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rod


Total Posts: 382
Joined: Nov 2006
 
Posted: 2014-11-14 03:23
Grothendieck has died:

Alexandre Grothendieck, ou la mort d'un génie qui voulait se faire oublier

Le plus grand mathématicien du XXe siècle est mort


tbretagn


Total Posts: 276
Joined: Oct 2004
 
Posted: 2014-11-14 11:15
I was thinking yesterday about him. Very sad

Et meme si ce n'est pas vrai, il faut croire en l'histoire ancienne

polysena


Total Posts: 1064
Joined: Nov 2007
 
Posted: 2014-11-14 13:48
Interesting,tbretagn so was I, suddenly out of the blue yesterday I thought about him (quite irrational ).. and got the news today...it was nice to know he was still alive albeit silent.. now he is definitely silent... un être véritablement hors du commun mais si solitaire, si seul dans son génie  (Récoltes et Semailles)... espérons qu'il rencontre des esprits à sa pointure au Paradis des mathématiciens! Poly

Свобода - это то, что у меня внутри. (Ленинград и Кипелов - "Свобода")

pj


Total Posts: 3453
Joined: Jun 2004
 
Posted: 2014-11-14 16:21
Requiescat in Pace

OFFENDERS WILL BE TERMINATED

chiral3
Founding Member

Total Posts: 5087
Joined: Mar 2004
 
Posted: 2014-11-14 16:58
It would be interesting to find out what he has been up to. I know many people want to respect his privacy, but if the rumor-mill is accurate he has piles of notebooks.

Anyway, RIP.

Nonius is Satoshi Nakamoto. 物の哀れ

polysena


Total Posts: 1064
Joined: Nov 2007
 
Posted: 2014-11-26 15:31

A Grothendieck

 

Article in French with some late fotos (interesting material)

Article by P. Cartier on AG that is on the critical side and not just laudatio to a genius


Свобода - это то, что у меня внутри. (Ленинград и Кипелов - "Свобода")

jslade


Total Posts: 1182
Joined: Feb 2007
 
Posted: 2014-11-29 05:03
A great man for certain. If I could do a minor threadjack: is there such a thing as "algebraic topology for dummies?" I am going through Armstrong and Croom's point topology for dummies books, but the hill looks mighty steep.

"Learning, n. The kind of ignorance distinguishing the studious."

chiral3
Founding Member

Total Posts: 5087
Joined: Mar 2004
 
Posted: 2014-11-30 02:23
I am not the best person to answer this, but I asked a similar question a while back. My recollection is I was given Karoubi for k-theory and Hatcher for Algebraic Geometry

HERE

and

MAYBE THIS

Nonius is Satoshi Nakamoto. 物の哀れ

rod


Total Posts: 382
Joined: Nov 2006
 
Posted: 2014-11-30 03:38
@jslade

You may want to take a look at Prof. Robert Ghrist's work, namely, his notes on applied topology:

Elementary Applied Topology



Ghrist appears to be an interesting man. He is a mechanical engineer by training who managed to become an applied topologist. A few years ago, he did some interesting work on the algebraic topology of sensor networks.

urnash


Total Posts: 566
Joined: Sep 2006
 
Posted: 2014-11-30 20:49
Thanks everybody for the links. Very very interesting. Pity that I'll only have time to read them around 2040.

But now back to Grothendieck, he doesn't deserve someone like me hijacking his thread.

Jim Simmons is finalizing a c. 1000 page tome on Renaissance trading methods and is looking for a publisher. He was turned down by Wiley. They said he's taxing the attention span of their readers -- LongTheta

chiral3
Founding Member

Total Posts: 5087
Joined: Mar 2004
 
Posted: 2014-12-01 14:48
I remembered something else that is cool

THE QUILLEN NOTEBOOKS

Nonius is Satoshi Nakamoto. 物の哀れ

Nonius
Founding Member
Nonius Unbound
Total Posts: 12785
Joined: Mar 2004
 
Posted: 2015-03-13 10:02
Won't create a new RIP thread and this is a bit belated....one of the more brilliant guys I knew during my college days died in 2014. Anyone here ex topology know this guy?
Geoff Mess

Chiral is Tyler Durden

il_vitorio


Total Posts: 105
Joined: Aug 2014
 
Posted: 2015-03-13 15:17
I do not know him but from what it is written it hears like a pretty nice guy, hope that he is in a better place.

One of my most productive days was throwing away 1000 lines of code.

Nonius
Founding Member
Nonius Unbound
Total Posts: 12785
Joined: Mar 2004
 
Posted: 2015-03-15 15:05
he was definitely perversely high IQ and maybe a bit insane (he does strike a resemblance to the Unibomber). I remember once he got into a fist fight over mathematics.

