User thomas riepe - MathOverflow

Grothendieck tapes

2009-10-26T11:25:34Z

Illusie mentions tape recordings of Grothendieck explaining his trace formula and more. Are they or similar recordings online? I guess, even if (what I doubt) everything he thought about that is somewhere in print, it would give an interesting insight in his way of thinking.

Where to start reading into p-adic non-abelian Hodge theory?

2010-04-12T12:23:25Z

I'm curious about Faltings' "A p-adic Simpson correspondence ". Do you know more detailed, introductory, expositions, surveys, texts of seminars on that?

Edit: Annette Werner's survey "Vector Bundles on Curves over C_p" seems to be related.

Edit: The first part of a "new approach for the p-adic Simpson correspondence, closely related to the original approach of Faltings, but also inspired by the work of Ogus and Vologodsky on an analogue in characteristic p>0". An other related article.

Edit: today new in arxiv - "Non-abelian Hodge theory for algebraic curves over characteristic p"

japanese/chinese for mathematicians?

2009-10-29T09:43:30Z

I'd like to learn to read math articles in Japanese or Chinese, but I am not interested in learning these languages from usual textbooks. Exist suitable texts, specialized for the needs for reading mathematics? What do you suggest?

I look for something similar to "Russian for the mathematician", which was very usefull when I was interested in some russian articles. In the language books I know, most of the vocabulary is irrelevant for reading mathematics, but needed terminology is missing. A collection of mathematical vocabulary and training texts with translation would be usefull. I know good books, e.g. Bowring "An introduction to modern Japanese" or Lewin "Textlehrbuch der japanischen Sprache" and could read articles about history or humanities after having read them, but not mathematics (resulting in forgetting the language by lack of training).

Edit: F. Orgogozo's dictionary. (BTW, giving the direct link did not work, app. jap./chin. characters not accepted within a url by the MO-software)

Edit: Zagier's dictionary.

"Modular forms from Feynman integrals "?

2011-06-20T18:36:36Z

I would like to learn more about the background of this talk, but found no text on that theme. Do you know more? Edit: An interesting talk by Miranda Cheng (slides).

Edit: A talk today on the theme, has anyone a text or slides?: http://www.mpim-bonn.mpg.de/de/node/4590

"Motivic structure on higher homotopy of non-nilpotent spaces" ?

2013-04-30T06:04:21Z

Has anyone an idea where one can read more about Deepam Patel's talk "Motivic structure on higher homotopy of non-nilpotent spaces" http://www.ihes.fr/~abbes/SGA/patel.html ?

Edit/Answer: The video is here: http://www.ihes.fr/~abbes/SGA/suron-kika-passe.html

What is about J. v. Neumann's "continuous geometry"?

2013-04-16T09:25:11Z

I am curious about von Neumann's "continuous geometry" ( http://press.princeton.edu/titles/6267.html ) , but found no recent text or survey on it. Does anyone know the book and would be so nice to share the impression and (if, and) how the concept of such geometries fits into contemporary tries to generalize geometry?

Answer by Thomas Riepe for Great mathematics books by pre-modern authors

2013-04-19T11:02:10Z

Heinrich Weber's 3vol Algebra (but the 3rd vol is not entirely correct). And "the short version": Miller, Blichfeld, Dickson "Theory and Appl. of finite Groups".

current status of crystalline cohomology?

2010-01-13T11:42:50Z

The great references given on Ilya's question make me wonder about the current status of the many conjectures and open questions in Illusie's survey from 1994 on crystalline cohomology. Obviously (just compare Illusie's survey from 1975 with that above or with Chambert-Loir's survey from 1998), there is very intense work on that and the connections between the various cohomology theories attacking the case "l=p". Some more recent surveys only on Fontaine's p-adic Hodge theory are already linked to in the answers to Ilya's question, Le Stum's book (Errata) covers rigid chohomology. Among the open issues mentioned in Illusie's survey are finiteness theorems, crystalline coefficients, geometric semistability, the identity of characteristic polynomials of the Frobenius of different theories,... What is the current status of these? Which new theories have been created the past decade, how fit they together and which new questions emerged?

Edit: U. Jannsen talked recently on "a refinement of crystalline cohomology by using the theory of so-called gauges as introduced earlier by Mazur and Kato and certain syntomic sheaves." Unfortunately I found no preprint on that. Edit: Jannsen on (slides) "a cohomology theory in characteristic p which reﬁnes the crystalline cohomology – and works well for torsion" and "a sheaf theory which generalizes the Dieudonné theory – and works well for torsion."

