Truncation Functors

Let A be an abelian category, and let D = D(A) be the derived category. For any complex A• in A, and n ∈ Z, we let τ_≤nA• be the truncated complex

··· → Aⁿ⁻² → Aⁿ⁻¹ → ker(Aⁿ → Aⁿ⁺¹)→ 0 → 0 → ··· , and dually we let τ_≥nA be the complex

··· → 0 → 0 → coker(Aⁿ⁻¹ → Aⁿ) → Aⁿ⁺¹ → Aⁿ⁺² → ···

Note that

H^m(τ_≤nA•) = H^m(A•) if m ≤ n

= 0 if m > n

and that

H^m(τ_≥nA•) = H^m(A•)  if m ≥ n

= 0 if m < n

One checks that τ_≥n (respectively τ_≤n) extends naturally to an additive functor of complexes which preserves homotopy and takes quasi-isomorphisms to quasi-isomorphisms, and hence induces an additive functor D → D. In fact if D_≤n (respectively D_≥n) is the full subcategory of D whose objects are the complexes A• such that H^m(A•) = 0 for m > n (respectively m < n) then we have additive functors

τ_≤n : D → D_≤n ⊂ D

τ_≥n : D → D_≥n ⊂ D

together with obvious functorial maps

iⁿ_A : τ_≤n A• → A•

jⁿ_A : A• → τ_≥n A•

The preceding iⁿ_A , jⁿ_A induce functorial isomorphisms

Hom_D≤n (B•,τ_≤nA•) →^~ Hom_D(B•, A•) (B• ∈ D_≤n) —– (1)

Hom_D≥n (τ_≥nA•,C•) →^~ Hom_D(A•,C• ) (C• ∈ D_≥n) —– (2)

Bijectivity of (1) means that any map φ : B• → A• (in D) with B• ∈ D_≤n factors uniquely via i_A := iⁿ_A

Given φ, we have a commutative diagram

and since B• ∈ D_≤n, therefore i_B is an isomorphism in D, so we can write

φ = i ◦ (τ_≤nφ ◦ i⁻¹_B),

and thus (1) is surjective.

To prove that (1) is also injective, we assume that i_A ◦ τ_≤n φ = 0 and deduce that τ_≤n φ = 0. The assumption means that there is a commutative diagram in K(A)

where s and s′′ are quasi-isomorphisms, and f/s = τ_≤nφ

Applying the (idempotent) functor τ_≥n, we get a commutative diagram

Since τ_≤ns and τ_≤ns′′ are quasi-isomorphisms, we have

τ_≤nφ = τ_≤n f/τ_≤ns = 0/τ_≤ns′′ = 0

as desired.

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Bernard Cache’s Earth Moves: The Furnishing of Territories (Writing Architecture)

Take the concept of singularity. In mathematics, what is said to be singular is not a given point, but rather a set of points on a given curve. A point is not singular; it becomes singularized on a continuum. And several types of singularity exist, starting with fractures in curves and other bumps in the road. We will discount them at the outset, for singularities that are marked by discontinuity signal events that are exterior to the curvature and are themselves easily identifiable. In the same way, we will eliminate singularities such as backup points [points de rebroussement]. For though they are indeed discontinuous, they refer to a vector that is tangential to the curve and thus trace a symmetrical axis that constitutive of the backup point. Whether it be a reflection of the tan- gential plane or a rebound with respect to the orthogonal plane, the backup point is thus not a basic singularity. It is rather the result of an operation effectuated on any part of the curve. Here again, the singular would be the sign of too noisy, too memorable an event, while what we want to do is to deal with what is most smooth: ordinary continua, sleek and polished.

On one hand there are the extrema, the maximum and minimum on a given curve. And on the other there are those singular points that, in relation to the extrema, figure as in-betweens. These are known as points of inflection. They are different from the extrema in that they are defined only in relation to themselves, whereas the definition of the extrema presupposes the prior choice of an axis or an orientation, that is to say of a vector.

