What is the most difficult mathematical subject?

[Lucky Luciano]

I specialize in algebraic geometry and I think that is a VERY hard field. Within that field it seems that:

Motivic Cohomology is the most difficult intellectually challenging thing I have ever encountered - Hands down. Unbelieveable stuff.

There are some sharp people here, so I was wondering if anyone here finds another topic to be extremely challenging? Mathematics is so broad and huge now that an expert in one area can be completely ignorant in another area and that is why I ask.

[ewagner]

1. I'm with Asimov

He said that mathematics to him was like a cruel mistress in that he was madly in love with it (her) but she deserted him in his time of need.

Same with me.......fascinated but hopelessly inept.

[freetobegay]

2. Trying to figure out how Repukes retain power.

[wryter2000]

3. Matrix algebra

Okay, I'm no mathematician, but I did learn calculus in high school. Put a matrix in front of me, and I'm lost.

[mmm]

4. Please tell us a little about Motivic Cohomology

[Lucky Luciano]

9. Where do I begin?

Actually, I wrote this more for me to test my understanding of things. I don't expect anyone to understand a word. It is kind of fun to read how ridiculous it all gets though. Why do they study this ridiculous thing? The reason is because a smart guy name Vladimir Voevodsky came along and invented much of it and used it to prove some highly nontrivial conjectures...the applications for proving many other conjectures are popping up all over. The trouble is that it takes years to understand this stuff (I don't understand it yet) before one can makes sense of it well enough to apply. Voevodsky won a Fields medal in 2002 for his work by solving the celebrated Milnor conjecture (Milnor was another Fields medalist) using these methods. I think he is the smartest person in the world now. Here is something interesting about Voevodsky - it kind of makes me laugh:

http://groups.google.com/groups?q=Voevodsky+group:alt.*&hl=en&lr=&ie=UTF-8&group=alt.*&selm=31B4CD5D.3874%40math.harvard.edu&rnum=9


edit to say the link does not work, but you may need to copy and paste it since DU messes things up with the "." in the link above.

Here is what he wrote:

"From: Vladimir Voevodsky ([e-mail])
Subject: Re: Announce: Mind Books
View: Complete Thread (14 articles)
Original Format
Newsgroups: alt.drugs.psychedelics
Date: 1996/06/04


Guy wrote:
> It doesn't matter. Would you take the risk of letting someone you didn't
> know in on the secret that you do drugs? In fact, here's a question for
> you: if they are legit, then why don't they know better than to post shit
> like this? Don't they realize that people are afraid of the oppressive
> drug laws? Buying too much indoor gardening equipment will get you
> reported to the cops by the store you bought it from. It's a dangerous
> world out there for those who don't do what their big brother tells them
> to.

It is not exactly clear to me what this is supposed to mean.
Fisrt of all selling/buying any kind of legally printed books
is a perferctly legitimate thing to do. I have probably half
of the titles from "Mind books" catalog in my home library (all bought
from usual bookstores) and it would never occur to me to hide it from
anybody DEA included. In fact, in a highly hypothetical case of the
beforementioned organization setting up a mailorder company to sell
books on psychedelics :-) the only reason I could see not to buy from
this company is that I would not want to support them with my money.

Besides, the person who posted the original ad was and is active in this
newsgroup in a manner which makes it very unlikely that he is affiliated
in
any way with a law enforcement agency.

I would really suggest you to be a little more careful in your posts
since what you said is meaningless and offensive to someone who seems to
be doing his best to provide interested
people with information both in his posts and through the mailorder
company in question.

Best,
Vladimir."

yep - this world class genius loves his drugs! LOL

I would need to define what a "category" including "objects" and "morphisms" and I would need to define what a "functor" is.

I would first need to explain "groups," "rings," "fields,". Also I would need to defines "varieties," and "sheaves" for the "Zariski topology" along with a number of other things just to define what a "scheme" is. See Hartshorne's "Algebraic Geometry"

Then there are a bunch of things I would need to do to define the category of "smooth" scheme.

Then I would need to define the category of "finite correspondences" for smooth schemes. Then I would need to define "Grothendieck topologies" and a particular example of one called the "Nisnevich topology." Then I would need to define what a "sheaf" is for these kinds of topologies.

