Other Should I Become a Mathematician?

[mathwonk]

yes the book modern geometries by james R smart (is that a joke?) has it all on both sides of one page in the appendix.

but the "one page every 8 weeks" density seems high for presentation in the first course.

 

[quasar987]

Hey, everybody, I just decided that I want to be a mathematician! Yay! Actually, I just realized that I'm not happy doing anything else. So, yeah, I'm going to go read this thread now.

Has anyone yet touched on mathematical logic or areas relating to language, e.g., model theory, proof theory?

GOnna be taking 'Logic' next semester. Why do you ask?

 

[mathwonk]

Logic reminds me again of eucldean geometry, where the logic is complicated by our over familiarity with the subject matter.

and the irony is that as a lifelong profesional algebraic geometer i know hardly anything about plane euclidean geometry.

What I have learned is roughly this (about the logic). Consider a set of statements ("axioms").

they are "consistent" iff one cannot deduce a statement of form P and notP from them, iff there exists a "model" universe in whiuch all the statements are true of the model.

Even this is probably wrong, but I am a beginner in logic.

Questions one asks about axiom sets include:

are they consistent?i.e. does at least one model exist?

does more than one model exist? i.e. do they fully characterize some one model geometry?

e.g. if you look at the postulates given in the list of postulates for geometry in Harold Jacobs book 3rd edition, you will see they all hold not only in the euclidean plane, but also in euclidean 3 space.

hence it is imposible to prove from them the theorem of pasch, that a line which meets one side of a triangle away from a vertex, must also meet another side. but this property is cruciaL to all plane geometry of triangles. or that two circles which meet at a point which is not collinear with their centers must meet a second time.

i also recall SAS congruence being a theorem from high school, but it is properly an axiom, since there exist geometry models in which all other axioms of protractor geometry hold, including pasch, but in which SAS is false.

another question is whether axioms are independent, i.e. given one of them, can it be proved from assuming only the others? if not then apparently there is a model in which all the others hold but this one does not, and vice versa.

this makes it really cool and fun to look at various different models, and see what is true of each one.

e.g. if you assume all euclidean postulates except the euclidean parallel postulate, then it seems there can by triangles whose angles do not add up to 180 degrees.

and although mathematicians searched unfruitfully for thousands for evid3nce as to whether the parallel postulate was indeed independent of the others (it is), the almost trivial example of "table top geometry" i.e. lines on a table top that reach from one edge to another, almost give an example.

I.e. they immediately satisfy ll other postulates except the ability to lay off infinitely many copies of a line segment on any line, but this can be rescued if you just realize that you can change the meaning of length as you get closer to the edge, so that you never fall off.

I.e. think of walking along a line, and that you walk slower if it gets colder. Then just drop the temperature near the edge of the table. then you can take as many steps as you want along a line without going off the table if you keep walking slower and slower, i.e. if it gets colder and colder.

 

[mathwonk]

honest rosewater, please read the advice on fields medalist terry tao's webpage. that is much better than anything I wrote here.

 

[mathwonk]

this baragar's preface: so far so good:

From the Inside Flap
Preface for the Instructor and Reader
I never intended to write a textbook and certainly not one in geometry. It was not until I taught a course to future high school teachers that I discovered that I have a view of the subject which is not very well represented by the current textbooks. The dominant trend in American college geometry courses is to use geometry as a medium to teach the logic of axiomatic systems. Though geometry lends itself very well to such an endeavor, I feel that treating it that way takes a lot of excitement out of the subject. In this text, I try to capture the joy that I have for the topic. Geometry is a fun and exciting subject that should be studied for its own sake.

 

[honestrosewater]

yes the book modern geometries by james R smart (is that a joke?) has it all on both sides of one page in the appendix.

Haha. Oh, I see.

GOnna be taking 'Logic' next semester. Why do you ask?

Because that's what I'm most interested in at the moment. I meant to ask whether it has been talked about yet (as some other subjects have) in this thread.

Logic reminds me again of eucldean geometry, where the logic is complicated by our over familiarity with the subject matter.

