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?

Notes on Set Theory

So, I've been writing notes about mathematics and have been pondering about the problem "What is a mathematician's notebook like?"

Elements of Mathematics

I've decided to start writing a series of cohesive notes on mathematics. This is partially inspired by Bourbaki, among others.

So, I just finished some notes on set theory (Fascicles 0: Naive Foundations). It's around 120 pages of notes, and 10 pages of front matter.

I think that, like Donald Knuth, I'll try to write fascicles that are roughly 128 pages in length. This will limit my focus, and allow self-contained "quanta" of mathematical writing to be released.

One issue that came up: how to come up with good exercises?

I asked one of my professors. He said, "Ah, that's a good question Alex. I don't have a good answer. However, some times I add exercises which are examples I cannot get to in class. The real answer is: you'll know when you start teaching."

However, in Atish Bagchi and Charles Wells' Varieties of Mathematical Prose, they note that there are really 6 types of examples in mathematical writing:

1. An "Easy Example" is one that can be verified quickly.
2. A "Motivating Example" is one that is given before some definition. In my notes, for example, the discussion of a span of sets is a motivational example for binary relations.
3. A "Delimiting Example" is an example with the smallest possible number of elements or an example with degenerate structure.
4. A "Deceptive Non-Example" is one which innocently resembles something to be true, but really beneath the skin level is false.
5. An "Elucidating Example" is one that really clarifies various aspects to a notion or definition.
6. An "Inventory Example" is just a grocery list of useful examples that are not too general or specific.

I suppose that exercises follow this as well? It seems like the best exercises should have some sort of "moral" or "lesson". Frequently one finds "Prove or find a counter-example: ..." type of exercises, but is there any trick to coming up with good exercises?

A Mathematician's Notebook

It is the stuff of folk-lore that all mathematicians have cherished notebooks. But what do they do with them? I started reading The Kourovka Notebook (a notebook of unsolved problems in group theory), and noticed that V. I. Arnold had Online Problem Lists.

I also observed that Robert Wilson also keeps online a list of Research Problems.

I suppose, in short, a mathematician's notebook is focused more on problems than notes. This is what characterizes a good mathematician: the ability to come up with good exercises, good problems, good questions.

Of course, presenting them and coming up with solutions are also vital. It's not good if no one understands what you are asking! Donald Knuth, et al.'s Mathematical Writing lecture notes are an excellent introduction, they tackle various rules to bear in mind on pp. 3–8.

Gel'fand wrote in Pavel Etingof et al.'s The Unity of Mathematics: In Honor of the Ninetieth Birthday of I.M. Gelfand Progress in Mathematics 244 (2006) pp. xiv, that:

The next question was: How can I work at my age? The answer is very simple. I am not a great mathematician. I speak seriously. I am just a student all my life. From the very beginning of my life I was trying to learn. And for example now, when listening to the talks and reading notes of this conference, I discover how much I still do not know and have to learn. Therefore, I am always learning. In this sense I am a student—never a "Führer."

I forgot who said it, but I think it's best to have some applications in mind. Even if it's something along the lines of "How do I [accomplish some pure mathematical calculation]?" My use of "applications" is far more liberal than it appears!

Arnold's lecture On teaching mathematics had the opinion that: "Since scholastic mathematics that is cut off from physics is fit neither for teaching nor for application in any other science, the result was the universal hate towards mathematicians — both on the part of the poor schoolchildren (some of whom in the meantime became ministers) and of the users."

More liberally I think it is fair to say that mathematics is the language of science. So to really come up with good exercises, a mathematician should be constantly reading scientific literature — even popular texts on scientific topics.

But the short, and honest, answer is: I don't know how to come up with good problems. If anyone has any tips or advice, please share!

Case Studies

Curiously Knuth writes on pg 21 of his notes on mathematical writing:

Exercises are some of the most difficult parts of a book to write. Since an exercise has very little context, ambiguity can be especially deadly; a bit of carefully chosen redundancy can be especially important. For this reason, exercises are also the hardest technical writing to translate to other languages.

Copyright law has changed, making it technically necessary to give credit to all previously published exercises. Don says that crediting sources is probably sufficient (he doesn't plan to write every person referenced in the exercises for his new book, unless the publisher insists). Tracing the history of even well-known theorems can be difficult, because mathematicians have tended to omit citations. He recently spent four hours looking through the collected works of Lagrange trying to find the source of "Lagrange's inequality," but he was unsuccessful. Considering the benefit to future authors and readers, he's not too unhappy with the new law.

We can dispense with some of our rhetorical guidelines when writing the answers to exercises. Answers that are quick and pithy, and answers that start with a symbol, are quite acceptable.

Curiously, Terry Tao writes on page 2 that:

A good problem should be inviting enough to naturally draw the student in to attempt it, challenging enough that the student recognises that something nontrivial has to be done, and clean enough that the insight provided by the solution is not obscured by a mass of time-consuming calculations. It is here that a student can finally get a taste of how one's mathematical power increases when a tool is understood and used correctly, and how one can use the current class material to revisit earlier subjects and clarify them even further.