Then I would need to define a bunch of homological algebra terms. I would need to define "Ext" and "Tor" functors and derived functors in general. I would need to define "additive" categories and "abelian" categories. Then I would need to define categories of complexes over abelian categories and additive categories. Then I would need to define the "homotopy" categories associated to these complexes. Then I would need to define triangulated categories. Then I would need to define what a "quotient" category and "derived" category are.

Then I would need to explain the Mayer-Vietoris property and the "homotopy" property for complexes. Then I would have enough information to define the category of "geometric motives." I would need to define the "tensor structure" on this category. Then I would need to define what is called the "Tate Object" in the category of geometric motives. I would also need to take any smooth scheme, X and define what the "motive of X" is. Then I would be able to define the "(p,q) motivic cohomology of X" to be the group of morphisms from the motive of X to the p-fold tensor of the Tate Object (which is a complex) where we shift it as a complex by q.

Then....if this has not sent you away screaming yet, we note that this category of geometric motives is almost impossible to work with, so we have a clever trick of embedding it into the category of "motives" which is more convenient because it is an abelian category. For this we need to define what a "simplex" is in algebraic geometry. Actually there is a whole mess of stuff with "representable functors", triangulated catgories, homtopy categories, derived actegories for complexes of sheaves in the category of smooth correspondences with the Nisnevich topology and what not going on here - then we choose a subcategory of an appropriate derived category based on the homology being homotopy invariant:

and this will finally define what a "motive" is.

Then since a geometric motive is also a motive, we can still define motivic cohomology in the same way, but instead be working in the more "convenient" abelian category of motives, where things are still highly nontrivial, but workable. In the geometric motives case, it is all set up nice in theory, but is utterly unworkable.

[phantom power]

12. You had me at "hello"...

[papau]

13. topology and tensors, oh my!, Fields and rings and groups, oh my!

and we have definitions.

Seems to leave out the why we are creating this Harry Potter world (this is the part I do not understand and due to that lack of understanding, I left this stuff to work in the less creative actuarial world).

But I am impressed with those than see a path to a solution - even a "wrong" path - that uses these things as tools to get the answer.

[NNadir]

14. Well we chemists have to play with groups when we're young,

and we learn the Woodward-Hoffman rules and ruminate importantly on the conservation of orbital symmetry, and Mobius inversions in molecular orbitals, then we grow old and drift into esoterica of our own creation. There's nothing more privileged in life, I think, that to be allowed the time for esoterica.

Although I wouldn't pretend to have a clue about what you're talking about, it sounds like this business is great fun of the highest kind, frustrating fun. You've certainly managed to intimidate me, but I'll bet it's a beautiful game where you are frequently intimidated yourself. That, I would guess, would be your point.

After a few decades of not thinking too much about higher (or, depending on your prospective, lower) math, I've started dusting off some old math books because I'm trying to teach myself some nuclear engineering, and have been struggling, not all that successfully, at understanding a chapter entitled, "Spherical Harmonics (PL) methods in one-dimensional geometries." I'm trying to deal with these fellows Mark and Marshak and their damned Boundary Conditions. Maybe you would be in a better position than I to explain to me who Mark and Marshak are, why their names both start with "Mar," how come they're so conditional about their boundaries, and whether or not we can keep these guys from hurting each other over the matter. :-)

I probably will never have the opportunity to appreciate what you do, but I am very glad you are doing it. Keep up the good work!

[Lucky Luciano]

17. You nailed it!

"...it sounds like this business is great fun of the highest kind, frustrating fun. You've certainly managed to intimidate me, but I'll bet it's a beautiful game where you are frequently intimidated yourself. That, I would guess, would be your point."

I am very intimidated! I mean, I understand most of what I learned so far, but I just finished my doctorate and I feel like I will not be able to write any papers on this particular branch of the subject without at LEAST five more intense years of training. My advisor (Alexander Merkurjev) is actually giving a class on the subject, but after two quarters he has only been able to describe the Motivic Cohomology for the simplest of all geometric objects - a point! Actually, not even that much - he was able to explain some of the motivic cohomologies (The (n,n) cohomology) for a point - and the proof he did was quite sophisticated! The (n,0) and (n,1) cohomologies are known, but the general (p,q) cohomologies are still unsolved - even for a fucking point! All those n, p, and q are integers btw.