Maybe it helps to step back and consider other logics (as you might other geometries).

What I have learned is roughly this (about the logic). Consider a set of statements ("axioms").

they are "consistent" iff one cannot deduce a statement of form P and notP from them, iff there exists a "model" universe in whiuch all the statements are true of the model.

Even this is probably wrong, but I am a beginner in logic.

Right, that is a theorem of model theory: a theory's consistency and its having a model are equivalent. Although, come to think of it, that might be due to completeness (or just a restatement of it), so I should say it's specifically a theorem of first-order model theory (which is usually what is meant, I think).

Questions one asks about axiom sets include:

are they consistent?i.e. does at least one model exist?

does more than one model exist? i.e. do they fully characterize some one model geometry?

Yes, I think consistency, completeness (syntactic and semantic variations), and independence (of the axioms) are three big, basic properties that you want to know about a theory. Whether it is categorical (i.e., has exactly one model up to isomorphism) might be another.

e.g. if you look at the postulates given in the list of postulates for geometry in Harold Jacobs book 3rd edition, you will see they all hold not only in the euclidean plane, but also in euclidean 3 space.
[snip]
I.e. think of walking along a line, and that you walk slower if it gets colder. Then just drop the temperature near the edge of the table. then you can take as mnay steps as you want along a line without going off the table if you keep walking slower and slower, i.e. if it gets colder and colder.

Ah, you got independence. Thanks for the ideas. I guess I am really hungry for some (useful) problems to solve, or I'm ready to start accumulating solutions. I imagine you've heard of George Carr's http://books.google.com/books?id=FTgAAAAAQAAJ". This is the book of theorems, definitions, and such that Ramanujan got (and kept) his hands on. I was looking at it the other day, and I find it quite handy, as just a source of lots of problems to solve (theorems to prove), laid out in somewhat logical progressions. Does anyone know of another, perhaps more recent, book like this? I'm not looking for a full treatment of any subject or a "how to solve problems" book. I'd like just a list of theorems with whatever additional notes are necessary.

I suppose I already have my guy for model theory, if anyone else is looking: http://www.maths.qmul.ac.uk/~wilfrid/" . He's super. He's good for logic too.

 

[honestrosewater]

honest rosewater, please read the advice on fields medalist terry tao's webpage. that is much better than anything I wrote here.

Yeah, I've seen it. I assume you mean his http://www.math.ucla.edu/~tao/advice.html" (a very memorable phrase). I guess I didn't mention that I've loved math and been around it for a while. I'm just now deciding to give up and dive in.

 

[mathwonk]

i recommend the book by gelbaum and olmsted, counter examples in analysis. it was really fascinating to me as a freshman to see all the exotic things that are true about the reals.

Counterexamples in Analysis
Bernard R. Gelbaum|John M.H. Olmsted
Bookseller: WebBookStore
(Pittsburgh, PA, U.S.A.) Price: US$ 7.17
[Convert Currency]
Quantity: > 20 Shipping within U.S.A.:
US$ 3.95
[Rates & Speeds]
Book Description: Dover Publications, 2003. Paperback. Book Condition: Brand New. Brand new as book, not a remainder, no marks. As published by Dover Publications. Paperback edition. Book Size: Length: 8.27 inches, Width 5.43 Height inches 0.55 Inches. Book weight is 0.57 pounds. This book will require no additional postage. Orders processed on AbeBooks Monday - Friday and ships 6 days a week. Synopsis: These counterexamples deal mostly with the part of analysis known as "real variables." The 1st half of the book discusses the real number system, functions and limits, differentiation, Riemann integration, sequences, infinite series, more. The 2nd half examines functions of 2 variables, plane sets, area, metric and topological spaces, and function spaces. 1962 edition. Includes 12 figures. Barcode/UPC of the book/13 digit ISBN # 9780486428758. 10 digit ISBN # 0486428753. Brand New. Bookseller Inventory # 9780486428758_N

 

[Werg22]

I have a simple question: what kind of people are mathematicians?

