summer programs for advanced kids, some free!

[mathwonk]

These programs sound fantastic to me. There was nothing like this known to me when I was a kid. Even when my kids came along there were no free ones.

 

http://mitadmissions.../prepare/summer

 

 

here are more!

 

http://www.ams.org/p...l/emp-mathcamps

 

 

here re some photos from epsilon camp. I am in some from 2011. You can see how much fun we had, and how young the kids are who learned geometry from Euclid. among many other things.

 

http://www.epsiloncamp.org/gallery.php

 

 

here I am holding the chair for one of my best scholars.

 

http://www.epsiloncamp.org/index.php

[mathwonk]

If you want some more information about epsilon camp, here is a summary of what we did online for the past two months, just warming up to the actual camp, which starts Monday.

 

 

Well, camp is almost here, and the problems forum is winding down, at least for now.  We have treated there, as you may have followed, solving both quadratic and cubic equations and applications to problems, then ideas of multiplicities of roots combined with the concept of crossing or not crossing the X axis at a root, with application to the concept of a tangent line, both as used by Descartes, Euclid, and ultimately Newton.  This constitutes a small conceptual introduction to theory of equations and differential calculus.

 

Then we used the ideas learned about tangent lines, to solve some optimization problems, and to give an inductive proof of the rule of signs stated by Descartes and usually named for him, but actually proved by Abbe' de Gua it seems. 

 

Next we took an excursion to the "LaGrange interpolation formula" which allows one to write a unique polynomial of degree at most n and having any desired values at any n+1 different points.  This shows that the graph of a polynomial of degree n is completely determined by any n+1 distinct points on the graph.  Conversely, given any n+1 points in the plane, with different X coordinates, there is a unique graph of a polynomial of degree n passing through all of them. 

 

This generalizes the familiar fact that through any two distinct points there passes a unique line (although for lines the 2 points do not need to have distinct X coordinates.  Perhaps if we used more general equations in both X and Y we could also partially eliminate this in the case of curves of higher degree.  No I don't think so, but this leads us to the theory of plane curves, a topic for some other discussion. )

 

Then, we began a discussion of the familiar formula for summing up the first n integers, and generalized it to summing the first n squares, and beyond.  We learned, without full detail, how to see inductively that the formula for the sum of the first n kth powers, i.e. for 1^k + 2^k + .....+n^k,  always has form n^(k+1)/(k+1) + terms of degree lower than k+1. 

 

With just these partial formulas for such sums of powers, we are able to find the area under the graph of any monomial Y = X^k, between X=0 and X=any X.  The "moving area function" for that region was in fact X^(k+1)/(k+1), mimicking the lead term of the formula for the sum of the first n kth powers.   By subtracting we get the area between any two points X=a and X=b, and by additivity and linearity of area, this allows us to evaluate the area under the graph of any polynomial.

 

Moreover it shows us that the derivative (slope of tangent line) for the area formula, equals the height of the original graph whose area is being sought.  I.e. the slope of the graph of the area function is the height of the original function, or the derivative of the area function is the height function.  This is called one of the fundamental theorems of calculus, and we have proved it for polynomials.

 

The final topic, which is not complete, generalizes this area calculation to a calculation of volumes.  The open question there is what is the derivative of the volume function?  I.e. how is the derivative of the moving volume function for a solid related to the geometry of that solid?  Or, as we move slightly to the right on our solid, i.e. in the X direction, by what factor does the volume increase? 

 

A hint was that for areas, the derivative of the area function was the size (length) of the leading edge of the moving region whose area was being measured, i.e. the length of the part of that region having X coordinate exactly equal to X.  So for volume presumably it has something to do with the geometry of the part of the solid having X coordinate equal to X also.

 

Let me know if you have a conjecture or solution.

[daijobu]

 

The final topic, which is not complete, generalizes this area calculation to a calculation of volumes.  The open question there is what is the derivative of the volume function?  I.e. how is the derivative of the moving volume function for a solid related to the geometry of that solid?  Or, as we move slightly to the right on our solid, i.e. in the X direction, by what factor does the volume increase? 