Chiral is Tyler Durden

benji


Total Posts: 197
Joined: Feb 2005
 
Posted: 2015-06-17 19:01
A victory for mathematics: an agreement has just been reached and the (at least) 20,000 pages of A.G.'s notes (his 'scribbles') will be scanned. Timeline of the release of these is still unclear but at least progress is being made.

Rashomon


Total Posts: 202
Joined: Mar 2011
 
Posted: 2015-06-17 21:33
jslade: I think Hatcher should work on a dummy. (It worked on me, so.) He will tell you "This part will be hard" and "This part will be easy", which really helps me know what to skip, come back to, etc.

The fundamental concept I think is that of homotopy:



which can be thought of as "time" or as adding another geometric dimension X → X×[0,1].

but you find out that this "obvious" concept doesn't extend well. So you need to invent H₁, H₂, etc.





Start with topology conceived this way: start with a 1-D skeleton in ambient 2-D — ☮, the standard picture of the free group on 2 generators,



the drawing of highways in your road map,



butterfly's veins that were pressed flat in your keepsake book.






img credit. Ghrist and INPERC also have nice pictures of this kind of stuff.

And you can watch MacPherson draw the "growing ε balls".










covering spaces with other spaces*


Let's say you were trying to cover S¹ ⊂ 𝔼² with such amoeba-blobs. You could use a lot of small ones or a couple big ones, and yet they would all have in common the thing that gives rise to a winding number (whereas [0,∞) ⊂ 𝔼², [0,1] ⊂ 𝔼² and 𝔼¹ ⊂ 𝔼² don't look like that) — which is what π₁ is meant to capture. (We use S¹ because it's simpler than the Mercedes-Benz sign: in fact you could make ☮ by sticking together "S¹ cells".) π₂ captures something higher-dimensional.



















duality

AT made me realize I never "got" the duality from linear algebra, which mathematicians take for granted (it shows up in Spivak's DG1 ch 4 as taken for granted). And then you need this (and need to understand it beyond "reverse a category's arrows", imo) to get stuff like co-boundary or co-homology. A couple tricks that helped me with that part:



  1. thinking in types (like a programmer). What's the difference between int x and function() { return(int x); }?

  2. any portfolio adjustment is the movement of a covector (row vector). So is the change in any weighting scheme, or, for the ∞-dimensional case, integrating against a kernel.

  3. Seriously. Keep the data types in mind when you read something Lens spaces are one of the "weird" examples to look up. A filtration is a way of breaking a space up into pieces: (and see §0.7 of http://canyon23.net/math/1985thesis.pdf for a CT diagram to go with. Compact manifolds can be thought of as spheres with handles so S², T², T²#T², T²#T²#T², where # is knot+sum. Likewise each of these is homeomorphic to the configuration space of a bunch of popsicle sticks or paint-wiping sticks (some long) which have been bolted together à la








    .

    Crank rockers = windshield wipers work this way.
    crank rocker

    This is an example from Ghrist EAT although I believe the observation is Thurston's.




    As for fibrations Niles Johnson has all you need for the fundamental weird example. Wikipedia will talk about π: E → B and give some examples that might make sense to a physicist. (How about wind vectors over the USA? Then take a "section" Chicago ⊂ USA; this induces a natural definition for "the wind over Chicago".) He has a short paper on Hopf fibration using (and explaining) quaternions, and a 1-hr undergrad. math. club lecture. But mainly watch his 3-minute rock 'n' roll SAGE video for the pictures of indexing a bunch of S¹'s by an S². The video also shows how ℝ is one of the circles: the old "circle with infinite diameter" trick.






    E.A.T. is a wonderful book, as is Rob's student's thesis, but it's actually not as easy despite the title. I would recommend co-reading it for sure.



    AT is really elementary and simple, but it's simple the way Arch Linux is, or riding a bicycle: not easy or quick to learn, but once you've got it it sticks for good.





    HTH. Hatcher's is in my opinion a close-to perfect or perfect book (given the choice of topic).







    Does it have any applications to trading? Well, I remember the "Kolmogorov-Smirnov" thread where you mentioned Kalman filters or something as basically like a geek fantasy: I know some complicated thing therefore money. Unless markets are more interconnected than I think, the invariants from AT are probably not quite useful. But learning some simple outside math does wonders for taking away the geek power of the reverse Kolmogorov-Smirnov filter.


    FWIW, although the Gunnar Carlson comp-top stuff made a big splash academically, I personally do not think it has paid dividends with industrial applications. Sure, it's huge to make point-set topology computable, but imho (and I think Ghrist agrees) point-set covers too many aberrations. The invariants of AT may be useful somewhere but I think maybe more as a "the dark mansion has the lights on now" kind of thing. For example the free group is the biggest—ok, good to know that there is a biggest.





    Added: Ghrist agrees, the point-set stuff is extra hard because C° is so full of pathologies. It really makes it less interesting. Yet when outsiders look at topology it's always the Munkres type GT stuff first: axiomatic, and impossible to grasp any kind of gist or value.