Inviting, challenging, clean — these are the mark of a good exercise. Perhaps, then, it is not too far of a stretch to suggest these are also the mark of a good example?

In an attempt to answer my own questions, let me list a number of "problem books".

Kirby's Problems in Low-Dimensional Topology

A collection of problems in knot theory, surfaces, 3-manifolds, 4-manifolds, complexes, graphs, TQFTs, etc.

The Open Problems Project

Open mathematical problems relating to computational geometry.

Arnold's Problem List (February 1998).

11 Problems on various analysis topics.

Arnold's Problem List (September 1998).

Dealing with Pseudoring of Lie Algebras, Quaternionic matrices, Quaternionic vector bundles, Contact version of Liouville's theorem, Hofer fields, Uncertainty Principle for Lattices, Singularities of curves in contact geometry, causality in terms of linking, triangulation of knots, "Nekrasov's Discriminant", among other topics covered. I like how it is simple, but covers slightly different disciplines (e.g. relativity and knot theory).

Two Problems by Arnold (September 2001).

Dealing with Betti numbers of parabolic sets, and caustics of periodic functions.

Arnold's Problem List (January 2002).

12 pages of fascinating problems, mostly relating to aspects of curves, etc.

Future Directions

It is kind of open what I am going to do with this notebook/blog. I don't really know myself. Personally, I prefer TeX as my markup language as opposed to html, which is why I don't write as much as I should.

(Rant: TeX is just so much more convenient! It actually allows me to change notation with the flip of the wrist! For example, consider \let\propersubset=\subsetneq, then I simply use A\propersubset B and if I hate the notation...well, one line of code changed! With html, I have to change everything by hand. Or counters...html has no counter macros, grr...)

It would be nice to "categorify" Bourbaki's work. Take that with a grain of salt! What I mean by this is to be as comprehensive as Bourbaki (I'm reading through his Algebra right now), presenting definitions as e.g. a group object instead of a group, a magma object instead of a magma, etc.

That is to say, present the same material "internalized" in an arbitrary category. So for an example of this, consider the following definition:

Definition 1. A Magma Object consists of an object M in a monoidal category C equipped with a morphism μ:M⊗M→M.

I must confess that such an endeavor is appealing to me, but I might do it in LaTeX (taking advantage of its flexibility).

Short Summary

I think I will probably end up writing up a cohesive collection of notes — in the spirit of Bourbaki — that is self contained covering all of mathematics (from Set Theory and Category Theory foundations to...whatever!). However, I think I'll TeX it up, and post it online.

(As an aside, there is an interesting project called LuaTeX (there is also LuaLaTeX). There is no memory limits for it, and it has embedded Lua code. Perhaps it would be interesting to use this when writing notes involving numerical calculations?)

A Remark on Reading Bourbaki

A very brief and small remark that may be helpful to those that are studying Bourbaki. The first volume of the Elements of Mathematics (The Theory of Sets) is more or less useless.

The only useful (and in my humble opinion, coherent) parts of that book is the "Summary of Results".

Although, if one were really "hardcore", one would have a collection of e.g. composition notebooks to write notes on the series. By giving actual explanation and examples, it should expand the size of the text several fold.

Also, it helps to create a "cheat sheet" of notation. Bourbaki used bizarre notation since, I assume, they had to work with typewriters.

Unfortunately, no one uses their notation. So, we are forced to come up with a "Rosetta stone" to translate their notation into modern notation.

Some guidelines for your "Rosetta Stone for Bourbaki" might be:

1. Include the book and page numbers where it is first introduced or defined.
2. Include definitions, since those too are "Bourbaki-dependent".
3. Have a separate "Stone" for each book.

I just thought that it may be helpful to someone trying to read through this ancient tome…

Addendum Tuesday August 16, 2011 at 01:10:11PM (PDT)

After some more research, I found out that the bizarre system Bourbaki uses in The Theory of Sets is really something called "Epsilon Calculus."

Math Overflow had a discussion on Bourbaki's epsilon calculus which is useful, and the Stanford Encyclopedia of Philosophy's page is instructive.

I doubt that this system was intended to be fully used, since (as some pointed out in the math overflow discussion) "even trivial proofs require an astonishing number of steps directly from axioms. Existence of the empty set can be proved with 11,225,997 steps and transfinite recursion can be proved with 11,777,866,897,976 steps."

The internet encyclopedia of philosophy states:

The growing awareness of the larger meaning and significance of epsilon calculi has only come in stages. Hilbert and Bernays introduced epsilon terms for several meta-mathematical purposes, as above, but the extended presentation of an epsilon calculus, as a formal logic of interest in its own right, in fact only first appeared in Bourbaki's Elements de Mathematique (although see also Ackermann 1937-8). Bourbaki's epsilon calculus with identity (Bourbaki, 1954, Book 1) is axiomatic, with Modus Ponens as the only primitive inference or derivation rule. Thus, in effect, we get:

(X ∨ X) → X,
X → (X ∨ Y),
(X ∨ Y) → (Y ∨ X),
(X ∨ Y) → ((Z ∨ X) → (Z ∨ Y)),
Fy → FεxFx,
x = y → (Fx ↔ Fy),
(x)(Fx ↔ Gx) → εxFx = εxGx.