 

[J77]

I have a simple question: what kind of people are mathematicians?

You have to have a third nipple, but don't tell anyone...

 

[pivoxa15]

I have a simple question: what kind of people are mathematicians?

From my experience and hearing from what they say, all types. But one thing that unites them is that they are perfectionists. Are there any mathematicians who are not perfectionists?

However perfectionists interested in analytical objects tend to produce personalities that are introverted and so not too socially oriented. However there are exceptions and some are more extroverted. Perfectionists interested in other things might be very different.

 

[fopc]

Thank you for the suggestions. I have already ruled out Hartshorne, Euclid and Beyond; Millman and Parker; Moise, Elementary Geometry from an Advanced standpoint; Modern Geometries by James Smart;... all as excellent but too difficult.

I see you gave a review of the second edition of Millman's book at Amazon.
I'm trying to find out more about the first edition.
Can you make a comaprison between the two editions?

 

[mathwonk]

i have not seen but one edition of millman parker, but all they say that added to the second edition was a collection of "expository exercises" to implement the program "writing across the curriculum". so it makes almost no difference to the presentation and the way I teach the course, if that's all they did.

by the way i am reconsidering hartshorne, and some other books recommended here like Baragar and Bass et al, if I can find them. Thanks very much for the sugestions!, and looking for review copies in libraries, since the publishers make it so hard to get review copies.

after all one can teach out of anything if you handle it well in class. actually most sudents don't read the book anyway in calculus at least, so better to give them a good book and hope they read it than a bad book they claim they can read.

 

[mathwonk]

as to what kind of people mathematicians are, it seems fair to call them perfectionists, but others exist as well.

when i grade papers, i always write a lot of detailed comments on them, and yet 99% of the students never come to look at them, so all the hundreds? of hours spent doing that over the last 30 years are wasted. but i do it anyway, for the one in a hundred who might want to see them.

we don't do things to a standard that will pass muster from others, but to a standard of our own choosing. no one may ever see or read what we do, but we do it to our own standard of perfection anyway. when i write a paper it usually goes through dozens of iterates, some just changing a few words, some just removing superfluous spaces between words.

not everyone is like this. i think this perfectionism is an enemy in many cases to creativity, and some of the most creative people just try to forge ahead, not nit picking their own work at every stage. indeed this is essential. that's why math is so hard, it ideally requires both sides of the brain, real hard creative work, followed by very precise critical review.

just the subject of plane geometry we are talking about here has gone through what, 2,000 years? of critical review by mathematicians and still geting fresh looks, like Hartshorne's book from 2005. they want to get it right.

now I am finally beginning to think my students at least should not be held to this standard and I allow a lot of leeway.

 

[mathwonk]

mathematicians come in all stripes. there is no personality litmus test for one. If you enjoy doing math, and preferably have some success at it, or can learn to, you can be one too.

to see the variety just look at a photo of g. perelman and compare to a photo of eli cartan, and compare their biographies.

http://www.englishrussia.com/?p=250

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cartan.html

 

[mathwonk]

werg, the best high school geometry teacher i know, Steve Sigur, of the Paideia school in atlanta, taught me the basic principle of geometry prep for the SAT's "angles that look equal are equal".

 

[Werg22]

Thanks for the advice, but I took my SAT's a long time ago already! They're not much to prepare for nowadays anyway; the problems are more about intuition than analysis. This said, even if what you said is true, I'd still be inclined to verify the value of the angles at every problem; you never know!

 

[mathwonk]

i think, rather than advice, it was agreement with post 600.

 

[Werg22]

Oh I see. I have a poor short memory, I'm afraid.

 

[mathwonk]

well i had trouble remembering myself, but that seemed plausible. its like a password you make up. it seems so clever and memorable at the time and later, huhh?

 

[Werg22]

I only keep one password for everything. Should someone discover it, I'd be in deep trouble.

 

[honestrosewater]

I only keep one password for everything. Should someone discover it, I'd be in deep trouble.