With your spherical harmonics, I know nothing about this, but it sounds like some kind of differential equation method if I were to guess. I know the least about O/PDEs...but I could learn them on a need to know basis.

Alas, I don't know if academics is for me...it does not provide the adrenaline I require - I am an adrenaline fiend...I may be sucked into the world of finance and hedge funds while I keep "real" mathematics as a hobby - that is ok - I treated it as a hobby my whole time in graduate school since I did far less work than most. If I ever amass 5 or 10 million, then I will travel the world for the rest of my life with a great wife and do mathematical research, again, as a hobby...I will also try to learn a number of languages, go on risky adventures (That adrenaline thing again), and make friends the world over! That is my real goal!

[NNadir]

18. I'm going to guess that you suffer from being relatively young.

I on the other hand am decidedly middle aged, bordering on old, actually. As you age, your life grows more complex; I have a home; and a lovely wife; and young children; and an established career; some money worries; a hell of a lot of responsibility in a frightening time, the age of Cheney-Bush-Nader.

My career has gone in a direction in which I am seldom required to exercise much mathematical muscle. More or less I ended up in an industrial chemistry, specifically industrial pharmaceutical chemistry, a matter which is far more concerned (properly I think) with reproducibility and regulation than it is with mathematics. The statistical methods of process control are deliberately functional and theoretical considerations take a back seat to the bureaucratically established and proven methods.

To be honest, I kind of miss the days that I felt compelled to analyze NMR spin systems in complex macromolecules by diagonalizing the determinants of the associated Hamiltonians, or constructing molecular orbitals from their basis sets and connectivity matrices, blah, blah, blah. I once got to spend a month messing around with Bose-Einstein statistics as I tried to make thoroughly mixed bags of differing chemicals for the least possible money. I never thought I would say this, but I miss having my hands smell (repulsively) of aromatic selenols; I miss phosgene. I miss radiation.

As for my problem with spherical harmonics, you have guessed the nature of the problem exactly. I'm sure you'd find it prosaic than I do as you're far fresher than I am; it is merely a matter of generating Legendre polynomials and manipulating them in an appropriate manner. This is something I haven't really done in a decade or two, so it seems more daunting to me than it would to a fresher man. I think I'd do better at it were it not for my need to indulge myself in the pleasures of my wife and children. It is a lucky man who must choose between two beautiful things, and while I dabble in my reactor hobby, the beautiful things I have chosen, foremost at least, is to try to be the best husband I can to my wife and the best father I can be to my two magnificient boys.

Anyway thank you for generating an interesting thread. It was a pleasure to read.
















So may it be with you.

[midnight armadillo]

21. Spherical harmonics

Just think of them as the spherical coordinate version of sines and cosines in cartesian coords (like a Fourier decomposition) and all those Y_l^m's and Legendre polynomials will start to make a perverse sort of sense :-)

[NNadir]

22. Thanx. The text I was reading assumed fluid knowledge of the subject.

Thus it was written in an obtuse fashion.

I've looked in some other books and it isn't all that bad.

[phantom power]

5. I thought that proofs regarding fields over polynomials were hard

But it sounds to me as though you are well past that. Does algebraic geometry have to do with fields or groups over geometric objects?

We did some difficult proofs in computational complexity theory. I recall one epic session where the entire class *and* the professor were trying to explain the logic of a proof to ourselves. Wish I could recall what theorem that was.

[Lucky Luciano]

11. I'll get back to you.

I have to go now though.

[Dudley_DUright]

6. Vladimir Voevodsky won the Fields Medal

(the mathematician's equivalent of the Nobel Prize which is only awarded every four years) in 2002 for his work on Motivic Cohomology. Voevodsky works at the Princeton Institute for Advanced Study (where Einstein use to hang out). In my own field of physics, the mathematics underlying string theory is also unbelievably nasty. Edward Witten, who is also at Princeton's Institute for Advanced Study, won the 1990 Fields Medal for his work in this area.

[Lucky Luciano]

10. lol!

I just posted some stuff about Voevodsky below! String theory is a bitch too - indeed! I would like to learn a bit more about that, but I don't know much physics now. I like manifolds and such quite a bit though!

[donco6]

7. Infinite square wells.

Still can't figure them out from Calc III.

