Black Hole Entropy in terms of Mass. Note Quote.

If M-theory is compactified on a d-torus it becomes a D = 11 – d dimensional theory with Newton constant

G_D = G₁₁/L_d = l⁹₁₁/L_d —– (1)

A Schwartzschild black hole of mass M has a radius

R_s ~ M^(1/(D-3)) G_D^(1/(D-3)) —– (2)

According to Bekenstein and Hawking the entropy of such a black hole is

S = Area/4G_D —– (3)

where Area refers to the D – 2 dimensional hypervolume of the horizon:

Area ~ R_s^D-2 —– (4)

Thus

S ~ 1/G_D (MG_D)^(D-2)/(D-3) ~ M^(D-2)/(D-3) G_D^1/(D-3) —– (5)

From the traditional relativists’ point of view, black holes are extremely mysterious objects. They are described by unique classical solutions of Einstein’s equations. All perturbations quickly die away leaving a featureless “bald” black hole with ”no hair”. On the other hand Bekenstein and Hawking have given persuasive arguments that black holes possess thermodynamic entropy and temperature which point to the existence of a hidden microstructure. In particular, entropy generally represents the counting of hidden microstates which are invisible in a coarse grained description. An ultimate exact treatment of objects in matrix theory requires a passage to the infinite N limit. Unfortunately this limit is extremely difficult. For the study of Schwarzchild black holes, the optimal value of N (the value which is large enough to obtain an adequate description without involving many redundant variables) is of order the entropy, S, of the black hole.

Considering the minimum such value for N, we have

N_min(S) = MR_s = M(MG_D)^1/D-3 = S —– (6)

We see that the value of N_min in every dimension is proportional to the entropy of the black hole. The thermodynamic properties of super Yang Mills theory can be estimated by standard arguments only if S ≤ N. Thus we are caught between conflicting requirements. For N >> S we don’t have tools to compute. For N ~ S the black hole will not fit into the compact geometry. Therefore we are forced to study the black hole using N = N_min = S.

Matrix theory compactified on a d-torus is described by d + 1 super Yang Mills theory with 16 real supercharges. For d = 3 we are dealing with a very well known and special quantum field theory. In the standard 3+1 dimensional terminology it is U(N) Yang Mills theory with 4 supersymmetries and with all fields in the adjoint repersentation. This theory is very special in that, in addition to having electric/magnetic duality, it enjoys another property which makes it especially easy to analyze, namely it is exactly scale invariant.

Let us begin by considering it in the thermodynamic limit. The theory is characterized by a “moduli” space defined by the expectation values of the scalar fields φ. Since the φ also represents the positions of the original DO-branes in the non compact directions, we choose them at the origin. This represents the fact that we are considering a single compact object – the black hole- and not several disconnected pieces.

The equation of state of the system, defined by giving the entropy S as a function of temperature. Since entropy is extensive, it is proportional to the volume ∑³ of the dual torus. Furthermore, the scale invariance insures that S has the form

S = constant T³∑³ —– (7)

The constant in this equation counts the number of degrees of freedom. For vanishing coupling constant, the theory is described by free quanta in the adjoint of U(N). This means that the number of degrees of freedom is ~ N².

From the standard thermodynamic relation,

dE = TdS —– (8)

and the energy of the system is

E ~ N²T⁴∑³ —– (9)

In order to relate entropy and mass of the black hole, let us eliminate temperature from (7) and (9).

S = N²∑³⁽(E/N²∑³))^3/4 —– (10)

Now the energy of the quantum field theory is identified with the light cone energy of the system of DO-branes forming the black hole. That is

E ≈ M²/N R —– (11)

Plugging (11) into (10)

S = N²∑³(M²R/N²∑³)^3/4 —– (12)

This makes sense only when N << S, as when N >> S computing the equation of state is slightly trickier. At N ~ S, this is precisely the correct form for the black hole entropy in terms of the mass.

Advertisement

Game’s Degeneracy Running Proportional to Polytope’s Redundancy.

For a given set of vertices V ⊆ R^K a Polytope P can be defined as the following set of points:

P = {∑_i=1^|V|λ_iv_i ∈ R^K | ∑_i=1^|V|λ_i = 1; λ_i ≥ 0; v_i ∈ V}

Polytope is an intersection of boundaries that separate the space into two distinct areas. If a polytope is to be defined as an intersection of half spaces, then for a matrix M ∈ R^mxn, and a vector b ∈ R^m, polytope P is defined as a set of points

P = {x ∈ Rⁿ | M_x ≤ b}

Switching over to a two-player game, (A, B) ∈ R^mxn2_>0, the row/column best response polytope P/Q is defined by:

P = {x ∈ R^m | x ≥ 0; xB ≤ 1}

Q = {y ∈ Rⁿ | Ay ≤ 1; y ≥ 0}

The polytope P, corresponds to the set of points with an upper bound on the utility of those points when considered as row strategies against which the column player plays.