Indeed, a maximum or a minimum is a point where the tangent to the curve is directed perpendicularly to the axis of the ordinates [y-axis]. Any new orientation of the coordinate axes repositions the maxima and the min- ima; they are thus extrinsic singularities. The point of inflection, however, designates a pure event of curvature where the tangent crosses the curve; yet this event does not depend in any way on the orientation of the axes, which is why it can be said that inflection is an intrinsic singularity. On either side of the inflection, we know that there will be a highest point and a lowest point, but we cannot designate them as long as the curve has not been related to the orientation of a vector. Points of inflection are singularities in and of themselves, while they confer an indeterminacy to the rest of the curve. Preceding the vector, inflection makes of each of the points a possible extremum in relation to its inverse: virtual maxima and minima. In this way, inflection represents a totality of possibilities, as well as an openness, a receptiveness, or an anticipation……

Bernard Cache Earth Moves The Furnishing of Territories

The Womb of Cosmogony. Thought of the Day 30.0

Nowhere and by no people was speculation allowed to range beyond those manifested gods. The boundless and infinite UNITY remained with every nation a virgin forbidden soil, untrodden by man’s thought, untouched by fruitless speculation. The only reference made to it was the brief conception of its diastolic and systolic property, of its periodical expansion or dilatation, and contraction. In the Universe with all its incalculable myriads of systems and worlds disappearing and re-appearing in eternity, the anthropomorphised powers, or gods, their Souls, had to disappear from view with their bodies: — “The breath returning to the eternal bosom which exhales and inhales them,” says our Catechism. . . . In every Cosmogony, behind and higher than the creative deity, there is a superior deity, a planner, an Architect, of whom the Creator is but the executive agent. And still higher, over and around, withinand without, there is the UNKNOWABLE and the unknown, the Source and Cause of all these Emanations. – The Secret Doctrine

Many are the names in the ancient literatures which have been given to the Womb of Being from which all issues, in which all forever is, and into the spiritual and divine reaches of which all ultimately returns, whether infinitesimal entity or macrocosmic spacial unit.

The Tibetans called this ineffable mystery Tong-pa-nnid, the unfathomable Abyss of the spiritual realms. The Buddhists of the Mahayana school describe it as Sunyata or the Emptiness, simply because no human imagination can figurate to itself the incomprehensible Fullness which it is. In the Eddas of ancient Scandinavia the Boundless was called by the suggestive term Ginnungagap – a word meaning yawning or uncircumscribed void. The Hebrew Bible states that the earth was formless and void, and darkness was upon the face of Tehom, the Deep, the Abyss of Waters, and therefore the great Deep of kosmic Space. It has the identical significance of the Womb of Space as envisioned by other peoples. In the Chaldaeo-Jewish Qabbalah the same idea is conveyed by the term ‘Eyn (or Ain) Soph, without bounds. In the Babylonian accounts of Genesis, it is Mummu Tiamatu which stands for the Great Sea or Deep. The archaic Chaldaean cosmology speaks of the Abyss under the name of Ab Soo, the Father or source of knowledge, and in primitive Magianism it was Zervan Akarana — in its original meaning of Boundless Spirit instead of the later connotation of Boundless Time.

In the Chinese cosmogony, Tsi-tsai, the Self-Existent, is the Unknown Darkness, the root of the Wuliang-sheu, Boundless Age. The wu wei of Lao-tse, often mistranslated as passivity and nonaction, imbodies a similar conception. In the sacred scriptures of the Quiches of Guatemala, the Popol Vuh or “Book of the Azure Veil,” reference is made to the “void which was the immensity of the Heavens,” and to the “Great Sea of Space.” The ancient Egyptians spoke of the Endless Deep; the same idea also is imbodied in the Celi-Ced of archaic Druidism, Ced being spoken of as the “Black Virgin” — Chaos — a state of matter prior to manvantaric differentiation.

