 To send me email, click shapiro@math.ohio-state.edu.

The Ross Mathematics Program is a six week residential summer program for high school students talented in mathematics. It is truly an excellent experience for ambitious young people with mathematical interests.
Here's a link to the Ross Program Web Page.

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Further information for D. Shapiro:

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SHORT ARTICLES for high school (and middle school) students.

These mathematical articles are listed below (as pdf files). Your comments are welcome.

TOWERS OF POWERS.

We consider questions in   Z/kZ,  the system of integers modulo k.
For instance, the sequence 11, 22, 33, . . . , nn is eventually periodic (mod k). What is its minimal period?
Which integers c can be expressed as   c = xx (mod k) ?
Do similar properties hold for the sequence 111, 222, 333, . . . in   Z/kZ ?
For fixed n we also investigate properties of the sequence   n,   n^n = nn,   n^(n^n) = nnn, . . . , when reduced (mod k).

Details appear in the paper:  Iterated Exponents, published in the journal Integers: Electronic Journal Of Combinatorial Number Theory 7 (2007) #A23.

SUMS OF SQUARES IDENTITIES.

Suppose K is a field and let DK(n) be the set of nonzero elements of K which are expressible as a sum of n squares in K. Certainly DK(1) is a subgroup of the multiplicative group of K since it is just the set of all nonzero squares. The set DK(2) is also a subgroup because it is closed under multiplication. That closure is clear from the following 2-square identity:

(x12 + x22)(y12 + y22) = (z12 + z22),
where z1 = x1y1 + x2y2 and z2 = x1y2 - x2y1.
Question. For which other values of n is the set DK(n) a subgroup for every K?
That is, when is there an n-square identity?

Euler recorded a 4-square identity, which is related to the later discovery (invention?) of quaternions by Hamilton in 1843. Soon afterwards Graves and Cayley found the octonions, an 8-dimensional (non-associative) algebra whose norm provides an 8-square identity. In 1898 Hurwitz used linear algebra to answer our question, proving his "1, 2, 4, 8 Theorem". Further details on the history of this problem and its generalizations appear in the following lecture notes on "Products of Sums of Squares". Those expository lectures were part of a mini-course given at the Universidad de Talca (Chile) in December 1999.

Lecture 1 (dvi), (pdf) : Introduction and History.

Lecture 2 (dvi), (pdf) : Integer Compositions.

Lecture 3 (dvi) , (pdf) : Formulas over Arbitrary Fields.

Those notes provide an introduction to the more extensive treatment of this subject given in the book:
D. B. Shapiro, Compositions of Quadratic Forms, W. de Gruyter Verlag, 2000.
This book is available electronically here.