An affine combination of points z₁,….z_k in some Euclidean space is of the form ∑_i=1^kλ_iz_i, where λ₁, …, λ_k are reals with ∑_i=1^kλ_i= 1. It is called a convex combination, if λ_i ≥ 0 ∀ i. A set of points is convex if it is closed under forming convex combinations. Given points are affinely independent if none of these points are an affine combination of the others. A convex set has dimension d iff it has d + 1, but no more, affinely independent points.

A polyhedron P in R^d is a set {z ∈ R^d | C_z ≤ q} for some matrix C and vector q. It is called full-dimensional if it has dimension d. It is called a polytope if it is bounded. A face of P is a set {z ∈ P | c^⊤z = q₀} for some c ∈ R^d, q₀ ∈ R, such that the inequality c^⊤z ≤ q₀ holds for all z in P. A vertex of P is the unique element of a zero-dimensional face of P. An edge of P is a one-dimensional face of P. A facet of a d-dimensional polyhedron P is a face of dimension d − 1. It can be shown that any nonempty face F of P can be obtained by turning some of the inequalities defining P into equalities, which are then called binding inequalities. That is, F = {z ∈ P | c_iz = q_i, i ∈ I}, where c_iz ≤ q_i for i ∈ I are some of the rows in C_z ≤ q. A facet is characterized by a single binding inequality which is irredundant; i.e., the inequality cannot be omitted without changing the polyhedron. A d-dimensional polyhedron P is called simple if no point belongs to more than d facets of P, which is true if there are no special dependencies between the facet-defining inequalities. The “best response polyhedron” of a player is the set of that player’s mixed strategies together with the “upper envelope” of expected payoffs (and any larger payoffs) to the other player.

Nondegeneracy of a bimatrix game (A, B) can be stated in terms of the polytopes P and Q as no point in P has more than m labels, and no point in Q has more than n labels. (If x ∈ P and x has support of size k and L is the set of labels of x, then |L ∩ M| = m − k, so |L| > m implies x has more than k best responses in L ∩ N. Then P and Q are simple polytopes, because a point of P, say, that is on more than m facets would have more than m labels. Even if P and Q are simple polytopes, the game can be degenerate if the description of a polytope is redundant in the sense that some inequality can be omitted, but nevertheless is sometimes binding. This occurs if a player has a pure strategy that is weakly dominated by or payoff equivalent to some other mixed strategy. Non-simple polytopes or redundant inequalities of this kind do not occur for “generic” payoffs; this illustrates the assumption of nondegeneracy from a geometric viewpoint. (A strictly dominated strategy may occur generically, but it defines a redundant inequality that is never binding, so this does not lead to a degenerate game.) Because the game is nondegenerate, only vertices of P can have m labels, and only vertices of Q can have n labels. Otherwise, a point of P with m labels that is not a vertex would be on a higher dimensional face, and a vertex of that face, which is a vertex of P, would have additional labels. Consequently, only vertices of P and Q have to be inspected as possible equilibrium strategies. Algorithmically, if the input is a nondegenerate bimatrix game, and output is an Nash equilibria of the game, then the method employed for each vertex x of P − {0}, and each vertex y of Q − {0}, if (x, y) is completely labeled, the output then is the Nash equilibrium (x · 1/1^⊤x, y · 1/1^⊤y).

Why Can’t There Be Infinite Descending Chain Of Quotient Representations? – Part 3

 

For a quiver Q, the category Rep(Q) of finite-dimensional representations of Q is abelian. A morphism f : V → W in the category Rep(Q) defined by a collection of morphisms f_i : V_i → W_i is injective (respectively surjective, an isomorphism) precisely if each of the linear maps f_i is.

There is a collection of simple objects in Rep(Q). Indeed, each vertex i ∈ Q₀ determines a simple object S_i of Rep(Q), the unique representation of Q up to isomorphism for which dim(V_j) = δ_ij. If Q has no directed cycles, then these so-called vertex simples are the only simple objects of Rep(Q), but this is not the case in general.

If Q is a quiver, then the category Rep(Q) has finite length.

Given a representation E of a quiver Q, then either E is simple, or there is a nontrivial short exact sequence

0 → A → E → B → 0

Now if B is not simple, then we can break it up into pieces. This process must halt, as every representation of Q consists of finite-dimensional vector spaces. In the end, we will have found a simple object S and a surjection f : E → S. Take E₁ ⊂ E to be the kernel of f and repeat the argument with E₁. In this way we get a filtration

… ⊂ E₃ ⊂ E₂ ⊂ E₁ ⊂ E

with each quotient object Eⁱ⁻¹/Eⁱ simple. Once again, this filtration cannot continue indefinitely, so after a finite number of steps we get E_n = 0. Renumbering by setting E_i := Eⁿ⁻ⁱ for 1 ≤ i ≤ n gives a Jordan-Hölder filtration for E. The basic reason for finiteness is the assumption that all representations of Q are finite-dimensional. This means that there can be no infinite descending chains of subrepresentations or quotient representations, since a proper subrepresentation or quotient representation has strictly smaller dimension.

