# Problems

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Information pertinent to many of the problems listed below can be found in the various references on page 3.

• Problem 1

Can the symplectic group SP(n) be embedded in euclidean space with codimension less than n for for n>=3?

• Problem 2

The affine quadric in complex n-space is defined by the equation z_1^2 + ... + z_n^2 =1 and has the same homotopy type as the real sphere. What is the first element in the homotopy groups of spheres which cannot be represented by a polynomial map of affine quadrics? What about complex analytic maps?

• Problem 3

Which elements in the stable homotopy groups of spheres are representable by stable framings on a hypersurfaces? For any given positive integer k is there an element in the stable homotopy groups of spheres which cannot be represented by a stable framing on a manifold embedded in codimension k?

• Problem 4

Every square matrix A over the quaternions has a quaternionic eigenvalue l in the sense that A- lI is not invertible. Which other skew fields have this property?

• Problem 5

Let a(d) denote the sum of the digits in the binary expansion of the positive integer d. Much research has been done on the the distribution of primes with a fixed a value. For example, Fermat primes correspond to the case a=2. A number which frequently occurs in the hit problem for the Steenrod algebra is denoted by m(d) which means the smallest number k for which d can be represented as a sum of k powers of 2 minus 1. What is the distribution of primes having a fixed m value?. The case m =1 corresponds to Mersenne primes. How many primes are there less than 2^n with m value about n/2?

• Problem 6

The twin primes problem asks for successive primes p,p+2. With reference to the previous problem, it is easy to check that for any d we have m(d+1)= m(d)+1 or m(d)-1. So here is a generalisation of the twin primes problem. Are there infinitely many successive primes p,q satisfying the condition | m(p) - m(q)| <= 2?

• Problem 7

Cohomology of topological spaces and algebras of invariants of subgroups of general linear groups acting on polynomials over finite fields, both provide interesting sources of graded modules over the Steenrod algebra. The two areas have an intersection but are not identical. There exist algebras of invariants, for example Dickson algebras, which cannot be realised as the cohomolgy of spaces. A fundamental problem is to find minimal bases for these modules over the Steenrod agebra. In one aspect of this problem we can ask if the number of generators in any degree is bounded independently of the degree. This is the next best thing to finite generation. We shall call such modules g-bounded (the use of the unqualified word `bounded' might cause confusion since the homogeneous summands of the module itself need not be bounded.) The classic case is the polynomial algebra in n variables of degree 1. This is known to be g-bounded but the proof is not easy. Experimental evidence in the case of algebras of invariants seems to suggest that a Hopf module over the Steenrod algebra is g-bounded if it is finitely generated as an algebra. Is this really true?

• Problem 8

With reference to the problem 7 one can also ask for degrees in which no generators are needed. In the case of the polynomial algebra in n variables such degrees are determined by the condition m(d)>n, where m has the meaning explained in problem 5. It turns out that the same criterion works for the algebra of symmetric functions. Can this criterion be generalised to arbitrary Hopf modules over the Steenrod algebra?