunction Lemma(using vector bundles), both the gap (of inaccurate use of PL block bundles) in [MW16] can be recovered, andthe proof can be shortened.
N DIFFERENT RELIABILITY STANDARDS IN CURRENT MATHEMATICAL RESEARCH 7 this is a corollary of Theorem I, and it is somewhat well-known to be (Kalai showed thisfirst in his paper [Ka91]). I did not list all corollaries of that theorem explicitly, becausethere are too many and the derivations assuming my main theorem are done better by othersbefore me, but the Grunbaum is actually derived explicitly, as it follows from Theorem I andCorollary 4.8. (I also remark that after that corollary).Best, Karim ...(AS to KA, 5.04.2020)
Dear Karim,Thank you for your renewed interest in my critical remarks on arXiv:1812.10454. (
Addedlater: this is [Ad18] . ) I sent you all those remarks in January, 2019. I’m afraid they are nottaken into account in [v4] Tue, 2 Jul 2019. ( Added later: this v4 is [Ad18] . ) The proof ofGKS seems to be contained in (3) in p. 7, where you attribute the implication ‘Theorem I(1) ⇒ GKS’ to [Ka91] and write ‘We will give a simpler, self-contained proof of this implicationin Section 4.6’. Your paper suggests that Theorem I(1) was not known in 1991, even asa conjecture. Thus implication ‘Theorem I(1) ⇒ GKS’ could not be proved in [Ka91]. Section 4.6 contains no head ‘Proof of the implication Theorem I(1) ⇒ GKS’. Section 4.6has only one formally stated result, Corollary 4.8, which is not the implication ‘TheoremI(1) ⇒ GKS’.It would be nice if you could update arxiv version adding a proof of the implication‘Theorem I(1) ⇒ GKS’ under the name ‘Proof of the implication ‘Theorem I(1) ⇒ GKS”.If you send me a project of such an update, I am willing to read it and tell you if I havefurther questions.Best, A. Reviewing peer review system
Remark 2.1. (a) Being a reliable reference in the first approximation corresponds to ‘acceptas is’ recommendation for a peer review journal. Such a recommendation allows furtherchanges that need not be checked by a referee (unless changes are significant).(b) Sometimes ‘official’ publicly spread peer review standards differ from practice. This iseasy to see even to an outsider who does not has a vast experience of reading and/or writingreferee reports. E.g. ‘official’ peer review standards include reasonable checking of noveltyof results. However, I do not know any math journal which requires that submitted papersshould be publicly available on arxiv. If a paper is not publicly available on arxiv, then Iconsider novelty of its results not being reasonably checked. I do not know any oppositepublic statement by a journal (or a journal editor). Added later.
Indeed, the statement of Theorem I(1) is not present in [Ka91], so the implication isnot explicitly proved in [Ka91] (nor even explicitly stated there). It is still possible that [Ka91] containssomething from which the implication is clear, so that the implication should be attributed to [Ka91]. Areliable reference would prove the implication and explain why the implication is attributed to [Ka91], witha reference to a particular place in the 25-page paper [Ka91]. So when I found this gap in version 1 of [Ad18]I thought the gap is very minor. As of January 2021, this is at least 2-years gap and at least half-a-page gapin the sense of Remark 1.3.b. Still, I do not assert that the gap cannot be filled. We had some further correspondence in which Karim neither stated that according to his reliabilitystandards the paper [Ad18] is a reliable reference for a proof of the GKS conjecture, nor provided a mod-ification that would make the paper [Ad18] such a reliable reference (see though Remark 1.3.a). Most ofour letters were public, so they are available upon request. The letters will be published here only if thejudgement of this footnote is questioned and so a justification by their publication is required.
A. SKOPENKOV (c) How to modify an anonymous peer review system so that it would not be used topromote unreasonable opinions which do not stand open discussion? (See examples of suchmisuse in the last section of [Sk16] and in cases [Sk16, Sk18] of Remark 2.2.) See e.g. transparent anonymous peer review: (d) Let us make our decisions (e.g. editorial decision or distribution of credits) carefullystudying the publications in question, not blindly believing people we know.
Remark 2.2.
