Poincaré and Geometry of Curvature. Thought of the Day 60.0

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It is not clear that Poincaré regarded Riemannian, variably curved, “geometry” as a bona fide geometry. On the one hand, his insistence on generality and the iterability of mathematical operations leads him to dismiss geometries of variable curvature as merely “analytic”. Distinctive of mathematics, he argues, is generality and the fact that induction applies to its processes. For geometry to be genuinely mathematical, its constructions must be everywhere iterable, so everywhere possible. If geometry is in some sense about rigid motion, then a manifold of variable curvature, especially where the degree of curvature depends on something contingent like the distribution of matter, would not allow a thoroughly mathematical, idealized treatment. Yet Poincaré also writes favorably about Riemannian geometries, defending them as mathematically coherent. Furthermore, he admits that geometries of constant curvature rest on a hypothesis – that of rigid body motion – that “is not a self evident truth”. In short, he seems ambivalent. Whether his conception of geometry includes or rules out variable curvature is unclear. We can surmise that he recognized Riemannian geometry as mathematical, and interesting, but as very different and more abstract than geometries of constant curvature, which are based on the further limitations discussed above (those motivated by a world satisfying certain empirical preconditions). These limitations enable key idealizations, which in turn allow constructions and synthetic proofs that we recognize as “geometric”.

Algorithmic Subfield Representation of the Depth of Descent Tree

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A finite field K admits a sparse medium subfield representation if

– it has a subfield of q2 elements for a prime power q, i.e. K is isomorphic to Fq2k with k ≥ 1;

– there exist two polynomials h0 and h1 over Fq2 of small degree, such that h1Xq − h0 has a degree k irreducible factor.

We shall assume that all the fields under consideration admit a sparse medium subfield representation. Furthermore, we also assume that the degrees of the polynomials h0 and h1 are uniformly bounded by a constant δ. Any finite field of the form Fq2k with k ≤ q + 2 admits a sparse medium subfield representation with polynomials h0 and h1 of degree at most 2.

In a field in sparse medium subfield representation, elements will always be represented as polynomials of degree less than k with coefficients in Fq2. When we talk about the discrete logarithm of such an element, we implicitly assume that a basis for this discrete logarithm has been chosen, and that we work in a subgroup whose order has no small irreducible factor to limit ourselves to this case.

Proposition: Let K = Fq2k be a finite field that admits a sparse medium subfield representation. Under the heuristics, there exists an algorithm whose complexity is polynomial in q and k and which can be used for the following two tasks.

1. Given an element of K represented by a polynomial P ∈ Fq2[X] with 2 ≤ deg P ≤ k − 1, the algorithm returns an expression of log P (X ) as a linear combination of at most O(kq2) logarithms logPi(X) with degPi ≤ ⌈1/2 degP⌉ and of log h1(X).

2. The algorithm returns the logarithm of h1(X) and the logarithms of all the elements of K of the form X + a, for a in Fq2.

Let P(X) be an element of K for which we want to compute the discrete logarithm. Here P is a polynomial of degree at most k − 1 and with coefficients in Fq2. We start by applying the algorithm of the above Proposition to P. We obtain a relation of the form

log P = e0 log h1 + ei log Pi,

where the sum has at most κq2k terms for a constant κ and the Pi’s have degree at most ⌈1/2 degP⌉. Then, we apply recursively the algorithm to the Pi’s, thus creating a descent procedure where at each step, a given element P is expressed as a product of elements, whose degree is at most half the degree of P (rounded up) and the arity of the descent tree is in O(q2k). At the end of the process, the logarithm of P is expressed as a linear combination of the logarithms of h1 and of the linear polynomials, for which the logarithms are computed with the algorithm in the above Proposition in its second form.

We are left with the complexity analysis of the descent process. Each internal node of the descent tree corresponds to one application of the algorithm of the above Proposition, therefore each internal node has a cost which is bounded by a polynomial in q and k. The total cost of the descent is therefore bounded by the number of nodes in the descent tree times a polynomial in q and k. The depth of the descent tree is in O(log k). The number of nodes of the tree is then less than or equal to its arity raised to the power of its depth, which is (q2k)O(log k). Since any polynomial in q and k is absorbed in the O() notation in the exponent, we obtain the following result.

