Problems

I've decided to keep a list of problems to think about that are not solved to my knowledge, or that I want to work out explicitly for myself. These are random in subject, but interesting...

I've noted some other people's math problems while looking for the Kourovka notebook that may be interesting to some people. I would just like people to note the wiki Open Problem Garden: help it grow!

List 1 (5:02 PM on Sunday, May 8, 2011)

1. Stuff, Structure, and Properties related Problems
1. Bourbaki formalised the notion of structure, specifically there are three mother structures: topological, algebraic, or ordered. It is easy to see that algebraic and ordered structures fit into the "stuff, structure, properties" paradigm; what about topological structure?
   
   Sunday August 28, 2011 at 05:36:04PM (PDT): It appears that if we have a mathematical gadget, we consider the category Gad consisting of these gadgets and "gadgetomorphisms", we may equip a given mathematical gadget with some topological structure if Gad is a topos. (Is there a weaker condition to consider equipping some gadget a topology?)
   
   We can construct the topology by considering all subobjects of a given object, and demanding the inclusions from the pullback diagram to the subobject classifier are "continuous" with respect to the topology...then construct the topology in this manner.
2. Bourbaki's Set Theory (Springer–Verlag, 2004) notes that "A given species of structure therefore does not imply a well-defined notion of morphisms" (pp 272). Is this true for the Baez–Dolan notion of "stuff, structure, properties"?
3. Stuff, structure, and properties enables internalisation in the "obvious way" — the first problem is to make this rigorous. The second problem: can we internalise topological structure?
   
   Sunday August 28, 2011 at 05:39:51PM (PDT): one problem is that topological spaces are "nonfirstorderizable". See Anand Pillay's "First Order Topological Structures and Theories" (Journal of Symbolic Logic 52 3 (1987) 763–778, JSTOR) for more on making topology something first order (ish), and that would be a step towards internalisation.
   
   Tuesday October 25, 2011 at 08:19:24AM (PST): figured out a solution to this, posted it to my notebook [code.google.com].
2. In a few papers by R J Low (e.g. "Twistor linking and causal relations" Class. Quantum Grav. 7 (1990) 177; "Spaces of causal paths and naked singularities" Class. Quantum Grav. 7 (1990) 943; and "Twistor linking and causal relations in exterior Schwarzschild space" Class. Quantum Grav. 11 (1994) 453) linking is related to causality. Arnold pointed out this would be interesting to investigate with knot theory. What are knots telling us?
   
   Additionally, Roger Penrose's Techniques of Differential Topology in Relativity (J. W. Arrowsmith Ltd., Bristol: England 1972) discusses the space of causal curves in section 6. This is a worth-while supplement (complement?) to Low's papers.
   
   Answer: Vladimir Chernov (Tchernov) and Yuli B. Rudyak's "Linking and causality in globally hyperbolic spacetimes" (arXiv:math/0611111v3 [math.GT]) appears to solve this problem.
3. Categorification (Vertically) of Various Mathematical Gadgets
1. Can we categorify number theory? Specifically the Dedekind η function?
   
   Tom Leinster discusses the Möbius inversion of a category at the n-Category Cafe.
2. Can we categorify the Jacobi theta function?
3. When we (vertically) categorify a sheaf, we get a stack. If we consider a vertex operator algebra as a functor, we may view it as almost like a Vect-valued sheaf. If we look at it in this light, can we vertically categorify a vertex operator algebra analogous to categorifying a sheaf to obtain a stack?
   
   Moreover, Baez and Dolan's "From Finite Sets to Feynman Diagrams" (in Mathematics Unlimited — 2001 and Beyond 1, eds. Bjorn Engquist and Wilfried Schmid, Springer, Berlin 2001, pp. 29–50 arXiv:math/0004133v1 [math.QA]) gives us a categorification of ladder operators; can we pull-back the construction of vertex operator algebras to obtain the same categorification of them? Would it make a big, nice diagram commute between these different approaches?
4. Is there any hope of classifying 2-groups?
5. Concerning the whole topology of X encoded in C(X) = Hom(X,ℂ) idea.
   1. Consider the group Hom(X,ℂ) of continuous complex-valued functions on a surface X. What data can we recover, and how do we recover the topological information of X from this C(X) algebra alone?
   2. A related problem: study Hom(𝕆ℙ²,ℂ) of continuous complex-valued functions on the octonionic plane 𝕆ℙ². What sort of topological information do we recover?
   3. If X is a smooth manifold and we restrict C(X) to real-valued functions — how does it relate to Morse theory? Is this whole scheme really just a "poor man's Morse theory"?
   Just a few thoughts about this. We know that the set of Morse functions is dense in the set of smooth functions on a manifold. So the set of Morse functions is dense on a subset of Re(C(X)).
   
   Answer: Gennadi Sardanashvily provided a solution to this confusion "relating Morse functions with the smooth real-valued functions on a manifold" idea:

   No, there is no direct relation between the differential calculus over the ring of smooth real functions on a smooth manifold X and the Morse theory. Of course, the both constructions deal with smooth functions, but in different aspects.

Sunday May 8, 2011 at 10:17:14PM (PDT)

1. Concering infinite-dimensional differential geometry and Lie theory.
   1. How do we define an infinite-dimensional manifold? What does infinite-dimensional differential geometry look like?
   2. How do we define an infinite-dimensional Lie group? What is its tangent space to the identity element? And, of course, the structure theory of infinite-dimensional Lie theory, and the problem of categorifying all of this…
   3. With L_∞-algebras (which are vertical categorification of Lie algebras — that is, we work with chain complexes and relax the equalities to homotopy equivalences), how does it relate to categorification of Kac–Moody Algebras? How does all this relate to categorifying Vertex Operator Algebras outlined before?
2. It seems that vertical categorification amounts to internalisation in Cat. (a) Is this a good definition? (b) Would providing a "rigorous method" to do one induce a method to do the other?
   
   Answer to (a): this gives us a strict categorification, a better approach would be: consider categorifying a "mathematical gadget". We have the category Gad of "mathematical gadgets". We consider a category object internal to Gad, and that gives us a (vertically) categorified version of our mathematical gadget. If we consider a groupoid object internal to Gad, we get a groupoidified (or "horizontally categorified") version of the gadget.
3. Concering biology (or what Baez calls, I think, "Green Mathematics").
   1. Vernon Avila's Biology: Investigating life on earth (Jones and Bartlett (1995) pp. 11–18) gives a number of axioms for biology, namely (1) Cells are the basic unit of life; (2) New species and inherited traits are the product of evolution; (3) Genes are the basic unit of heredity; (4) An organism regulates its internal environment to maintain a stable and constant condition; (5) Living organisms consume and transform energy.
      
      This is a little bit sloppy but could we make it (more) rigorous? Is there some stronger axiomatic schemata which describes biology? What theorems can we derive from this?
      
      For example, how would we model a cell? It is tempting to model it as a category, so that we can really use the objects as various states of the cell. This is kind of like modeling it as a dynamical system (see Lawvere's Conceptual Mathematics for this). How do we model an organism? Tissues and organs? Species (of life)? Etc. etc. etc.
   2. What is the topology of DNA? What does it tell us? Can we use the machinery of knot theory to say anything interesting with DNA? Doesn't DNA exhibit a sort of hysteresis? On a related note...what would the moduli space of DNA look like?