I made up a simple algorithm for making up passwords that are acronyms formed from a sentence based on something persistent about the specific site (or whatever). Sentences are easier to remember, I find, and you don't actually have to remember it anyway since you can just rerun the program that generated it. Although, I suppose you could always forget the program or how to execute it, since it's just some instructions in your head. But if you write it in your native language, you'll have bigger problems if you ever forget how to execute it.

Also, with acronyms, you avoid the common spelling patterns of words (based on a language's http://en.wikipedia.org/wiki/Phonotactics" ), which can rule out a lot of combinations and make others more or less likely.

I'm not actually worried about anyone guessing my passwords, by the bye. I just like language and solving problems.

 

[mathwonk]

i also have one universal password,

qerii23849504434528888nmartw@!&@@

but i still keep forgetting it.

 

[Werg22]

I find it easier to remember a sequence of numbers or a word by associating them to either an idea or an image. For example, I remember one of my friend's phone # by remembering "the inverse of my regional code, two similar numbers (69) and the day of st-valentine (14)". If I remember these steps, I remember the number. Same thing goes with formulas: I like to remember formulas conceptually rather than just as expressions.

 

[morphism]

I have two base passwords, and I use different arrangements of each. Like uppercasing, adding numbers, switching o's with 0's and i's with 1's, etc. It usually takes me a couple of tries to finally figure out what the password should be!

 

[fopc]

i have not seen but one edition of millman parker, but all they say the added to the second edition was a collection of "expository exercises" to implement the program "writing across the curriculum". so it makes almost no difference to the presentation and the way I teach the course, if that's all they did.

by the way i am reconsidering hartshorne, and some other books recommended here like Baragar and Bass et al, if I can find them Thanks very much for the sugestions!, and looking for review copies in libraries, since the publishers make it so hard to get review copies.

after all one can teach out of anything if you handle it well in class. actually most sudents don't read the book anyway in calculus at least, so better to give them a good book and hope they read it than a bad book they claim they can read.

Thanks. First edition is about $50 cheaper.

By the way, I wouldn't give up on Smart.
Yes, the axioms from high school geometry (32 of them) are relegated to an appendix along with other axiom sets (Hilbert, Birkhoff), but you could supplement this material with you own notes. Also, how much time can you devote
to it in a quarter (or semester) anyway?
If I understand your comments, your class will be made
up primarily of college students intending to be *high school math teachers*. I'd say if that's their goal, then it should be expected that they'll come to class with
prerequisites satisfied (which at the very least should include good understanding of the
axioms from *high school geometry*).

 

[mathwonk]

well that is true, they should, but in fact they don't. this is the problem facing the teacher today.

by the way i found a copy of the first edition of millman parker in a library yesterday and compared the two editions for you. the first edition is only 15 pages shorter than the second, and has the same chapter headings, and every single chapter section has the same title.

oh yes and the quality of the paper was superior in the first edition and the print was larger and more readable. so the first edition seems to be a better book, as is usual.

 

[fopc]

well that is true, they should, but in fact they don't. this is the probl;em facing the teacher today.

by the way i found a copy of the first edition of millman parker in a library yesterday and compared the two editions for you. the first edition is only 15 pages shorter than the second, and has the same chapter headings, and every single chapter section has the same title.

oh yes and the quality of the paper was superior in the first edition and the print was larger and more readable. so the first edition seems to be a better book, as is usual.

Ordered a copy for $29.25. Thanks for information.
Price not too bad compared to the $79.50 price tag on the current ed.

 

[mathwonk]

i also like Hartshorne's recent book, geometry: euclid and beyond.

edit: I now recommend everyone to learn plane geometry from Euclid with Hartshorne as a guide.

 

[quasar987]

Did you ever try your luck at the Hodge conjecture mw?

 

[mathwonk]

well i did think about it once and had a proposal for proving it false, but did not try doing the heavy lifting to see if it worked. I told it to some much smarter people more expert in the topic and had the pleasure at least of seeing them think about it seriously.

It is a very hard problem. it says that something very unusual only happens in a geometrically restricted situation. So most of the time it holds vacuously. And in all reasonable situations where the hypotheses hold, it has been shown the conclusion does as well.