Edit: Go Yamashita talked about "La Theorie de Hodge p-adique pour varietes ouverts" avoiding Falting's almost etale extensions. Unfortunately I found no text where one can read that.

Edit: A short note by Bhargav Bhatt and Aise Johan de Jong on a shortened proof of the comparison theorem between crystalline and de Rham cohomology.

Edit: A new proof of the semistability conjecture by Beilinson and a definition of derived crystals by Gaitsgory and Rozenblyum.

Edit: A p-adic derived de Rham cohomology by Bhargav Bhatt, giving "derived de Rham descriptions of the usual period rings and related maps in p-adic Hodge theory" and "a new proof of Fontaine's crystalline conjecture and Fontaine-Jannsen's semistable conjecture".

Edit: A "a new cohomology theory in characteristic p>0, the so called F-gauge cohomology, a cohomology with values in the category of so-called F-gauges, which refines the cristalline cohomology" by Fontaine, Jannsen.

Status of Beilinson conjectures?

2013-04-06T10:41:54Z

(I hesitate to post that question here, but I received on answer on FB:) Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks currently on, look like?

What are "perfectoid spaces"?

2011-05-22T20:30:25Z

This talk is about a theory of "perfectoid spaces", which "compares objects in characteristic p with objects in characteristic 0". What are those spaces, where can one read about them?

Edit: A bit more infos can be found in Peter Scholze's seminar description and in Bhargav Bhatt's.

Edit: Peter Scholze posted yesterday this beautiful overview on the arxiv.

Edit: Peter Scholze posted today this new survey on the arxiv.

What is inter-universal geometry?

2009-10-17T10:22:21Z

I wonder what Mochizuki's inter-universal geometry and his generalisation of anabelian geometry is, e.g. why the ABC-conjecture involves nested inclusions of sets as hinted in the slides, or why such inclusion structures should be simpler if they are between categories , how that relates to F_1. It seems to me that his basic idea is that algebraic geometry has in general a kind of semantic feedback-loop, what sounds very beautifull, if it were true. His view of Grothendieck/Deligne's idea of using the section conjecture for indirect proving finiteness statements seems to me as if he would say "The first part of that is just the first jump into the feedback-loop".

Edit: A nice link was jut given in: http://mathoverflow.net/questions/106321/mochizukis-proof-and-siegel-zeros

Edit: A relatively new survey

Grothendieck's manuscript on topology

2011-11-28T14:32:44Z

Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible (+ Thanks to Jonathan Chiche who points - see his comment below - that it is not so clear if that idea was made a reality by now): http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-maths_830399

On p. 185 - 186 of the 3rd volume of Winfried Scharlau's Grothendieck biography, a handwritten text from 1986 by Grothendieck on foundations of topology, different from the concepts of topoi or tame topology, is shortly described. Scharlau doubts if it could be turned into a readable text, but perhaps someone knows the texts and has ideas about it?

Edit: Acc. to Winfried Scharlau's book, Grothendieck described his work in a letter to Jun-Ichi Yamashita as: "some altogether different foundations of 'topology', starting with the 'geometrical objects' or 'figures', rather than starting with a set of 'points' and some kind of notion of 'limit' or (equivalently) 'neighbourhoods'. Like the language of topoi (and unlike 'tame topology'), it is a kind of topology 'without points' - a direct approach to 'shape'. ... appropriate for dealing with finite spaces... the mathematics of infinity are just a way of approximating an understanding of finite agregates, whose structures seem too elusive or too hopelessly intricate for a more direct understanding (at least it has been until now)." Scharlau gives a copy of one page of the manuscript (at p. 188) and obviously has a copy of the complete text and remarks (on p. 199) that Grothendieck wrote a in 1983 letter about that theme to Z. Mebkhout.

Edit: In the meantime I could read a letter by Grothendieck about that, a summary: He started thinking from time to time about that ca. in the mid-1970's, the motivation was roughly that dissatisfaction with the usual topology which he expressed in the Esquisse, and looking at stratifications of moduli-"spaces" is his new starting point. Maybe, but not expressed in the letter or the Esquisse, the ubiquity of moduli problems in algebraic geometry (e.g. expressed in the beginning of Lafforgue's text ) is an other motivation. He describes his guiding ideas on new foundations of topology as more complicated than the guiding ideas behind the new foundations of algebraic geometry of EGA, SGA. A main test of his concepts now would be a "Dévissage"-theorem on "startified obstructions"(?) in terms of equivalences of categories. He has a precise heuristic formulation of that which helped him to find a "dévissage" corresponding to Teichmueller groups (probably what now is called "Grothendieck-Teichmueller group"?) which are related to stratifications "at infinity" of Deligne-Mumford moduli stacks.

