Projective Planes

1. Find a finite projective plane of non-prime- power order.
2. Must planes of prime order be Desarguesian?
3. Find an analogue of Segre's Theorem for cubics, i.e. find a combinatorial characterization of cubic curves in finite Desarguesian planes.
4. Classify translation planes of order 32.
5. Is every finite partial linear space embeddable in some finite projective plane?
6. Prove that every Desarguesian semibiplane lifts to a unique (Desarguesian) plane. [Currently this is known only in the case of prime order. The general case should be do-able; I just haven't given it enough attention.]
7. Prove that ℵ₀-categorical projective planes do not exist.
8. Suppose a finite projective plane has the property that every quadrangle lies in a proper subplane. Must the plane be Desarguesian?
9. Must every finite non-Desarguesian projective plane contain a subplane of order 2?
10. Determine the weight enumerators of the codes of the classical planes. These are currently known only for small order planes (up to order 8 maybe?).

Unitals

1. A unital is a 2-(n³+1,n+1,1) design (with no assumption of embeddability in a finite plane). Known examples have order n equal to a prime power or n=6. Find other examples of composite order.

Generalised quadrangles

1. Does there exist a GQ(s,t) with finite s>2 and infinite t ?
2. Find, if possible, a finite generalised quadrangle Q with an automorphism τ of order 2, such that the quotient Q/τ lifts to a quadrangle other than Q.

Ovoids of Polar Spaces

1. Do ovoids exist in O⁺(10,q) for some prime power q?
2. Do ovoids exist in O⁺(8,25)?
3. Do ovoids exist in O(7,q), other than the known examples (for which q is a power of 3)?
4. Certain Weyl groups acting on finite orthogonal spaces are known to preserve ovoids when the ground field has prime order (see here). Why are there no analogues over prime power order?

Hadamard Matrices

1. Does there exist a Hadamard matrix of order 4n for every positive integer n? Currently the smallest open case is 4n=668.

Lattices and Sphere Packing

1. Show that for n sufficiently large, no lattice packing in ℝⁿ can attain the optimal density for packing of uniform spheres in ℝⁿ.
2. Some nice root lattices in dimensions ≤ 8 give rise to ovoids of finite orthogonal spaces by reduction mod p (see here). What happens for the Leech lattice? We know that this does not give analogous ovoids; but presumably some very interesting finite geometric structures must arise here.

Chromatic numbers of graphs

1. Find a subgraph of the Euclidean plane graph having chromatic number at least 6.
2. Is the chromatic number of the plane constructed over the field of complex numbers, finite? This graph has pairs of complex numbers as vertices, and the vertex (u,v) is adjacent to (z,w) whenever (z–u)²+(w–v)²=1.

Finite Fields

1. Let F be a finite field of order q, and consider the set of all functions f : F → F such that ∑_a∈F f(a) = 0. Which such functions also have the property that ∑_a∈F σ(a)f(a) = 0 for every permutation σ of F? (The obvious ones are all constant on a set of size q–2. There are q²(q–1)/²/2 such functions if q>3.)
2. Prove unconditionally (i.e. without assuming the Riemann hypothesis) that there exists a deterministic polynomial-time algorithm for finding a nonsquare mod p for a given odd prime p.

/ revised March, 2017