Here I present my own experience as an author. I omit cases before 2017 andthose cases which did not involve non-trivial discussion (i.e. those cases in which the paperwas accepted after a reasonable amount of changes required by referee).[Sk16]
A review of handling of [Sk16] in Russian Math. Surveys.
One of the referee reports misused anonymous peer review system to promote an unrea-sonable opinion which does not stand open discussion (see the last section of [Sk16]). Sincethe unreasonable report did not play a crucial role for the Editors’ decision, there is no needto justify my criticism by publishing my reply to the report.
I am grateful to an Editor S. Shlosman for publicly criticizing in [Sh18] the survey [Sk16]because this allows to see that the criticism exists and is unreasonable.
I am grateful to the Editors for publication of [Sk16] without correcting the passages crit-icized by an Editor S. Shlosman, and publishing S. Shlosman’s criticism in the same volume[Sh18]. This represents high standard of impartiality and freedom of scientific discussions.[CS16]A
A review of handling of [CS16] in Algebr. and Geom. Topol.
To appear.[CS16]M
A. Skopenkov’s review of handling of [CS16] in Moscow Math. J.
One of the referees (Alexey Zhubr) kindly disclosed his identity to facilitate the discussionof the paper. He presented important criticism and justly asked for a major revision.In response to his report made after such a revision the authors wrote to the Editors,‘The referee of our paper Alexey Zhubr kindly forwarded to us his report of 19.09.2019.We are grateful to him for a thorough reading of our paper, many specific suggestions hemade in earlier reports, and a recommendation (however reserved) to publish this paper.The critical conclusions of the latest report are not justified by references to the paper andsuggestions what could be done in a more clear way. So we can only state that we disagreewith that conclusions, that sections 3 and 4 are mostly hard to read because of complexityof the matter (not because of poor exposition), and that we invested several years to writethis paper in a clear way. In our opinion, a referee should be requested to justify his criticalconclusions only if the Editors think the conclusions could affect the acceptance of a paper.Otherwise the matter could be dropped.’Another referee presented minor but valuable critical remarks.The paper was accepted.Critical attention of Editors both to referee reports and to authors’ replies ensures highlevel of peer review.[Sk17]
A review of handling of [Sk17] in Israel J. Math.
To appear.[Sk18]
A review of handling of [Sk18] in Arnold Math. Journal. There were two initialreports on the paper. The first was positive, it contained important criticism and asked fora major revision. The second was negative, it misused anonymous peer review system topromote an unreasonable opinion which does not stand open discussion (see the last sectionof [Sk16]). In my reply to the Editors I justified the latter judgement by considering thereferee’s comments one by one. The Editors suggested a major revision (even before receivingmy criticism of the second report). I do not know whether the Editors sent my criticism to See https://scirev.org/journal/arnold-mathematical-journal/ . N DIFFERENT RELIABILITY STANDARDS IN CURRENT MATHEMATICAL RESEARCH 9 the second referee or not. I received no reply to my criticism from the second referee. Sincethe unreasonable report did not play a crucial role for the Editors’ decision, there is no needto justify my criticism by publishing my reply to the report.Critical attention of Editors both to referee reports and to authors’ replies ensures highlevel of peer review.[AKS]
A review of handling of [AKS] in Israel J. Math.
To appear.
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Invariants of graph drawings in the plane. Arnold Math. J., 6 (2020) 21–55; fullversion: arXiv:1805.10237.[Sk18o] *
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A short exposition of S. Parsa’s theorems on intrinsic linking and non-realizability.Discr. Comp. Geom.; full version: arXiv:1808.08363.[Sk19] *
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A short exposition of the Levine-Lidman example of spineless 4-manifolds,arXiv:1911.07330.[Sk20o]
A. Skopenkov.
On some results of S. Abramyan and T. Panov, arXiv:2005.11152.[Sk20e] *
A. Skopenkov.
Extendability of simplicial maps is undecidable, arXiv:2008.00492.[Skw] *
A. Skopenkov.
Whitney trick for eliminating multiple intersections, slides for talks at St Peters-burg, Brno, Kiev, Moscow, .[ST17]
A. Skopenkov and M. Tancer,
Hardness of almost embedding simplicial complexes in R d , Discr.Comp. Geom., 61:2 (2019), 452–463. arXiv:1703.06305., Discr.Comp. Geom., 61:2 (2019), 452–463. arXiv:1703.06305.