In many geometric and algebraic contexts, what is of interest in representations of a quiver Q are morphisms associated to the arrows that satisfy certain relations. Formally, a quiver with relations (Q, R) is a quiver Q together with a set R = {r_i} of elements of its path algebra, where each r_i is contained in the subspace A(Q)_aibi of A(Q) spanned by all paths p starting at vertex aiand finishing at vertex b_i. Elements of R are called relations. A representation of (Q, R) is a representation of Q, where additionally each relation r_i is satisfied in the sense that the corresponding linear combination of homomorphisms from V_ai to V_bi is zero. Representations of (Q, R) form an abelian category Rep(Q, R).

A special class of relations on quivers comes from the following construction, inspired by the physics of supersymmetric gauge theories. Given a quiver Q, the path algebra A(Q) is non-commutative in all but the simplest examples, and hence the sub-vector space [A(Q), A(Q)] generated by all commutators is non-trivial. The vector space quotientA(Q)/[A(Q), A(Q)] is seen to have a basis consisting of the cyclic paths a_na_n−1 · · · a₁ of Q, formed by composable arrows a_i of Q with h(a_n) = t(a₁), up to cyclic permutation of such paths. By definition, a superpotential for the quiver Q is an element W ∈ A(Q)/[A(Q), A(Q)] of this vector space, a linear combination of cyclic paths up to cyclic permutation.

Perverse Ideologies. Thought of the Day 100.0

Žižek (Fantasy as a Political Category A Lacanian Approach) says,

What we are thus arguing is not simply that ideology permeates also the alleged extra-ideological strata of everyday life, but that this materialization of ideology in the external materiality renders visible inherent antagonisms that the explicit formulation of ideology cannot afford to acknowledge. It is as if an ideological edifice, in order to function “normally,” must obey a kind of “imp of perversity” and articulate its inherent antagonism in the externality of its material existence.

In this fashion, Žižek recognizes an element of perversity in all ideologies, as a prerequisite for their “normal” functioning. This is because all ideologies disguise lack and thus desire through disavowal. They know that lack is there, but at the same time they believe it is eliminated. There is an object that takes over lack, that is to say the Good each ideology endorses, through imaginary means. If we generalize Žižek’s suggestion, we can either see all ideological relations mediated by a perverse liaison or perversion as a condition that simply helps the subjects relate to each other, when signification fails and they are confronted with the everlasting question of sexual difference, the non-representable dimension. Ideology, then, is just one solution that makes use of the perverse strategy when dealing with Difference. In any case, it is not pathological and cannot be determined mainly by relying on the role of disavowal. Instead of père-vers (this is a Lacanian neologism that denotes the meanings of “perversion” and “vers le père”, referring to the search for jouissance that does not abolish the division of the subject, her desire. In this respect, the père-vers is typical of both neurosis and perversion, where the Name-of-the-Father is not foreclosed and thereby complete jouissance remains unobtainable sexuality, that searches not for absolute jouissance, but jouissance related to desire, the political question is more pertinent to the père-versus, so to say, anything that goes against the recognition of the desire of the Other. Any attempt to disguise lack for instrumental purposes is a père-versus tactic.

To the extent that this external materialization of ideology is subjected to fantasmatic processes, it divulges nothing more than the perversity that organizes all social and political relations far from the sexual pathology associated with the pervert. The Other of power, this fictional Other that any ideology fabricates, is the One who disavows the discontinuities of the normative chain of society. Expressed through the signifiers used by leadership, this Other knows very well the cul-de-sac of the fictional view of society as a unified body, but still believes that unity is possible, substantiating this ideal.

The ideological Other disregards the impossibility of bridging Difference; therefore, it meets the perversion that it wants to associate with the extra-ordinary. Disengaging it from pathology, disavowal can be stated differently, as a prompt that says: “let’s pretend!” Pretend as if a universal harmony, good, and unity are feasible. Symbolic Difference is replaced with imaginary difference, which nourishes antagonism and hostility by fictionalizing an external threat that jeopardizes the unity of the social body. Thus, fantasy of the obscene extra-ordinary, who offends the conformist norm, is in itself a perverse fantasy. The Other knows very well that the pervert constitutes no threat, but still requires his punishment, moral reformation, or treatment.

Contact Geometry and Manifolds

Let M be a manifold of dimension 2n + 1. A contact structure on M is a distribution ξ ⊂ TM of dimension 2n, such that the defining 1-form α satisfies

α ∧ (dα)ⁿ ≠ 0 —– (1)

A 1-form α satisfying (1) is said to be a contact form on M. Let α be a contact form on M; then there exists a unique vector field R_α on M such that

α(R_α) = 1, ι_Rα dα = 0,

where ι_Rα dα denotes the contraction of dα along R_α. By definition R_α is called the Reeb vector field of the contact form α. A contact manifold is a pair (M, ξ) where M is a 2n + 1-dimensional manifold and ξ is a contact structure. Let (M, ξ) be a contact manifold and fix a defining (contact) form α. Then the 2-form κ = 1/2 dα defines a symplectic form on the contact structure ξ; therefore the pair (ξ, κ) is a symplectic vector bundle over M. A complex structure on ξ is the datum of J ∈ End(ξ) such that J² = −I_ξ.