The Orphic Mysteries taught of the Thrice-Unknown Darkness or Chronos, about which nothing could be predicated except its timeless Duration. With the Gnostic schools, as for instance with Valentinus, it was Bythos, the Deep. In Greece, the school of Democritus and Epicurus postulated To Kenon, the Void; the same idea was later voiced by Leucippus and Diagoras. But the two most common terms in Greek philosophy for the Boundless were Apeiron, as used by Plato, Anaximander and Anaximenes, and Apeiria, as used by Anaxagoras and Aristotle. Both words had the significance of frontierless expansion, that which has no circumscribing bounds.

The earliest conception of Chaos was that almost unthinkable condition of kosmic space or kosmic expanse, which to human minds is infinite and vacant extension of primordial Aether, a stage before the formation of manifested worlds, and out of which everything that later existed was born, including gods and men and all the celestial hosts. We see here a faithful echo of the archaic esoteric philosophy, because among the Greeks Chaos was the kosmic mother of Erebos and Nyx, Darkness and Night — two aspects of the same primordial kosmic stage. Erebos was the spiritual or active side corresponding to Brahman in Hindu philosophy, and Nyx the passive side corresponding to pradhana or mulaprakriti, both meaning root-nature. Then from Erebos and Nyx as dual were born Aether and Hemera, Spirit and Day — Spirit being here again in this succeeding stage the active side, and Day the passive aspect, the substantial or vehicular side. The idea was that just as in the Day of Brahma of Hindu cosmogony things spring into active manifested existence, so in the kosmic Day of the Greeks things spring from elemental substance into manifested light and activity, because of the indwelling urge of the kosmic Spirit.

Thermodynamics of Creation. Note Quote.

Just like the early-time cosmic acceleration associated with inflation, a negative pressure can be seen as a possible driving mechanism for the late-time accelerated expansion of the Universe as well. One of the earliest alternatives that could provide a mechanism producing such accelerating phase of the Universe is through a negative pressure produced by viscous or particle production effects. The viscous pressure contributions can be seen as small nonequilibrium contributions for the energy-momentum tensor for nonideal fluids.

Let us posit the thermodynamics of matter creation for a single fluid. To describe the thermodynamic states of a relativistic simple fluid we use the following macroscopic variables: the energy-momentum tensor T^αβ ; the particle flux vector N^α; and the entropy flux vector s^α. The energy-momentum tensor satisfies the conservation law, T^αβ_;β = 0, and here we consider situations in which it has the perfect-fluid form

T^αβ = (ρ+P)u^αu^β − P g^αβ

In the above equation ρ is the energy density, P is the isotropic dynamical pressure, g^αβ is the metric tensor and u^α is the fluid four-velocity (with normalization u^αu_α = 1).

The dynamical pressure P is decomposed as

P = p + Π

where p is the equilibrium (thermostatic) pressure and Π is a term present in scalar dissipative processes. Usually, it is associated with the so-called bulk pressure. In the cosmological context, besides this meaning, Π can also be relevant when particle number is not conserved. In this case, Π ≡ p_c is called the “creation pressure”. The bulk pressure,  can be seen as a correction to the thermostatic pressure when near to equilibrium, thus, it should be always smaller than the thermostatic pressure, |Π| < p. This restriction, however, does not apply for the creation pressure. So, when we have matter creation, the total pressure P may become negative and, in principle, drive an accelerated expansion.

The particle flux vector is assumed to have the following form

N^α = nu^α

where n is the particle number density. N^α satisfies the balance equation N^α_;α = nΓ, where Γ is the particle production rate. If Γ > 0, we have particle creation, particle destruction occurs when Γ < 0 and if Γ = 0 particle number is conserved.

