as for the ones that seem passingly interesting .. to me, anyway ..
// warning: massive speculation follows //
Mathematical Challenge 10: Algorithmic Origami and Biology
Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.
it seems that the convention is that isometric embeddings are defined as smooth, injective mappings between manifolds that preserve the (riemannian) metric [1]. on the other hand, the notion of embedding can be purely topological, so an isometric embedding should really be treated in the sense of metric geometry.
the problem sounds intriguing because it suggests to treat proteins like any other molecule .. which is to say, a discrete geometric object instead of something continuous, let alone smooth.
if they had intended someone to code all the symmetries and angular degrees of the various atoms and reduce it to numerical computation, then it would seem more like one of those big data kinds of problems. i don't think that's the point.
instead, i wonder if it has to do with some kind of "energy" .. in the sense that if a particular configuration of a protein is realised in physical space, then usually it's good for something. maybe it's even an optimal or close-to-optimal shape [2], so there would be some hope in finding a functional, some kind of "energy" for which the shape is minimal.
imagine that: a dirichlet problem for proteins!
Mathematical Challenge 15: The Geometry of Genome Space
What notion of distance is needed to incorporate biological utility?
i have no idea what this problem is even asking, but already i wonder what kind of geometry this kind of configuration space could obey. heck, it's not even clear to me what should be a point in this space: a gene? according to the wiki, they seem like finite subsequences that have been decided empirically to be distinct.
as a mathematician, that doesn't sound well-defined to me. \-:
despite the nomenclature of a "DNA sequence," could a sequence really encode all the geometry that serves biological function? what about genes that are related? now that i think about it, i have no idea what it means for two genes to be related.
all i imagine is a DNA sequence, strung out along a real line, and every relation between genes is like a wire connecting two points on the line. if the relation between genes could be quantified as more strongly or weakly correlated ..
.. then that sounds like a graph to me. i wonder if biologists treat DNA too one-dimensionally .. \-:
[1] every so often i run into a riemannian geometer, and suddenly i have to be careful to say "minimising geodesic" instead of 'geodesic' and "pointwise metric" instead of 'metric.' to be honest, sometimes i wish they'd start calling a riemannian metric a 'riemannian inner product' instead.
[2] .. which would only make sense if in the same medium or solvent, one sees essentially the same shapes all the time. it could be false from empirical evidence. i really don't know: according to the wiki, the configuration of a protein molecule could depend on the chemistry / structure of its surroundings, which only causes more complications ..
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