[MARALE]

8. calculus with imaginary numbers

I took this in college after my 3 calc. classes and differintal (sp) equations. Very interesting but working with calculus in two or more planes was hard to grasp.

[DEMVET-USMC]

15. They are all hard for me,I haven`t a clue as to what you are talking about

Did you read my responses to some the questions you asked ? I may be confusing you with someone else, I am not sure. Also, can I ask you questions having to do with math if I need to ? I would never abuse such a privilege and would only do so when I have exhausted other resources I am aware of. Please,if so, tell me how to do so :this forum`s mailbox feature if that is what it is called ? Regards Lucky, Oscar

[Bdog]

16. Partial Differential Equations

was my hardest math class...although I did make a B in that class.

Linear Algebra in real world applications is impossible if you don't use software or an engineering calculator.

[junker]

19. maxwell's field equations....pre heavyside

maxwell's field equations, pre-heavyside heavy-handed revisionism

what other math subset directly agrees with/expounds the concepts that arise within the lankhara sutra and actually pre-date buddha's exposition of the kalapa's? and deals with the 5/10 dimensional nature of life? ok, so bucky fullers' synergetics gets into a bit of it....but, maxwell was da' man..

[midnight armadillo]

20. Hard math

You crazy mathematicians... ;-)

The toughest math I ever encountered was the stuff involved in relativistic electromagnetism and quantum field theory, lots of contravariant and covariant matrices, 4-vectors, and the like. Ugh.
I never bothered with general relativity since the payoff wasn't worth the effort to me (my physics doctorate is in experimental biophysics and microscopy techniques).

The nice thing about math is that no matter how smart you are and how hard you work, there's always an infinitely tougher subject around the corner. One of these days I'll tackle tomography and differential geometry, as soon as I get out of my shitty postdoc into a real job and life settles down a little.

Did you ever read any of the biographies of Paul Erdos, like The Man Who Loved Only Numbers?

[Lucky Luciano]

27. You are right

There is always an infinitely tougher field around the corner...My advisor, who is a true superstar worries that continuing much further will eventually get to the point where doing anything will require so many years of background experience that someone burns out before they can produce innovative work...a mathematical saturation if you will and limit to how far we can go almost...that has been his feel since trying to tackle this motivic business...he has been trying to decipher it all for the last 8 years or so and he is only starting to get a firm grasp...the class he gave probably accelerates my own understanding very well because he is such a flawless lecturer and I don't have to hack through much of the introductory stuff the way he had to since he handed much of it to me on a silver platter. In any event, when lectruing, he NEVER makes a mistake and only needs to make cursory glances at his notes when lecturing and has perfect handwriting - an all around impressive mathemetician.

[_NorCal_D_]

23. Fractal Geometry/Chaos Theory

which primairly deals with patterns in nature. The concepts are frustrating to say the least. :scared:

http://www.duke.edu/~mjd/chaos/chaosh.html

[hunter]

24. Playing with computers and fractal geometry...

I haven't done this for quite a few years, but it is remarkably helpful.

I always enjoy math if I'm not facing some sort of deadline -- like an exam on Friday or some bone-headed promise to make a machine work by a certain date.

[struggle4progress]

25. If you really want a technical answer to the question, ...

you might want to check out Steve Simpson's reverse mathematics program: Simpson has been studying (for some years) the logical strengths of various well-known theorems.

If you don't want to stray too far into meaningless idealism, I think number theory is already impossibly hard (by Godel's theorem): so (unless you want to start comparing difficulty in terms of higher recursion theory) there may be a case to be made that it doesn't get any harder than number theory.

[Lucky Luciano]

26. Number Theory's deepest theorems may be proven

by some of the methods of Motivic Cohomology...number theorists are being told to learn about motives to solve their problems...a daunting task...number theory is tough too...number theory and algebraic geometry are often very much intertwined. The Riemann hypothesis is the granddaddy of all number theory problems...some people are wondering if it will be solvable...I tend to think it will be...if it is solved, it will be a radical revolution for all of mathematics.

That said, the class I attended today on my subject in question was really impressive...we proved some an amazing theorem and stated another that we do not have time to prove...all of this was Voevodsky's work...the guy is truly world class...There are an insane number of technical details to work on just to make the whole theory viable...I would wonder what Simpson might have to day about it because I have to imagine there are some technical mistakes that need to be fixed - but I bet any mistake will be fixed if one were to surface.