Let α be a contact form on M, with ξ = ker α and let κ = 1/2 dα. A complex structure J on ξ is said to be κ-calibrated if g_J [x](·, ·) := κ[x](·, J_x ·) is a J_x–Hermitian inner product on ξ_x for any x ∈ M.

The set of κ-calibrated complex structures on ξ will be denoted by C_α(M). If J is a complex structure on ξ = ker α, then we extend it to an endomorphism of TM by setting

J(R_α) = 0.

Note that such a J satisfies

J² =−I + α ⊗ R_α

If J is κ-calibrated, then it induces a Riemannian metric g on M given by

g := g_J + α ⊗ α —– (2)

Furthermore the Nijenhuis tensor of J is defined by

N_J (X, Y) = [JX, JY] − J[X, JY] − J[Y, JX] + J²[X, Y] for any X, Y ∈ TM

A Sasakian structure on a 2n + 1-dimensional manifold M is a pair (α, J), where

• α is a contact form;

• J ∈ C_α(M) satisfies N_J = −dα ⊗ R_α

The triple (M, α, J) is said to be a Sasakian manifold. Let (M, ξ) be a contact manifold. A differential r-form γ on M is said to be basic if

ι_Rα γ = 0, L_Rα γ = 0,

where L denotes the Lie derivative and R_α is the Reeb vector field of an arbitrary contact form defining ξ. We will denote by Λ^r_B(M) the set of basic r-forms on (M, ξ). Note that

dΛ^r_B(M) ⊂ Λ^r+1_B(M)

The cohomology H^•_B(M) of this complex is called the basic cohomology of (M, ξ). If (M, α, J) is a Sasakian manifold, then

J(Λ^r_B(M)) = Λ^r_B(M), where, as usual, the action of J on r-forms is defined by

J_φ(X₁,…, X_r) = φ(JX₁,…, JX_r)

Consequently Λ^r_B(M) ⊗ C splits as

Λ^r_B(M) ⊗ C = ⊕_p+q=r Λ^p,q_J(ξ)

and, according with this gradation, it is possible to define the cohomology groups H^p,q_B(M). The r-forms belonging to Λ^p,q_J(ξ) are said to be of type (p, q) with respect to J. Note that κ = 1/2 dα ∈ Λ^1,1_J(ξ) and it determines a non-vanishing cohomology class in H^1,1_B(M). The Sasakian structure (α, J) also induces a natural connection ∇^ξ on ξ given by

∇^ξ_X Y = (∇_X Y)^ξ if X ∈ ξ

= [R_α, Y] if X = R_α

where the subscript ξ denotes the projection onto ξ. One easily gets

∇^ξ_X J = 0, ∇^ξ_Xg_J = 0, ∇^ξ_X dα = 0, ∇^ξ_X Y − ∇^ξ_Y X = [X,Y]^ξ,

for any X, Y ∈ TM. Consequently we have Hol(∇^ξ) ⊆ U(n).

The basic cohomology class

c^B₁(M) = 1/2π [ρ^T] ∈ H^1,1_B(M)

is called the first basic Chern class of (M, α, J) and, if it vanishes, then (M, α, J) is said to be null-Sasakian.

Furthermore a Sasakian manifold is called α-Einstein if there exist λ, ν ∈ C^∞(M, R) such that

Ric = λg + να ⊗ α, where Ric is the Ricci Tensor.

A submanifold p: L ֒→ M of a 2n + 1-dimensional contact manifold (M, ξ) is said to be Legendrian if :

1) dim_RL = n,

2) p_∗(TL) ⊂ ξ

Observe that, if α is a defining form of the contact structure ξ, then condition 2) is equivalent to say that p^∗(α) = 0. Hence Legendrian submanifolds are the analogue of Lagrangian submanifolds in contact geometry.

Philosophy of Dimensions: M-Theory. Thought of the Day 85.0

Superstrings provided a perturbatively finite theory of gravity which, after compactification down to 3+1 dimensions, seemed potentially capable of explaining the strong, weak and electromagnetic forces of the Standard Model, including the required chiral representations of quarks and leptons. However, there appeared to be not one but five seemingly different but mathematically consistent superstring theories: the E8 × E8 heterotic string, the SO(32) heterotic string, the SO(32) Type I string, and Types IIA and IIB strings. Each of these theories corresponded to a different way in which fermionic degrees of freedom could be added to the string worldsheet.

Supersymmetry constrains the upper limit on the number of spacetime dimensions to be eleven. Why, then, do superstring theories stop at ten? In fact, before the “first string revolution” of the mid-1980’s, many physicists sought superunification in eleven-dimensional supergravity. Solutions to this most primitive supergravity theory include the elementary supermembrane and its dual partner, the solitonic superfivebrane. These are supersymmetric objects extended over two and five spatial dimensions, respectively. This brings to mind another question: why do superstring theories generalize zero-dimensional point particles only to one-dimensional strings, rather than p-dimensional objects?