What's about "quantum modular forms"?

2011-01-04T15:48:51Z

Do you know where one could read on "Modular Forms, K-theory and Knots"? The combination of themes sounds thrilling!

Edit: Zagier's paper on "quantum modular forms" will be published in Clay's volume dedicated to Connes anniversary.

Edit: Copies of the fascinating article circulate in the web. If asked by email, I would help finding it.

Edit: a survey + lecture notes by Ken Ono on harmonic Maass forms.

Questions about analogy between Spec Z and 3-manifolds

2009-11-04T12:40:03Z

I'm not sure if the questions make sense: Conc. primes as knots and Spec Z as 3-manifold - fits that to the Poincare conjecture? Topologists view 3-manifolds as Kirby-equivalence classes of framed links. How would that be with Spec Z? Then, topologists have things like virtual 3-manifolds, has that analogies in arithmetics?

Edit: New MFO report: "At the moment the topic of most active interaction between topologists and number theorists are quantum invariants of 3-manifolds and their asymptotics. This year’s meeting showed significant progress in the field."

Edit: "What is the analogy of quantum invariants in arithmetic topology?", "If a prime number is a knot, what is a crossing?" asks this old report.

An other such question: Minhyong Kim stresses the special complexity of number theory: "To our present day understanding, number fields display exactly the kind of order ‘at the edge of chaos’ that arithmeticians find so tantalizing, and which might have repulsed Grothendieck." Probably a feeling of such a special complexity makes one initially interested in NT. Knot theory is an other case inducing a similar impression. Could both cases be connected by the analogy above? How could a precise description of such special complexity look like and would it cover both cases? Taking that analogy, I'm inclined to answer Minhyong's question with the contrast between low-dimensional (= messy) and high-dimensional (= harmonized) geometry. Then I wonder, if "harmonizing by increasing dimensions"-analogies in number theory or the Langlands program exist.

Minhyong hints in a mail to "the study of moduli spaces of bundles over rings of integers and over three manifolds as possible common ground between the two situations". A google search produces an old article by Rapoport "Analogien zwischen den Modulräumen von Vektorbündeln und von Flaggen" (Analogies between moduli spaces of vector bundles and flags) (p. 24 here, MR). There, Rapoport describes the cohomology of such analogous moduli spaces, inspired by a similarity of vector bundles on Riemann surfaces and filtered isocrystals from p-adic cohomologies, "beautifull areas of mathematics connected by entirely mysterious analogies". (book by R., Orlik, Dat) As interesting as that sounds, I wonder if google's hint relates to the initial theme. What do you think about it? (And has the mystery Rapoport describes now been elucidated?)

Edit: Lectures by Atiyah discussing the above analogies and induced questions of "quantum Weil conj.s" etc.

This interesting essay by Gromov discusses the topic of "interestung structures" in a very general way. Acc. to him, "interesting structures" exist never in isolation, but only as "examples of structurally organized classes of structured objects", Z only because of e.g. algebraic integers as "surrounding" similar structures. That would fit to the guesses above, but not why numbers were perceived as esp. fascinating as early as greek antiquity, when the "surrounding structures" Gromov mentions were unknown. Perhaps Mochizuki has with his "inter-universal geometry" a kind of substitute in mind?

Edit: Hidekazu Furusho: "Lots of analogies between algebraic number theory and 3-dimensional topology are suggested in arithmetic topology, however, as far as we know, no direct relationship seems to be known. Our attempt of this and subsequent papers is to give a direct one particularly between Galois groups and knots."

A remark by Gromov on 4-manifolds

2012-08-05T21:29:47Z

Gromov remarks in a a survey on manifolds (p.12) that "it is hard to imagine that there are infinitely many non-diffeomorphic, but mutually homeomorphic, quotients of the hyperbolic 4-space by discrete isometry groups". What is the background of that?

Answer by Thomas Riepe for Not especially famous, long-open problems which anyone can understand

2012-07-14T11:35:17Z

N. M. Katz: "Simple Things we don't know": https://web.math.princeton.edu/~nmk/pisa16.pdf

Answer by Thomas Riepe for Fiction books about mathematicians?

2012-07-08T13:09:38Z

Definitely Maybe, where the mathematician is modeled after a famous russian mathematician.