The entropy flux vector is given by

s^α = nσu^α

where σ is the specific (per particle) entropy. Note that the entropy must satisfy the second law of thermodynamics s^α_;α ≥ 0. Here we consider adiabatic matter creation, that is, we analyze situations in which σ is constant. With this condition, by using the Gibbs relation, it follows that the creation pressure is related to Γ by

p_c = − (ρ+p)/3H Γ

where H = a ̇/a is the Hubble parameter, a is the scale factor of the Friedmann-Robertson-Walker (FRW) metric and the overdot means differentiation with respect to the cosmic time. If σ is constant, the second law of thermodynamics implies that Γ ≥ 0 and, as a consequence, particle destruction (Γ < 0) is thermodynamically forbidden. Since Γ ≥ 0, it follows that, in an expanding universe (H > 0), the creation pressure p_c cannot be positive.

Dissipations – Bifurcations – Synchronicities. Thought of the Day 29.0

Deleuze’s thinking expounds on Bergson’s adaptation of multiplicities in step with the catastrophe theory, chaos theory, dissipative systems theory, and quantum theory of his era. For Bergson, hybrid scientific/philosophical methodologies were not viable. He advocated tandem explorations, the two “halves” of the Absolute “to which science and metaphysics correspond” as a way to conceive the relations of parallel domains. The distinctive creative processes of these disciplines remain irreconcilable differences-in-kind, commonly manifesting in lived experience. Bergson: Science is abstract, philosophy is concrete. Deleuze and Guattari: Science thinks the function, philosophy the concept. Bergson’s Intuition is a method of division. It differentiates tendencies, forces. Division bifurcates. Bifurcations are integral to contingency and difference in systems logic.

The branching of a solution into multiple solutions as a system is varied. This bifurcating principle is also known as contingency. Bifurcations mark a point or an event at which a system divides into two alternative behaviours. Each trajectory is possible. The line of flight actually followed is often indeterminate. This is the site of a contingency, were it a positionable “thing.” It is at once a unity, a dualism and a multiplicity:

Bifurcations are the manifestation of an intrinsic differentiation between parts of the system itself and the system and its environment. […] The temporal description of such systems involves both deterministic processes (between bifurcations) and probabilistic processes (in the choice of branches). There is also a historical dimension involved […] Once we have dissipative structures we can speak of self-organisation.

Figure: In a dynamical system, a bifurcation is a period doubling, quadrupling, etc., that accompanies the onset of chaos. It represents the sudden appearance of a qualitatively different solution for a nonlinear system as some parameter is varied. The illustration above shows bifurcations (occurring at the location of the blue lines) of the logistic map as the parameter r is varied. Bifurcations come in four basic varieties: flip bifurcation, fold bifurcation, pitchfork bifurcation, and transcritical bifurcation. 

A bifurcation, according to Prigogine and Stengers, exhibits determinacy and choice. It pertains to critical points, to singular intensities and their division into multiplicities. The scientific term, bifurcation, can be substituted for differentiation when exploring processes of thought or as Massumi explains affect:

Affect and intensity […] is akin to what is called a critical point, or bifurcation point, or singular point, in chaos theory and the theory of dissipative structures. This is the turning point at which a physical system paradoxically embodies multiple and normally mutually exclusive potentials… 

The endless bifurcating division of progressive iterations, the making of multiplicities by continually differentiating binaries, by multiplying divisions of dualities – this is the ontological method of Bergson and Deleuze after him. Bifurcations diagram multiplicities, from monisms to dualisms, from differentiation to differenciation, creatively progressing. Manuel Delanda offers this account, which describes the additional technicality of control parameters, analogous to higher-level computer technologies that enable dynamic interaction. These protocols and variable control parameters are later discussed in detail in terms of media objects in the metaphorical state space of an in situ technology:

[…] for the purpose of defining an entity to replace essences, the aspect of state space that mattered was its singularities. One singularity (or set of singularities) may undergo a symmetry-breaking transition and be converted into another one. These transitions are called bifurcations and may be studied by adding to a particular state space one or more ‘control knobs’ (technically control parameters) which determine the strength of external shocks or perturbations to which the system being modeled may be subject.