During the “second superstring revolution” of the mid-nineties it was found that, in addition to the 1+1-dimensional string solutions, string theory contains soliton-like Dirichlet branes. These Dp-branes have p + 1-dimensional worldvolumes, which are hyperplanes in 9 + 1-dimensional spacetime on which strings are allowed to end. If a closed string collides with a D-brane, it can turn into an open string whose ends move along the D-brane. The end points of such an open string satisfy conventional free boundary conditions along the worldvolume of the D-brane, and fixed (Dirichlet) boundary conditions are obeyed in the 9 − p dimensions transverse to the D-brane.

D-branes make it possible to probe string theories non-perturbatively, i.e., when the interactions are no longer assumed to be weak. This more complete picture makes it evident that the different string theories are actually related via a network of “dualities.” T-dualities relate two different string theories by interchanging winding modes and Kaluza-Klein states, via R → α′/R. For example, Type IIA string theory compactified on a circle of radius R is equivalent to Type IIB string theory compactified on a circle of radius 1/R. We have a similar relation between E8 × E8 and SO(32) heterotic string theories. While T-dualities remain manifest at weak-coupling, S-dualities are less well-established strong/weak-coupling relationships. For example, the SO(32) heterotic string is believed to be S-dual to the SO(32) Type I string, while the Type IIB string is self-S-dual. There is a duality of dualities, in which the T-dual of one theory is the S-dual of another. Compactification on various manifolds often leads to dualities. The heterotic string compactified on a six-dimensional torus T⁶ is believed to be self-S-dual. Also, the heterotic string on T⁴ is dual to the type II string on four-dimensional K3. The heterotic string on T⁶ is dual to the Type II string on a Calabi-Yau manifold. The Type IIA string on a Calabi-Yau manifold is dual to the Type IIB string on the mirror Calabi-Yau manifold.

This led to the discovery that all five string theories are actually different sectors of an eleven-dimensional non-perturbative theory, known as M-theory. When M-theory is compactified on a circle S¹ of radius R₁₁, it leads to the Type IIA string, with string coupling constant g_s = R^3/2₁₁. Thus, the illusion that this string theory is ten-dimensional is a remnant of weak-coupling perturbative methods. Similarly, if M-theory is compactified on a line segment S¹/Z₂, then the E8 × E8 heterotic string is recovered.

Just as a given string theory has a corresponding supergravity in its low-energy limit, eleven-dimensional supergravity is the low-energy limit of M-theory. Since we do not yet know what the full M-theory actually is, many different names have been attributed to the “M,” including Magical, Mystery, Matrix, and Membrane! Whenever we refer to “M-theory,” we mean the theory which subsumes all five string theories and whose low-energy limit is eleven-dimensional supergravity. We now have an adequate framework with which to understand a wealth of non-perturbative phenomena. For example, electric-magnetic duality in D = 4 is a consequence of string-string duality in D = 6, which in turn is the result of membrane-fivebrane duality in D = 11. Furthermore, the exact electric-magnetic duality has been extended to an effective duality of non-conformal N = 2 Seiberg-Witten theory, which can be derived from M-theory. In fact, it seems that all supersymmetric quantum field theories with any gauge group could have a geometrical interpretation through M-theory, as worldvolume fields propagating on a common intersection of stacks of p-branes wrapped around various cycles of compactified manifolds.

In addition, while perturbative string theory has vacuum degeneracy problems due to the billions of Calabi-Yau vacua, the non-perturbative effects of M-theory lead to smooth transitions from one Calabi-Yau manifold to another. Now the question to ask is not why do we live in one topology but rather why do we live in a particular corner of the unique topology. M-theory might offer a dynamical explanation of this. While supersymmetry ensures that the high-energy values of the Standard Model coupling constants meet at a common value, which is consistent with the idea of grand unification, the gravitational coupling constant just misses this meeting point. In fact, M-theory may resolve long-standing cosmological and quantum gravitational problems. For example, M-theory accounts for a microscopic description of black holes by supplying the necessary non-perturbative components, namely p-branes. This solves the problem of counting black hole entropy by internal degrees of freedom.

Hypersurfaces

Let (S, C_S) and (M, C_M) be manifolds of dimension k and n, respectively, with 1 ≤ k ≤ n. A smooth map : S → M is said to be an imbedding if it satisfies the following three conditions.

(I1) Ψ is injective.

(I2) At all points p in S, the associated (push-forward) linear map (Ψ_p)_∗ : S_p → M_Ψ(p) is injective.

(I3) ∀ open sets O₁ in S, Ψ[O₁] = [S] ∩ O₂ for some open set O₂ in M. (Equivalently, the inverse map Ψ⁻¹ : Ψ[S] → S is continuous with respect to the relative topology on [S].)