[struggle4progress]

28. More generally, algebraic geometry seems to be useful, and

cohomology also has its uses.

But on general principles, I expect any technique to have limited efficacy: the boundary between the "solvable" and the "unsolvable" is highly irregular. Near that boundary, making slight changes in problems has unpredictable effects on problem difficulty, so that apparently minor variants of tractable problems become completely impossible.

Of course, everyone is interested if you resolve Riemann's conjecture. If I remember correctly, somebody claimed to have a proof in the late 1990s, but the general reaction was "Uh, yeah, sure, right."

[Lucky Luciano]

31. So what is your specialty?

What have you studied because you seem to know a thing or two...did you study Logic? I know nothing about the kind of logic that mathemeticians study...

[Dogmudgeon]

29. A brief, yet merciless, introduction to Motivic Cohomology

Some of the most significant developments in mathematics in the past year stem from a breakthrough achieved by Vladimir Voevodsky, and from work by Voevodsky and Andrei Suslin which builds on this breakthrough. Providing bridges across different areas of mathematics, this work constitutes a significant step toward resolving some questions that had eluded mathematicians for several decades. Voevodsky has been invited to present a plenary lecture about his work at the International Congress of Mathematicians, the most important meeting in the mathematical world which takes place every four years and will next be held in Berlin in August 1998.
On its most general level, the work of Voevodsky provides a new link between two circles of ideas in mathematics: the algebraic and the topological. The term algebra as used here refers to a much broader and deeper field than that studied by high school students. What mathematicians mean by algebra is, roughly speaking, a theory for studying the general structure of sets endowed with algebraic operations, like addition and multiplication of the integers {...-3, -2, -1, 0,1, 2, 3,...}.

There are many different kinds of algebraic objects, the most basic one being an abelian group. An abelian group is a set together with an operation on the elements of the set, where the operation has all the properties of addition of the integers. More complicated structures, such as commutative rings and fields, arise when one considers more than one operation and how the operations interact. The simplest example of a ring is the integers, together with the two operations of addition and multiplication. If one introduces a third operation, division, one obtains the rational numbers, i.e., fractions, and such a structure is formalized in the notion of a field. Other examples of rings are the sets of polynomials (in any number of variables) whose coefficients belong to a given field. For any finite set of polynomials one can look at the common zeros of those polynomials. This set of zeros is called the algebraic variety defined by the polynomials. Adding more polynomials will often cut down the size of the common zero set, yielding subvarieties of the algebraic variety. Subvarieties are sometimes referred to as algebraic cycles on the variety.

Two algebraic objects are said to be isomorphic if they have the same size and structure. This means there must be a way to match the elements of the two objects in a one-to-one correspondence so that the matching uses up all the elements, and so that the matching preserves the algebraic structure. This kind of matching is called an isomorphism. Two algebraic objects that are isomorphic are, from the algebraic viewpoint, exactly the same.

And if you liked that, you can find the whole enchilada on the web at http://www.ams.org/new-in-math/mathnews/motivic.html

--bkl
From a seminar given at the Yogi Berra Institute for Linguistics, Mathematics, Logic, and a Clean Baseball Business.

[Lucky Luciano]

30. My advisor was going to try to get me a post-doc with Suslin

but he is very ill these days from his diabetes. That is a post-doc I would have taken, but most likely hedge funds are my path...

In that article it discusses spectral sequences and K-Theory...Spectral sequences are pretty wild - they contain a tremendous amount of data and they can be very powerful objects. The difficulty in working with them can be all the bookkeeping that you must do - I always have to review them when using them for this reason - so much to keep track of. K-theory is also a very hard subject and very mysterious still after 30 years - Daniel Quillen defined it 30 years ago and got a Fields medal for it. The spectral sequence of motivic cohomology groups that converges to the K-theory of the object is one property we did not have time to prove....We could have done it with five more weeks maybe, but K-theory was not a prerequisite for the course - I have studied K-theory a bit and it would take a few weeks to define it properly - actually maybe more. Then maybe a few more weeks to prove the existence of the specrtral sequence in question.....