Let K = Fq2k be a finite field that admits a sparse medium subfield representation. Assuming the same heuristics as in the above Proposition, any discrete logarithm in K can be computed in a time bounded by

max(q, k)O(log k)

Obstruction Theory

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Obstruction is a concept in homotopy theory where an invariant equals zero if a corresponding problem is solvable and is non-zero otherwise. Let Y be a space, and assume for convenience that Y is n-simple for every n, that is, the action of π1(y) on πn(y) is trivial for every n. Under this hypothesis we can forget about base points for homotopy groups, and any map ƒ: S → Y determines an element of πn(Y).

Let B be a complex and A a subcomplex. Write Xn for A U Bn, where Bn denotes the n-skeleton of B. Let σ be an (n + I)-cell of B which is not in A. Let g = gσ be the attaching map σ. = Sn → Xn ⊂ B.

Given a map ƒ: Xn → Y, denote by c(ƒ) the cochain in Cn+1 (B, A; πn(Y)) given by c(ƒ): σ → [f º gσ]. Then it is clear that ƒ may be extended over Xngσ σ iff f º gσ is null-homotopic, that is, iff c(ƒ)(σ) = 0, and therefore that ƒ can be extended over Xn+1 = A ∪ Bn+1 if the cochain c(ƒ) is the zero cochain. It is a theorem of obstruction theory that c(ƒ) is a cocycle. It is called the obstruction cocycle or “the obstruction to extending ƒ over Bn+1

There are two immediate applications. First, any map of an n-dimensional complex K into an n-connected space X is null-homotopic.

Take (B, A) = (K x I, K x i) and define ƒ:A → X by the given map K → X on one piece and a constant map on the other piece; then ƒ can be extended over B because the obstructions lie in the trivial groups πi(X).

Second, as a particular case, a finite-dimensional complex K is contractible iff πi(K) is trivial for all i < dim K.

Suppose ƒ, g are two maps Xn → Y which agree on Xn-1. Then for each n-cell of B which is not in A, we get a map Sn → Y by taking ƒ and g on the two hemispheres. The resulting cochain of Cn(B, A; πn(Y)) is called the difference cochain of ƒ and g, denoted d(ƒ, g).

Sustainability of Debt

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For economies with fractional reserve-generated fiat money, balancing the budget is characterized by an exponential growth D(t) ≈ D0(1 + r)t of any initial debt D0 subjected to interest r as a function of time t due to the compound interest; a fact known since antiquity. At the same time, besides default, this increasing debt can only be reduced by the following five mostly linear, measures:

(i) more income or revenue I (in the case of sovereign debt: higher taxation or higher tax base);

(ii) less spending S;

(iii) increase of borrowing L;

(iv) acquisition of external resources, and

(v) inflation; that is, devaluation of money.

Whereas (i), (ii) and (iv) without inflation are essentially measures contributing linearly (or polynomially) to the acquisition or compensation of debt, inflation also grows exponentially with time t at some (supposedly constant) rate f ≥ 1; that is, the value of an initial debt D0, without interest (r = 0), in terms of the initial values, gets reduced to F(t) = D0/ft. Conversely, the capacity of an economy to compensate debt will increase with compound inflation: for instance, the initial income or revenue I will, through adaptions, usually increase exponentially with time in an inflationary regime by Ift.

Because these are the only possibilities, we can consider such economies as closed systems (with respect to money flows), characterized by the (continuity) equation

Ift + S + L ≈ D0(1+r)t, or

L ≈ D0(1 + r)t − Ift − S.

Let us concentrate on sovereign debt and briefly discuss the fiscal, social and political options. With regards to the five ways to compensate debt the following assumptions will be made: First, in non-despotic forms of governments (e.g., representative democracies and constitutional monarchies), increases of taxation, related to (i), as well as spending cuts, related to (ii), are very unpopular, and can thus be enforced only in very limited, that is polynomial, forms.

Second, the acquisition of external resources, related to (iv), are often blocked for various obvious reasons; including military strategy limitations, and lack of opportunities. We shall therefore disregard the acquisition of external resources entirely and set A = 0.

As a consequence, without inflation (i.e., for f = 1), the increase of debt

L ≈ D0(1 + r)t − I − S

grows exponentially. This is only “felt” after trespassing a quasi-linear region for which, due to a Taylor expansion around t = 0, D(t) = D0(1 + r)t ≈ D0 + D0rt.

So, under the political and social assumptions made, compound debt without inflation is unsustainable. Furthermore, inflation, with all its inconvenient consequences and re-appropriation, seems inevitable for the continuous existence of economies based on fractional reserve generated fiat money; at least in the long run.