This adds to a basis for the propositional calculus an epsilon axiom schema, then Leibniz' Law, and a second epsilon axiom schema, which is a further law of identity. Bourbaki, though, used the Greek letter tau rather than epsilon to form what are now called "epsilon terms"; nevertheless, he defined the quantifiers in terms of his tau symbol in the manner of Hilbert and Bernays, namely:

(∃x)Fx ↔ FεxFx,
(x)Fx ↔ Fεx¬Fx;

and note that, in his system the other usual law of identity, "x = x", is derivable.

The principle purpose Bourbaki found for his system of logic was in his theory of sets, although through that, in the modern manner, it thereby came to be the foundation for the rest of mathematics. Bourbaki's theory of sets discriminates amongst predicates those which determine sets: thus some, but only some, predicates determine sets, i.e. are "collectivisantes". All the main axioms of classical Set Theory are incorporated in his theory, but he does not have an Axiom of Choice as a separate axiom, since its functions are taken over by his tau symbol. The same point holds in Bernays' epsilon version of his set theory (Bernays 1958, Ch VIII).

Source: Internet Encyclopedia of Philosophy.

Over at the nLab, the entry on Choice Operators really helps explain what that pesky τ operator is in Bourbaki's Theory of Sets.

Haskell Annotations

I am starting to play around with the Haskell programming language, it is based off of category theory (well very loosely). There are a few questions I have for any Haskell gurus and category theory mathematicians, namely with regards to the existence of a terminal object in Hask (the category of Haskell data types and programs).

The underlying aim, I must confess, is to write up some abstract algebra in Haskell. For the scheme of things, I must give a reference:

• Saunders Mac Lane, Categories for the Working Mathematician, Springer-Verlag (2000).

The question I want to answer is this: to what extent can we use Haskell to model abstract algebra in a universal algebraic manner?

First let us review some concepts from category theory (and a little Haskell), then we will dive into my rambling questions.

Review Of Category Theory

To repent for my sins, I will review category theory first before launching into my twisted approach to Haskell.

Let us review two propositions, the first from chapter III "Universals and Limits, section 5 "Categories with Finite Products":

Proposition. If a category C has a terminal object T and a product diagram A ← A×B → B for any objects A,B∈C, then C has all finite products.

For an introduction to category theory in Haskell syntax, the reader is referred to:

• Alpheccar's Blog, "Category Theory and the category of Haskell programs", Part I June 18, 2007; Part II June 24, 2007; Part III: Algebras and Monads July 02, 2007.

There are a few reservations I have about those series of blog posts, for example in Part I the author states "In Haskell, a terminal object may be a phantom type: data T since T is containing only the element undefined." There is a different way to present the terminal object, it's the unit object ().

In fact, now that I think of it, the unit object () as the terminal object is more in line with category theory than the phantom type. If we refer to part II, we have the product correspond to (A,B). If we use Mac Lane's insight:

In particular, C has a product of no objects, which is simply a terminal object T in C, as well as a product for any two objects.

Source: Saunders Mac Lane, Categories for the Working Mathematician
Chapter III Universals and Limits, Section 5 "Categories with Finite Products".

It would follow that the terminal object would correspond to the unit object. Well, to be strict, it would be (,) which is isomorphic to ().

We use Mac Lane's other insight about monoids and groups from Chapter III, section 6 "Groups in Categories", which boils down to this: a monoid in C is a triple (C, μ:C×C→C, η:T→C) where T is the terminal object of C, μ is multiplication, and η is the "identity element" of C.

Using the jargon of "stuff, structure, and properties", we have the stuff be C an object of a category, the structure be two morphisms μ and η, and the properties be diagrams.

A group in C is a monoid (C, μ, η) together with a morphism ζ:C→C which makes a diagram commute (intuitively it sends each element of C to its (right) inverse).

Now for something slightly different: if we take the product of two objects A and B in Hask to be (A,B), then Hask is closed under finite products.

Conjecture. The category Hask is closed under finite products.

Haskell Question

Now, we have precisely enough knowledge to consider the questions relevant to abstract algebraic Haskell programming.

Naively, I would want to say that in Haskell typeclasses are structure-types. That is, we can write

TerminalObject :: ()

class Monoid C where
    mu :: C -> C -> C
    eta :: TerminalObject -> C

However, we cannot specify anything about the implementation of class Monoid. It is frustratingly close, we have the stuff C, we have the structure, but alas no properties!

Question 1: Is there any way around this or am I doomed? More precisely, is there any way to specify behavior without specifying implementation?

Now, consider the following additional code snippet:

class (Monoid C) => Group C where
    zeta :: C -> C

Question 2: Can I specify the behavior of zeta without specifying implementation, while assuming that mu and eta have been implemented already?

Haskell is so frustratingly close to being a good language, but it fails at some of these key things.