Several comments about the definition are in order. First, given any point p in S, (I2) implies that (Ψ_p)_∗[S_p] is a k-dimensional subspace of M_Ψ(p). So the condition cannot be satisfied unless k ≤ n. Second, the three conditions are independent of one another. For example, the smooth map Ψ : R → R² defined by (s) = (cos(s), sin(s)) satisfies (I2) and (I3) but is not injective. It wraps R round and round in a circle. On the other hand, the smooth map : R → R defined by (s) = s³ satisfies (I1) and (I3) but is not an imbedding because (Ψ₀)_∗ : R₀ → R₀ is not injective. (Here R₀ is the tangent space to the manifold R at the point 0). Finally, a smooth map : S → M can satisfy (I1) and (I2) but still have an image that “bunches up on itself.” It is precisely this possibility that is ruled out by condition (I3). Consider, for example, a map : R → R² whose image consists of part of the image of the curve y = sin(1/x) smoothly joined to the segment {(0, y) : y < 1}, as in the figure below. It satisfies conditions (I1) and (I2) but is not an imbedding because we can find an open interval O₁ in R such that given any open set O₂ in R², Ψ[O₁] ≠ O₂ ∩ Ψ[R].

Suppose(S, C_S) and (M, C_M) are manifolds with S ⊆ M. We say that (S, C_S) is an imbedded submanifold of (M, C_M) if the identity map id: S → M is an imbedding. If, in addition, k = n − 1 (where k and n are the dimensions of the two manifolds), we say that (S, C_S) is a hypersurface in (M, C_M). Let (S, C_S) be a k-dimensional imbedded submanifold of the n-dimensional manifold (M, C_M), and let p be a point in S. We need to distinguish two senses in which one can speak of “tensors at p.” There are tensors over the vector space Sp (call them S-tensors at p) and ones over the vector space M_p (call them M-tensors at p). So, for example, an S-vector ξ ̃^a at p makes assignments to maps of the form f ̃: O ̃ → R where O ̃ is a subset of S that is open in the topology induced by C_S, and f ̃ is smooth relative to C_S. In contrast, an M-vector ξ^a at p makes assignments to maps of the form f : O → R where O is a subset of M that is open in the topology induced by C_M, and f is smooth relative to C_M. Our first task is to consider the relation between S-tensors at p and M-tensors there.

Let us say that ξ^a ∈ (M_p)^a is tangent to S if ξ^a ∈ (id_p)_∗[(S_p)^a]. (This makes sense. We know that (id_p)_∗[(S_p)^a] is a k-dimensional subspace of (M_p)^a; ξ^a either belongs to that subspace or it does not.) Let us further say that η_a in (M_p)^a is normal to S if η_aξ^a =0 ∀ ξ^a ∈ (M_p)^a that are tangent to S. Each of these classes of vectors has a natural vector space structure. The space of vectors ξ^a ∈ (M_p)^a tangent to S has dimension k. The space of co-vectors η_a ∈ (M_p)^a normal to S has dimension (n − k).

Mathematical Reductionism: As Case Via C. S. Peirce’s Hypothetical Realism.

During the 20th century, the following epistemology of mathematics was predominant: a sufficient condition for the possibility of the cognition of objects is that these objects can be reduced to set theory. The conditions for the possibility of the cognition of the objects of set theory (the sets), in turn, can be given in various manners; in any event, the objects reduced to sets do not need an additional epistemological discussion – they “are” sets. Hence, such an epistemology relies ultimately on ontology. Frege conceived the axioms as descriptions of how we actually manipulate extensions of concepts in our thinking (and in this sense as inevitable and intuitive “laws of thought”). Hilbert admitted the use of intuition exclusively in metamathematics where the consistency proof is to be done (by which the appropriateness of the axioms would be established); Bourbaki takes the axioms as mere hypotheses. Hence, Bourbaki’s concept of justification is the weakest of the three: “it works as long as we encounter no contradiction”; nevertheless, it is still epistemology, because from this hypothetical-deductive point of view, one insists that at least a proof of relative consistency (i.e., a proof that the hypotheses are consistent with the frequently tested and approved framework of set theory) should be available.

Doing mathematics, one tries to give proofs for propositions, i.e., to deduce the propositions logically from other propositions (premisses). Now, in the reductionist perspective, a proof of a mathematical proposition yields an insight into the truth of the proposition, if the premisses are already established (if one has already an insight into their truth); this can be done by giving in turn proofs for them (in which new premisses will occur which ask again for an insight into their truth), or by agreeing to put them at the beginning (to consider them as axioms or postulates). The philosopher tries to understand how the decision about what propositions to take as axioms is arrived at, because he or she is dissatisfied with the reductionist claim that it is on these axioms that the insight into the truth of the deduced propositions rests. Actually, this epistemology might contain a short-coming since Poincaré (and Wittgenstein) stressed that to have a proof of a proposition is by no means the same as to have an insight into its truth.