His Master's Voice, where the main character is a mathematician.

K(F_1) = sphere spectrum?

2009-10-21T11:02:49Z

I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

Answer by Thomas Riepe for Nonstandard Methods ( or Model Theory ) and Arithmetic Geometry

2012-04-16T07:51:40Z

An ongoing conference on that: http://www.mpim-bonn.mpg.de/node/3515 (It would be great if slides or texts from it were put online)

What's about N. M. Katz' "over-world" of exp. sums?

2012-03-15T11:48:10Z

Having just read in N. M. Katz' beautiful old survey on exponential sums a d differential eq.s, I wonder what became out of his question (on p. 297 - 300) on a "general conceptual framework in which to think about the variations with p of exponential sums on arbitrary schemes of finite type over Z", which he calls "over-world"?

Do there exist modern expositions of Klein's Icosahedron?

2009-12-21T14:37:25Z

Reading Serre's letter to Gray , I wonder if now modern expositions of the themes in Klein's book exist. Do you know any?

Answer by Thomas Riepe for Most interesting mathematics mistake?

2009-10-17T14:55:58Z

Cantor's set theory - had he known the related paradoxes, he would probably not have started developing set theory.

What means "extended concepts of symmetry"?

2011-12-13T15:47:56Z

Where could one find a short description oft: "two mathematical extensions of the symmetry - to moduli spaces of sheaves and to derived categories", found here? Happen there interesting things like with the moduli spaces of pointed curves etc.?

Answer by Thomas Riepe for Books you would like to read (if somebody would just write them...)

2011-01-24T22:25:29Z

"Faltings explained" : Several of his articles are very hard to read and existing surveys on his concepts don't really fill the gap. I would like to read a book about his work, his themes, background ideas and techniques which is a readable walk through all that, something like Connes' "NCG"-book + Connes/Marcolli's "noncommutative garden".
"Morava explained" : The same as above on Morava's work, containing a (for the arithmetic geometry inclined reader) readable description of the homotopy theory background. With comments from Manin, Kontsevich and Connes, and a (sci-fi ?) chapter on how homotopy theory and number theory may mutually interfuse (e.g. through "brave new rings").
Mumford suggested in a letter to Grothendieck to publish a suitable edited selection of letters by Grothendieck to his friends, because the letters he received from him were "by far the most important things which explained your ideas and insights ... vivid and unencumbered by the customary style of formal french publications ... express(ing) succintly the essential ideas and motivations and often giv(ing) quite complete ideas about how to overcome the main technical problems ... a clear alternative (to the existing texts) for students who wish to gain access rapidly to the core of your ideas". (Found in the very beautifull 2nd collection)

Answer by Thomas Riepe for Best Algebraic Geometry text book? (other than Hartshorne)

2009-10-25T10:38:55Z

Macdonald "Algebraic geometry: Introduction to schemes" (not only about noetherian schemes), Dieudonné's two booklets with focus on the motivation and history, the first chapter in Demazure, Gabriel "Groupes algebraique I", Mumford's "red book".

Mumford suggested in a letter to Grothendieck to publish a suitable edited selection of letters from Grothendieck to his friends, because the letters he received from him were "by far the most important things which explained your ideas and insights ... vivid and unencumbered by the customary style of formal french publications ... express(ing) succintly the essential ideas and motivations and often giv(ing) quite complete ideas about how to overcome the main technical problems ... a clear alternative (to the existing texts) for students who wish to gain access rapidly to the core of your ideas". (Found in the very beautifull 2nd collection - when I got it from the library I could not stop reading in it, which happens to me rarely with such collections, despite the associated saga)

Answer by Thomas Riepe for What's the "Yoga of Motives"?

2009-10-23T23:16:25Z

I too would recommend to look into André's book very much, and several articles by Deligne, esp. "Hodge I", "Valeurs de fonctions de L et Périods Integrales", "A quoi servent les motifs?". I found Nekovar's slides and Barbieri-Viale's "Pamphlet" usefull too.

Edit: Goncalo Tabuada held a talk on "the construction of the categories of noncommutative motives (pure and mixed) in the spirit of Drinfeld Kontsevich's noncommutative algebraic geometry program. In the process, I will present the first conceptual characterization of Quillen's higher K-theory since Quillen's foundational work in the 70's" (link). Edit: New preprints (1, 2)

Edit: Nori's unpublished notes on motives.

ubiquitous modulicity?