So there are hundreds of papers out there saying "the hodge conjecture holds for cubic threefolds" or in some other case. But no one knows how to show it holds in general. One of my coworkers, Elham Izadi, has an inductive approach that may be useful.

Thanks for the suggestion I may be on it. It takes courage to work on something that hard.

 

[mathwonk]

here was my idea. the hodge conjecture is about recognizing the cohomology classes of algebraic subvarieties of a given algebraic variety.

I.e. every algebraic variety is a topological space, and if smooth, is a manifold. So it has a fairly computable cohomology group. recall that a homology or cohomology group is a group of equivalence classes of triangulable topological subspaces, where two classes are equivalent if their difference is the boundary of the class of a triangulable subspace, think submanifold, with boundary. All this is topology.

Now the analytic side of algebraic varieties allows one to represent all cohomology classes using differential forms, and by defining a metric, by differential forms which are harmonic, in the sense that the real parts of holomorphic functions are harmonic functions. This decomposition turns out to be independent of the choice of metric.

So then harmonic forms can be written as sums of terms involving dz's and dzbar's, and it turns out that the cohomology class of an algebraic subvariety always has the same number of dz as dzbar representatives, i.e. has "type (p,p)", for some p.

So as I understand it, which is minimally, the hodge conjecture asks if this is also a sufficient condition for algebraic representability of a cohomology class, i.e. that its harmonic representatives have class (p,p).

My idea was to look at the space, let's see now, its been so long ago, of hodge substructures of type (p,p), i.e. those which could be hodge structures of algebraic subvarieties. And in there to look at the subspace of actual geometric hodge structures, those which come from algebraic subvarieties.

So the Hodge conjecture is to see if those two are equal, or not. but if two subvarieties are equal, then their tangent cones are also equal, so my idea was to compute the tangent cones of these subvarieties at some interesting yet accessible element, and hopefully show they are different. That would disprove the Hodge conjecture. That would not win the prize money, but would settle it.

the reason this is an approachable tack is that tangent cones are far easier to compute than anything else.

 

[Vagrant]

I have just finished with high school and will be starting with engineering college in two months time.
I am a little weak in the following topics:
Functions,
Continuity and differentiability,
Permutations and combinations,
Equations and inequations.
I am looking for a book that will have more emphasis on theory and proofs, because I have a few books which contain problems for practice.
In school basically we were told how to deal with specific problems and given formulas.
Will "What is Mathematics?" by Courant and Robbins be a good choice?

 

[Werg22]

I have just finished with high school and will be starting with engineering college in two months time.
I am a little weak in the following topics:
Functions,
Continuity and differentiability,
Permutations and combinations,
Equations and inequations.
I am looking for a book that will have more emphasis on theory and proofs, because I have a few books which contain problems for practice.
In school basically we were told how to deal with specific problems and given formulas.
Will "What is Mathematics?" by Courant and Robbins be a good choice?

No, What is Mathematics can't be considered a rigourous textbook. Courant and Fritz John's books first out of three books is what you're looking for, though permutations and combinations will need you to be looking somewhere else. I must warn you though; Courant's text is no easy one, especially if it's your first dive into mathematical rigor.

 

[mathwonk]

i guess i agree that What is Math? is not a textbook, but it strikes me as mathematically and logically rigorous. I like it and think it has a lot to offer. It is not as dry as a regular textbook, and covers more topics. But the author Richard Courant, is a much better mathematician and scientist than most textbook authors.

Rigor is a relative concept. In the 1960's when set theoretic topology was growing in influence in textbooks, Fritz John rewrote Courant's book to make it more modern and "rigorous" by using more point set language, but to me the effect was more to make it less appealing.

Rephrasing the definition of continuity from epsilon /delta to the open set version in my opinion only makes it less intuitive and no more rigorous. But these are matters of taste. Surely there are discussions in What is M? that lack full details, but they are still valuable.