Attempts to disclose the ontology of mathematical objects reveal the following tendency in epistemology of mathematics: Mathematics is seen as suffering from a lack of ontological “determinateness”, namely that this science (contrarily to many others) does not concern material data such that the concept of material truth is not available (especially in the case of the infinite). This tendency is embarrassing since on the other hand mathematical cognition is very often presented as cognition of the “greatest possible certainty” just because it seems not to be bound to material evidence, let alone experimental check.

The technical apparatus developed by the reductionist and set-theoretical approach nowadays serves other purposes, partly for the reason that tacit beliefs about sets were challenged; the explanations of the science which it provides are considered as irrelevant by the practitioners of this science. There is doubt that the above mentioned sufficient condition is also necessary; it is not even accepted throughout as a sufficient one. But what happens if some objects, as in the case of category theory, do not fulfill the condition? It seems that the reductionist approach, so to say, has been undocked from the historical development of the discipline in several respects; an alternative is required.

Anterior to Peirce, epistemology was dominated by the idea of a grasp of objects; since Descartes, intuition was considered throughout as a particular, innate capacity of cognition (even if idealists thought that it concerns the general, and empiricists that it concerns the particular). The task of this particular capacity was the foundation of epistemology; already from Aristotle’s first premisses of syllogism, what was aimed at was to go back to something first. In this traditional approach, it is by the ontology of the objects that one hopes to answer the fundamental question concerning the conditions for the possibility of the cognition of these objects. One hopes that there are simple “basic objects” to which the more complex objects can be reduced and whose cognition is possible by common sense – be this an innate or otherwise distinguished capacity of cognition common to all human beings. Here, epistemology is “wrapped up” in (or rests on) ontology; to do epistemology one has to do ontology first.

Peirce shares Kant’s opinion according to which the object depends on the subject; however, he does not agree that reason is the crucial means of cognition to be criticised. In his paper “Questions concerning certain faculties claimed for man”, he points out the basic assumption of pragmatist philosophy: every cognition is semiotically mediated. He says that there is no immediate cognition (a cognition which “refers immediately to its object”), but that every cognition “has been determined by a previous cognition” of the same object. Correspondingly, Peirce replaces critique of reason by critique of signs. He thinks that Kant’s distinction between the world of things per se (Dinge an sich) and the world of apparition (Erscheinungswelt) is not fruitful; he rather distinguishes the world of the subject and the world of the object, connected by signs; his position consequently is a “hypothetical realism” in which all cognitions are only valid with reservations. This position does not negate (nor assert) that the object per se (with the semiotical mediation stripped off) exists, since such assertions of “pure” existence are seen as necessarily hypothetical (that means, not withstanding philosophical criticism).

By his basic assumption, Peirce was led to reveal a problem concerning the subject matter of epistemology, since this assumption means in particular that there is no intuitive cognition in the classical sense (which is synonymous to “immediate”). Hence, one could no longer consider cognitions as objects; there is no intuitive cognition of an intuitive cognition. Intuition can be no more than a relation. “All the cognitive faculties we know of are relative, and consequently their products are relations”. According to this new point of view, intuition cannot any longer serve to found epistemology, in departure from the former reductionist attitude. A central argument of Peirce against reductionism or, as he puts it,

the reply to the argument that there must be a first is as follows: In retracing our way from our conclusions to premisses, or from determined cognitions to those which determine them, we finally reach, in all cases, a point beyond which the consciousness in the determined cognition is more lively than in the cognition which determines it.

Peirce gives some examples derived from physiological observations about perception, like the fact that the third dimension of space is inferred, and the blind spot of the retina. In this situation, the process of reduction loses its legitimacy since it no longer fulfills the function of cognition justification. At such a place, something happens which I would like to call an “exchange of levels”: the process of reduction is interrupted in that the things exchange the roles performed in the determination of a cognition: what was originally considered as determining is now determined by what was originally considered as asking for determination.

The idea that contents of cognition are necessarily provisional has an effect on the very concept of conditions for the possibility of cognitions. It seems that one can infer from Peirce’s words that what vouches for a cognition is not necessarily the cognition which determines it but the livelyness of our consciousness in the cognition. Here, “to vouch for a cognition” means no longer what it meant before (which was much the same as “to determine a cognition”), but it still means that the cognition is (provisionally) reliable. This conception of the livelyness of our consciousness roughly might be seen as a substitute for the capacity of intuition in Peirce’s epistemology – but only roughly, since it has a different coverage.

Ricci-flow as an “intrinsic-Ricci-flat” Space-time.