 

A hint was that for areas, the derivative of the area function was the size (length) of the leading edge of the moving region whose area was being measured, i.e. the length of the part of that region having X coordinate exactly equal to X.  So for volume presumably it has something to do with the geometry of the part of the solid having X coordinate equal to X also.

 

Let me know if you have a conjecture or solution.

 

A plane?

[mathwonk]

yes, it is a plane section of the solid, whose volume is being considered. Or rather the area of that plane section. So the derivative at x, of the moving volume function for a solid, is the area of the plane section of that solid, perpendicular to the X axis, taken at X.

 

 

You did n't really have the whole statement of the problem to go on. We were considering the half ball generated by revolving the upper right quarter of a unit disc centered at the origin, around the X axis.

 

The volume function was V(X) = that portion of the volume of the half ball lying between X=0 and X=X, if that makes sense.

 

The derivative of that function at X is then V'(X) = ?? (for 0 ≤ X ≤ 1.)

[Bang!Zoom!]

My poor DH came in while I was reading this, and he said, "Whatchya reading?" So I began to read aloud to him your post...

 

I said, "So, is he talking about critical stress points or velocity issues depending on substances and pressures?"

 

He just sorta tilted his head and backed away slowly....

[mathwonk]

Here are this year's forum topics, some repetitions of last year:  I was kind of blown away this year because last year's students were back with even more enthusiasm and knowledge and I had trouble keeping up.  (Less calculus, more geometry and topology.)

 

 

returning students:

 

determined orders of all rotation groups of the 5 regular solids.

 

classified all finite subgroups of 3 dimensional rotations, without knowing they are rotation groups of regular solids to begin with, by classifying their action on “polesâ€.

 

classified all compact triangulated surfaces, oriented or not, and their representation using cross caps and handles.  showed homeomorphism of sphere with either three cross caps or with one cross cap and one handle. including discussion of how to define an orientation of a triangulated surface by giving compatible orderings of vertices.

 

discussed euler characteristics of surfaces, mainly by triangulating but also briefly by “morse theoryâ€.  determined the effect on the euler characteristic of adding a handle or cross cap to a surface.

 

classified all plane euclidean isometries, and determined the possible translation subgroups and associated quotient or “point groups†of all discrete subgroups of plane isometries.  discussed slightly the orbifolds resulting from modding out the plane by a discrete subgroup of isometries.  

 

 

 

new students:

 

solved quadratic and (reduced) cubic equations by “cardano’s formulaâ€

 

determined which primes are sums of 2 squares, using factorization by Gaussian integers, modular arithmetic.  including discussion of unique factorization of gaussian integers.

 

tangent lines, “Descartes’ rule of signsâ€.

 

[mathwonk]

By the way, this year epsilon camp started a sister progem: del;ta camp for 6-7 year olds.  I met a few of those amazing kids yesterday.  It was such a  trip when a supremely confident 7 year old girl plops down opposite me and asks energetically, "and what's YOUR name?"  making me feel as accepted as if i were another 7 year old.

[mathwonk]

next years epsilon camp, i.e. summer 2015, will be in st louis at wASHINGTON UNIVERSITY ST LOUIS.

 

http://www.epsiloncamp.org/#apply

 

 

But you probably all know this.   I wish it had been here in seattle again so i could vist but maybe it will benefit some of you to be more centrally located.

[mathwonk]

By the way, there is a superb summer program in Park City Utah, and overseen by the Institute for Advanced Study in Princeton.  It is not really suitable for very young kids, but extremely beneficial to a select group of older ones, college age and up.  This is worth knowing about I think, for its future value to some of the people here.  The program is so advanced that as a professional mathematician and Professor, I learned a lot from the lectures aimed at graduate and even undergraduate students.  This is really a good program in math. 

 

https://pcmi.ias.edu/program-index/2015

 

One big benefit is the inclusive nature, with a diverse group of participants.  I enioyed also interacting with the high school teachers there.  Once I met Irving Adler there and had lunch with him.  He was an academic who was so principled and brave that when obliged to sign a loyalty oath as a condition of employment at his institution in the 1950's he refused and earned a living writing popular books on math and science.  (I found a better article than his obituary.)

 

http://en.wikipedia.org/wiki/Irving_Adler