Here is a little example from What is M? The usual proof of uniqueness of prime factorization begins by developing the theory of the gcd and the lemma that a prime number cannot divide a product of two integers unless it divides one of the two factors.

Courant observes that the proof of prime factorization can be done without this lemma, if one observes that the lemma definitely holds for integers which do have prime factorization.

This way one is able to do the proof by induction, building up from cases where the lemma holds. It then follows as a corollary of uniqueness that the whole theory of gcd's goes through.

This argument as given by Courant, is not only completely rigorous, but contains insights one finds almost nowhere else. A typical textbook would merely present the usual theory of gcd's and then prime factorization, with or without perfect rigor.

E.g. the proof in Dummit and Foote, unlike Courant's, has a major logical gap, as I have observed elsewhere, although DF has the appearance of a rigorous text. But Courant is a master.

Although of course you are right that Courant is not written in the style of a usual textbook, still it is useful to read the masters no matter how they express themselves.

I myself have struggled for years with trying to write out this proof of unique prime factorization, troubled by the need to reorder the factors and give the induction in modern over precision, maybe using permutations notation.

Than I read Gauss, where it is done very clearly indeed, with only the amount of precision that illuminates the proof, and not so much irrelevant over- precision as serves only to obscure the argument.

So as long as you can provide any missing details, then Courant should be adequate, or even if not, it is a good introduction to topics one can read in more detail elsewhere later.

I agree though that as a student I sometimes found other treatments more understandable than Courant.

 

[fopc]

i also like hartshorne's recent book, geometry: euclid and beyond.

I can see why.
I read the first several pages of it that are available at Amazon.
Interesting, the Amazon list price is about $51.
I see copies out there (new or like new) for under $18.

I imagine you have major experience with his other monster textbook.

By the way, any experience with Goldblatt's book, "Topoi, the Categorical Analysis of Logic"?

 

[mathwonk]

no, i am not familiar with goldblatts book.

i think $50 is a good price and about right for Hartshorne. new copies of a book like that for $18 suggest something is amiss, i.e. that they are pirated, or "international editions: not intended for sale in the US.

So I avoid buying them.

 

[DeadWolfe]

What is the difference in those 'international editions". I know a guy who gets them and seems to have no problems with them. Are they illegal, or what's the deal with them?

 

[Chris Hillman]

Category Theory books?

By the way, any experience with Goldblatt's book, "Topoi, the Categorical Analysis of Logic"?

I found it valuable for some things, but the discussion of logic is IMO insufficiently clear. The best first book on category theory is Lawvere and Schanuel, Conceptual Mathematics. Good second books include

1. Saunders Mac Lane, Categories for the Working Mathematician,

2. Colin McLarty, Elementary Categories, Elementary Toposes,

3. Robert Geroch, Mathematical Physics (dont' be fooled by the title, it's really a reprise of standard undergraduate math major courses from the perspective of categories).

 

[fopc]

Thanks Chris. I know about the books you mentioned, except for Geroch.

Regarding Goldblatt, it's his focus on logic that got my interest.
But if the (logic) development is not sufficiently clear or weak, then I'm not too interested.

Incidently, I now see his book is available for viewing at:
http://historical.library.cornell.edu/math/

So I can check it out for myself.

 

[JeffN]

hello mathwonk,

in another thread, you listed four theorems in single variable calculus which you thought were the most important. The intermediate and extreme value theorems, as well as rolle's theorem and the mean value theorem. Can you explain why?

 

[mathwonk]

well i was trying to outline the content of a typical non theoretical calculus course for my class. the usual content covers 4 types of problems:

1) proving equations like x^3 = 2 have real solutions.
2) solving max/min problems.
3) graphing functions.
4) integrating to find areas and volumes.the first problem is solved by the intermediate value theorem, the second by the extreme value theorem, the third by the rolle theorem (which implies that a function can only change direction at a critical point, and can only change concavity at a second order critical point), and the 4th is covered by the corollary to the MVT which implies that a function is determined up to a constant by its derivative. (That implies that since the derivative of the area function is the height function, then you can find the area function by antidifferentiating the height function.)

actually, theoretically these theorems are not too different. the proof of the extreme value theorem is similar but a little more complicated than the proof of the intermediate value theorem, and the rolle thm is actually implied by the extreme value theorem, and MVT is a slight generalization of rolle, and is implied by rolle. thus really there are only two essentially different theorems there, the IVT and the EVT, but i called them 4 different thms because they have 4 different uses in the course, and they look different to the students.