Another useful example of bifurcation with respect to research in the neurological and cognitive sciences is Francesco Varela’s theory of the emergence of microidentities and microworlds. The ready-for-action neuronal clusters that produce microindentities, from moment to moment, are what he calls bifurcating “break- downs”. These critical events in which a path or microidentity is chosen are, by implication, creative:

On the basis of this fast dynamic, as in an evolutionary process, one neuronal ensemble (one cognitive subnetwork) finally becomes more prevalent and becomes the behavioral mode for the next cognitive moment. By “becomes more prevalent” I do not mean to say that this is a process of optimization: it resembles more a bifurcation or symmetry-breaking form of chaotic dynamics.

Varela’s “breakdown” is an event triggering diachronic emergence – a timeless interval of reserve potential that “restructures the virtual” (Protevi). Varela, with his deeply phenomenological leanings, and Deleuze and Guattari, with their nomadic surfaces, share a jagged cartography of concurrence. Their leanings vacillate (resonate) between difference and resemblance.

Local Lifts into Period Domains: Holonomies: Philosophies of Conjugacy. Part 2.

Let F = G^C/P be a flag manifold. Then there is a unique inner symmetric space G-space N associated to F together with a finite number of homogeneous fibrations F → N.

Let us emphasise that this construction depends on nothing but the conjugacy class of p ⊂ g^C and the choice of compact real form g. Equivalently, it depends solely on the choice of invariant complex structure on F.

Every flag manifold fibres over an inner symmetric space. Conversely, every inner symmetric space is the target of the canonical fibrations of at least one flag manifold. Let us now see how this story relates to the geometry of J(N).

So let p : F → N be a canonical fibration. By construction, the fibres of p are complex submanifolds of F and this allows us to define a fibre map i_p : F → J(N) as follows: at f ∈ F we have an orthogonal splitting of T_fF into horizontal and vertical subspaces both of which are invariant under the complex structure of F. Then dp restricts to give an isomorphism of the horizontal part with T_p(f)N and therefore induces an almost Hermitian structure on T_p(f)N : this is i_p(f) ∈ J_p(f)N. Such a construction is possible whenever we have a Riemannian submersion of a Hermitian manifold with complex submanifolds as fibres.

i_p : F → J(N) is a G-equivariant holomorphic embedding. This implies that i_p (F) is an almost complex submanifold of J(N) on which J is integrable.

If j ∈ Z ⊂ J(N) then G · j is a flag manifold canonically fibred over N. In fact, G · j = i_p(F ) for some canonical fibration p : F → N of a flag manifold F .

For this, the main observation is the following: at π(j), we have the symmetric decomposition g = k ⊕ q

with q ≅ T_π(j)N. If q⁻ is the (0,1)-space for j then [q⁻, q⁻] ⊕ q⁻

is the nilradical of a parabolic subalgebra p, where G · j is equivariantly biholomorphic to the corresponding flag manifold G^C/P. Each canonical fibration of a flag manifold gives rise to a G-orbit in Z for some inner symmetric G-space N and that all such orbits arise in this way. But, for fixed G, there are only a finite number of biholomorphism types of flag manifold (they are in bijective correspondence with the conjugacy classes of parabolic subalgebras of g^C) and each flag manifold admits but a finite number of canonical fibrations. Thus Z is composed of a finite number of G-orbits all of which are closed. In this way, we obtain a geometric interpretation of the purely algebraic construction of the canonical fibrations: they are just the restrictions of the projection π : J(N) → N to the various realisations of F as an orbit in Z.