Added: youtu.be/6eo9UhJnHZg ←cohomology as an algebraic invariant on surfaces. A map between shapes induces a pullback on algebra. Using it to prove that reversing the orientation of ℂℙ³ can't be diffeomorphic (easier to show in the cohomology of ℂℙ³ since it's a reduced form of the shape)

Added 2: youtu.be/NWMAR8Pb11c ←∃ some easier examples of coproduct when Rota applied the concept to combinatorics

"My hands are small, I know, but they're not yours, they are my own. And they're, not yours, they are my own." ~ Jewel

jslade


Total Posts: 1182
Joined: Feb 2007
 
Posted: 2015-06-20 09:06
Re: Ghrist: I somehow missed this suggestion, though I already had the book. Kind of data topology comix.

Wow, Rashomon, thanks for taking the time to make that nice post with diagrams and everything. Very generous of you.

I've got Hatcher in PDF form somewhere, but it gives me the heebie jeebies. Maybe I will grab a paper copy and go to town on it. FWIIW, Nash and Sen's "Topology and Geometry for Physicists" makes some things intelligible. They use examples which people of my educational handicaps find completely obvious. I'm not sure I actually learned anything, but when I talk to people who know topology, they don't look at me like I am completely retarded.

As for Gunnar's toys: there are actually a few use cases where they do things which can't be done otherwise.

"Learning, n. The kind of ignorance distinguishing the studious."

Maggette


Total Posts: 1138
Joined: Jun 2007
 
Posted: 2015-06-20 12:25
Hi,

I am planing to have alook at topology and especially computational topolgy as well. And I am probably the guy with the worst pen and paper math background ever on this phorum.

Thanks for the suggestions.

To me the following was recommended in my situation:



Computational Topology: An Introduction by Edelsbrunner and Harer

Ich kam hierher und sah dich und deine Leute lächeln, und sagte mir: Maggette, scheiss auf den small talk, lass lieber deine Fäuste sprechen...

Nonius
Founding Member
Nonius Unbound
Total Posts: 12785
Joined: Mar 2004
 
Posted: 2015-07-02 17:44
Nice, I'll take a look at that.

For people with some background in manifolds, Steenrod's Fiber Bundles is a good intro on the topic. For Homotopy Theory, Whitehead. Once you immerse (no pun intended) in a lot of this stuff, you'll see a lot of it is very category theoretical.

Chiral is Tyler Durden

Rashomon


Total Posts: 202
Joined: Mar 2011
 
Posted: 2015-07-03 06:18
jslade, not a worry. Consider it a minor repayment for many of your useful posts over years.

I saw some MIT class only get through 2 chapters of Hatcher in a semester -- if that gives you an idea of the time scope. I go in with the assumption that any math. reading will take 20x to 100x longer than fiction or something, like it was at least one sit-down to understand the retraction part. In other words I mean "For patient dummies".

www.math.upenn.edu/~ghrist/preprints/ATSN.pdf (huge PDF) is a nice visual tour also if you like pictures. Also maybe have a peek at https://tlovering.files.wordpress.com/2011/04/sheaftheory.pdf (the egg part) for why this all makes sense, and www.math.upenn.edu/~ghrist/preprints/barcodes.pdf for an overview of the concept, and like I mentioned http://inperc.com/wiki/index.php?title=Differential_forms which does things more with computer vision. And http://www.millersville.edu/~rumble/slides/MU-F&M4-4-2013.pdf

I'm very interested if you think the Carlsson stuff really enables something useful. You mind sharing what?

"My hands are small, I know, but they're not yours, they are my own. And they're, not yours, they are my own." ~ Jewel

benji


Total Posts: 197
Joined: Feb 2005
 
Posted: 2015-07-26 17:50
Very interesting article about Grothendieck's Tōhoku paper

FatChoi


Total Posts: 126
Joined: Feb 2008
 
Posted: 2015-07-28 02:04
For Algebraic Topology I would put in a word for anything by John Milnor. In particular, for a manifold oriented introduction
Topology from the differentiable viewpoint for lucidity, brevity, narrative drive and links to more classical mathematics
Morse theory for more examples and calculations
and
Characteristic classes for interpretation of cohomology groups and links with K Theory
The lectures forming the basis of Topology from the differentiable viewpoint are now on you tube thanks to the Simons foundation.

Nonius
Founding Member
Nonius Unbound
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Posted: 2015-07-28 16:37
I liked Milnors Morse Theory notes.

Chiral is Tyler Durden

LongTheta
The Snowman

Total Posts: 3149
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Posted: 2015-07-29 11:57
Anyone can read Milnor.

Time is on my side.

chiral3
Founding Member

Total Posts: 5087
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Posted: 2015-07-29 12:11
Grothendieck can't.

Nonius is Satoshi Nakamoto. 物の哀れ
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