 

[mathwonk]

here is a survey of the hodge conjecture by a friend of mine who does understand it:

http://math1.unice.fr/~beauvill/basically it asks for ( and proposes) a characterization of those homology classes on an algebraic variety, which arise as the fundamental classes of algebraic subvarieties.

 

[threetheoreom]

Hey mathwonk, your advice is well heeded .. i was wondering if you heard of the is book or ( or anyone for that matter ) Fundations and Fundemental Concepts of Mathematics?I am trying to wrk through it for the summer.

 

[mathwonk]

well there are extensive reviews at amazon, but the excerpts viewable there do not reveal much. this seems a book for the general public, apparently a good one.

 

[mathwonk]

there is a great algebraic geometry conference starting in paris on monday, unofficially in honor of my friend arnaud beauville's 60th b'day. i'll be there, let me know if you will, and maybe we can have lunch or something.

 

[mathwonk]

well i am in paris at the institut henri poincare, and it is wonderful to be here.

the city of paris alone is intellectually stimulating in a way one cannot at all believe coming from the US south. ~I passed a public bookstore today with the complete works of galois on display in the window, unheard of even in a typical university town in US.

And the leadoff talk in the conference was a wonderful account of recent work on determining when certain varieties constructed by group quotients, are rationally connected or not.

A variety is rationally connected if you can connect any pair of points by some rational curve, and this checkable property is conjectured to imply the variety is the image of a rational variety, which is unknown.

the distinction is between finding lots of maps from P^1 to the variety and finding one map from some large P^n onto it.

This is what I came for, the instant bringing up to date on interesting and current questions by masters, in a single hour.

there are also people sitting around discussing "political" matters like how to raise funds to support the education of mathematics students in the developing world, people having a wider impact than just by their own research program.

All this makes one ask what one could be doing to enlarge the reach of mathematics education, such as the keepers of the flame on this site are doing. bravo to them!

so it is both educational and inspiring to be here. best wishes to you all. hope to see some of you sometime at one of these meetings. If you are here and want to recognize me, I am the nerdy American touristy looking guy in a t shirt with the honoree's picture on the back and the conference poster on the front. come up and say hi.

 

[mathwonk]

Apparently MSRI in berkeley is having a big algebraic geometry session spring 2009, so think about coming there if you like the topic and want some immersion in it.

 

[mathwonk]

i just heard a nice talk by a friend, Fabrizio Catanese, vastly generalizing an old result of serre on the action of Galois groups on algebraic surfaces.

if you have an algebraic surface defined by equations with complex numbers, and you change those numbers to their complex conjugates, would you think it changes the surface much? actually it only changes the complex struture and not the differentiable structure so at least the fundamental group does not change.

but what if the equation is in terms of algebraic numbersa and you let the galois group of the algebraic numbers act on it? Catanese showed that for every non trivial element of that Galois group, except complex conjugation, there is a surface whose fundamental group is changed by the Galois action.

somehow that seems odd. for one thing there is an algebraic form of the fundamental group, which turns out to be the completion of the topological one, and these groups do not change under Galois action, so one gets a large collection of groups that are different but whose completions are isomorphic.

lovely talk, very concrete, with all the surfaces constructed explicitly as group quotients of products of explicit plane curves with very simple equations. it was very apporpriate at this conference dedicated to Beauville, since these surfaces generalize a construction of Beauville.

 

[mathmuncher]

Sounds like you're having a blast. Have a safe return trip.

Anyhow, I just came across a very light-hearted joke that some of you oughta like:

Why do so many math majors confuse Halloween and Christmas?
Because Oct 31 is Dec 25.