NOTE: This is not a well-maintained web page. I have recently deleted some of the problems, some because they were solved, others because they were stupid.
Below, I list some of my favourite unsolved problems. But first, a few warnings. Many of these problems have been posed by other people. I will try to give proper attributions, but I am likely to miss someone's name eventually. Many of the problems I know of were posed by Chris Godsil. Secondly, the problems are all confined to areas in which I work. That is, the list is rather narrow in scope and may not seem thematic.
Q: Given (X,S) and (Y,T), does there exist such a locally injective simplicial map?
Exercise: This problem is NP-complete in general.
Challenges: Find polynomially solvable subclasses of this problem. Find easy-to-verify sufficient conditions for the existence of such a map.
Application to numerical integration: Let X={(i,j) : 1 <=i<=s, 1<=j<=k}
and let
S={ {(i,j): j<= t_{i} } :
t_{1} + . . . + t_{s} <= t}.
Let T={ B subset of Y: |B| <= t}. Find modest
conditions on |Y|, s, k which guarantee the existence of our
function f.
This is a generalisation of the concept of a quantum error-correcting code. In that special case, the graph consists of all 4-ary n-tuples with those pairs at Hamming distance less than some fixed d joined by an edge. But this is not the full story; the configuration of spaces associated to a quantum code is intricately tied to the so-called "error group".