A Ricci flow solution {(M^m, g(t)), t ∈ I ⊂ R} is a smooth family of metrics satisfying the evolution equation

∂/∂t g = −2Rc —– (1)

where M^m is a complete manifold of dimension m. We assume that sup_M |Rm|_g(t) < ∞ for each time t ∈ I. This condition holds automatically if M is a closed manifold. It is very often to put an extra term on the right hand side of (1) to obtain the following rescaled Ricci flow

∂/∂t g = −2 {Rc + λ(t)g} —– (2)

where λ(t) is a function depending only on time. Typically, λ(t) is chosen as the average of the scalar curvature, i.e. , 1/m ∱Rdv or some fixed constant independent of time. In the case that M is closed and λ(t) = 1/m ∱Rdv, the flow is called the normalized Ricci flow. Starting from a positive Ricci curvature metric on a 3-manifold, Richard Hamilton showed that the normalized Ricci flow exists forever and converges to a space form metric. Hamilton developed the maximum principle for tensors to study the Ricci flow initiated from some metric with positive curvature conditions. For metrics without positive curvature condition, the study of Ricci flow was profoundly affected by the celebrated work of Grisha Perelman. He introduced new tools, i.e., the entropy functionals μ, ν, the reduced distance and the reduced volume, to investigate the behavior of the Ricci flow. Perelman’s new input enabled him to revive Hamilton’s program of Ricci flow with surgery, leading to solutions of the Poincaré conjecture and Thurston’s geometrization conjecture.

In the general theory of the Ricci flow developed by Perelman in, the entropy functionals μ and ν are of essential importance. Perelman discovered the monotonicity of such functionals and applied them to prove the no-local-collapsing theorem, which removes the stumbling block for Hamilton’s program of Ricci flow with surgery. By delicately using such monotonicity, he further proved the pseudo-locality theorem, which claims that the Ricci flow can not quickly turn an almost Euclidean region into a very curved one, no matter what happens far away. Besides the functionals, Perelman also introduced the reduced distance and reduced volume. In terms of them, the Ricci flow space-time admits a remarkable comparison geometry picture, which is the foundation of his “local”-version of the no-local-collapsing theorem. Each of the tools has its own advantages and shortcomings. The functionals μ and ν have the advantage that their definitions only require the information for each time slice (M, g(t)) of the flow. However, they are global invariants of the underlying manifold (M, g(t)). It is not convenient to apply them to study the local behavior around a given point x. Correspondingly, the reduced volume and the reduced distance reflect the natural comparison geometry picture of the space-time. Around a base point (x, t), the reduced volume and the reduced distance are closely related to the “local” geometry of (x, t). Unfortunately, it is the space-time “local”, rather than the Riemannian geometry “local” that is concerned by the reduced volume and reduced geodesic. In order to apply them, some extra conditions of the space-time neighborhood of (x, t) are usually required. However, such strong requirement of space-time is hard to fulfill. Therefore, it is desirable to have some new tools to balance the advantages of the reduced volume, the reduced distance and the entropy functionals.

Let (M^m, g) be a complete Ricci-flat manifold, x₀ is a point on M such that d(x₀, x) < A. Suppose the ball B(x₀, r₀) is A⁻¹−non-collapsed, i.e., r^−m₀|B(x₀, r₀)| ≥ A⁻¹, can we obtain uniform non-collapsing for the ball B(x, r), whenever 0 < r < r₀ and d(x, x₀) < Ar₀? This question can be answered easily by applying triangle inequalities and Bishop-Gromov volume comparison theorems. In particular, there exists a κ = κ(m, A) ≥ 3^−mA^−m−1 such that B(x, r) is κ-non-collapsed, i.e., r^−m|B(x, r)| ≥ κ. Consequently, there is an estimate of propagation speed of non-collapsing constant on the manifold M. This is illustrated by Figure

We now regard (M, g) as a trivial space-time {(M, g(t)), −∞ < t < ∞} such that g(t) ≡ g. Clearly, g(t) is a static Ricci flow solution by the Ricci-flatness of g. Then the above estimate can be explained as the propagation of volume non-collapsing constant on the space-time.

However, in a more intrinsic way, it can also be interpreted as the propagation of non-collapsing constant of Perelman’s reduced volume. On the Ricci flat space-time, Perelman’s reduced volume has a special formula

V((x, t)r²) = (4π)^-m/2 r^-m ∫_M e^{-d2(y, x)/4r2} dv_y —– (3)

which is almost the volume ratio of B_g(t)(x, r). On a general Ricci flow solution, the reduced volume is also well-defined and has monotonicity with respect to the parameter r², if one replace d²(y, x)/4r² in the above formula by the reduced distance l((x, t), (y, t − r²)). Therefore, via the comparison geometry of Bishop-Gromov type, one can regard a Ricci-flow as an “intrinsic-Ricci-flat” space-time. However, the disadvantage of the reduced volume explanation is also clear: it requires the curvature estimate in a whole space-time neighborhood around the point (x, t), rather than the scalar curvature estimate of a single time slice t.

Obstruction Theory

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 π_n(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 π_n(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 = g_σ 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; π_n(Y)) given by c(ƒ): σ → [f º g_σ]. Then it is clear that ƒ may be extended over Xⁿ ∪ _gσ σ iff f º g_σ 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 π_i(X).

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

Suppose ƒ, g are two maps Xⁿ → Y which agree on X^n-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; π_n(Y)) is called the difference cochain of ƒ and g, denoted d(ƒ, g).