2011-11-19T10:21:32Z

On the one hand, as mentioned here, basically "everything" in algebraic geometry could be seen in the context of "moduli problems" - on the other hand, Grothendieck's few remarks on a possible "tame topology" tell that he wondered about some general principle behind the stratifications of known moduli "spaces".

This makes me wonder:
When did re-interpretations as moduli problems turn out to be helpfull?
How is "tame topology" used then?
... and how develops tame topology, e.g. Grothendieck mentions in his "Esquisse" something on "tubular neighbourhoods of subtopoi" - what's that?
Which role play moduli problems in derived or else generalised geometry?

more on "Transalgebraic Theories" (a 19th century yoga)?

2011-10-23T10:08:17Z

Among the talks at occasion of the Galois Bicentennial, one is about "Transalgebraic Theories". Unfortunately I found only this article describing that fascinating idea as " an extremely powerful 'philosophical' principle that some mathematicians of the XIXth century seem to be well aware of. In general terms we would say that analytically unsound manipulations provide correct answers when they have an underlying transalgebraic background." Do you know more?

Edit: This text tells a few words more (e.g. "This philosophy can be linked to Kronecker’s ”Judgendtraum” and Hilbert’s twelfth problem, which seems to have remained largely misunderstood.") and refers to a manuscript "Transalgebraic Number Theory". Has someone a copy?

Hilbert's 3rd problem,number theory, motives, cyclic homology,...

2011-10-28T09:08:36Z

This talk by Jinhyun Park connects a lot of interesting themes, making me curious to read more about that. Do you know where?

Answer by Thomas Riepe for Free, high quality mathematical writing online?

2011-09-10T10:15:38Z

The website of the Leibniz award offers a free online collection: link, and there is the preprint server of the IHES.

Comment by Thomas Riepe

2013-05-16T10:49:48Z

Wonderful - thanks!

Comment by Thomas Riepe

2013-05-10T09:06:34Z

Thanks, Iker! And welcome to MO!

Comment by Thomas Riepe

2013-05-02T07:39:49Z

@David: Yes, the talk is very good, so one could see it as answer, but I wait until the preprints are free available. @Andy: Thanks!

Comment by Thomas Riepe

2013-04-30T12:00:34Z

I took the email adress from the papers on that site too, it did not work.

Comment by Thomas Riepe

2013-04-30T07:08:41Z

The email adress given in his papers (at least in those I looked up) did not work.

Comment by Thomas Riepe

2013-04-26T14:47:34Z

Yes, I found that too, but it looks to me more like a nice try of an AI system to simulate mathematical on a rhetoric level.

Comment by Thomas Riepe

2013-04-19T10:59:40Z

I never tried to read it. It would be great if you tell us more about what of his themes and sections you find most interesting to read!

Comment by Thomas Riepe

2013-04-07T09:23:07Z

Thanks too, François!

Comment by Thomas Riepe

2013-04-06T18:30:09Z

@Jonathan: I would upvote your comment if you explain it (isn't MO just another such site, specialized on mathematics?).

Comment by Thomas Riepe

2013-04-06T18:21:42Z

By the way, one of the causes of my curiosity is a similar impression as you express: Having noticed the conceptual work on BSD etc. - i.e. "the other side of the special-values-questions" -, I failed to notice similar things the "Beilinson side".

Comment by Thomas Riepe

2013-04-06T16:24:31Z

Thanks, Andreas!

Comment by Thomas Riepe

2013-04-06T16:07:40Z

@Marc and Jonathan: Yes, I had posted the question on facebook, but had so far received no answer (probably because I failed to find some recent survey or article and therefore an answer would be too simple; the question came up for me from random bedside yesterday reading some old article on these conjectures, i.e. the typical sort of things for social network sites. It is then too simple for MO, but the weight of curiosity ...).

Comment by Thomas Riepe

2013-03-15T09:01:46Z

Thanks! I had thought that idea is already made real.

Comment by Thomas Riepe

2013-02-13T00:42:03Z

@Margaret Friedland - this may be interesting for Winfried Scharlau or Leila Schneps who work on a Grothendieck bio.

Comment by Thomas Riepe

2013-02-12T15:40:13Z

@Jonny Evans and arsmath - I only tell what I perceived. As said, I do not think it is worth the effort to try to analyze that, because the interesting issue is IMO a different one. One cause of a dislike of 'star'-mathematicians by the others just comes from the selfperception of the business: If one thinks, mathematics strives for complicated proofs for special statements, one would find work like Grothendieck's very absurd. And as most mathematicians think, 'the difference' between the people comes from IQ + background knowledge, they may be upset if such causes would not show up.