For each non-compact real form G_R of a complex semisimple group Lie group G^C, there is a unique Riemannian symmetric space G_R/K of non-compact type. The corresponding involution is called the Cartan involution of G_R. Consider now the orbits of such a G_R on the various flag manifolds F = G^C/P. Those orbits which are open subsets of F are called flag domains: an orbit is a flag domain precisely when the stabilisers contain a compact Cartan subgroup of G_R. It turns out that the presence of this compact Cartan subgroup is precisely what we need to define a canonical element of g_R and thus an involution of g_R just as in the compact case. However the involution is not necessarily a Cartan involution (i.e. the associated symmetric space need not be Riemmanian). In case that the involution is a Cartan involution, the flag domain is a canonical flag domain which is then exponentiated such that the involution gets to a Riemannian symmetric space of non-compact type and a canonical fibration of canonical flag domain over it.

Let X be a flag manifold or canonical flag domain and π : X → N be a homogeneous fibration onto a Riemannian symmetric space. If φ : M → X is a holomorphic map of a Kahler manifold with image tangent to the super-horizontal distribution then π ◦ φ : M → N is harmonic.

Maps φ : M → X satisfying such hypotheses of being super-horizontal are holomorphic maps.

For instance, let X = SU(n + 1)/S(U(1) × ··· × U(1)) = F(1,…,1; C_n+1). There are n+1 homogeneous fibrations π_i : X → CPⁿ, i = 1,…,n, with π₀ holomorphic and π_n anti-holomorphic. Now a super-horizontal holomorphic map φ : S² → X is essentially just the Frenet frame of the holomorphic map π₀ ◦ φ : S² → CPⁿ while each π ◦ φ is a harmonic map S² → CPⁿ. It is the content of the classification theorem for harmonic 2-spheres in CPⁿ that all such harmonic maps arise in this way. On the non-compact side of the fence, the super-horizontal distribution is precisely that which defines the infinitesimal period relation. Thus the (local lifts of) period maps are precisely the super-horizontal holomorphic maps into period domains.

Holonomies: Philosophies of Conjugacy. Part 1.

Suppose that N is an irreducible 2n-dimensional Riemannian symmetric space. We may realise N as a coset space N = G/K with G^τ ⊂ K ⊂ (G^τ)₀ for some involution τ of G. Now K is (a covering of) the holonomy group of N and similarly the coset fibration G → G/K covers the holonomy bundle P → N. In this setting, J(N) is associated to G:

J(N) ≅ G ×_K J (R²ⁿ)

and if K/H is a K-orbit in J(R²ⁿ) then the corresponding subbundle is G ×_K K/H = G/H and the projection is just the coset fibration. Thus, the subbundles of J(N) are just the orbits of G in J(N).

Let j ∈ J (N). Then G · j is an almost complex submanifold of J (N) on which J is integrable iff j lies in the zero-set of the Nijenhuis tensor N^J.

This focusses our attention on the zero-set of NJ which we denote by Z. In favourable circumstances, the structure of this set can be completely described. We begin by assuming that N is of compact type so that G is compact and semi-simple. We also assume that N is inner i.e. that τ is an inner involution of G or, equivalently, that rankG = rankK. The class of inner symmetric spaces include the even-dimensional spheres, the Hermitian symmetric spaces, the quaternionic Kähler symmetric spaces and indeed all symmetric G-spaces for G = SO(2n+1), Sp(n), E₇, E₈, F₄ and G₂. Moreover, all inner symmetric spaces are necessarily even-dimensional and so fit into our framework.

Let N = G/K be a simply-connected inner Riemannian symmetric space of compact type. Then Z consists of a finite number of connected components on each of which G acts transitively. Moreover, any G-flag manifold is realised as such an orbit for some N.

The proof for the above requires a detour into the geometry of flag manifolds and reveals an interesting interaction between the complex geometry of flag manifolds and the real geometry of inner symmetric spaces. For this, we begin by noting that a coset space of the form G/C(T) admits several invariant Kählerian complex structures in general. Using a complex realisation of G/C(T) as follows: having fixed a complex structure, the complexified group G^C acts transitively on G/C(T) by biholomorphisms with parabolic subgroups as stabilisers. Conversely, if P ⊂ G^C is a parabolic subgroup then the action of G on G^C/P is transitive and G ∩ P is the centraliser of a torus in G. For the infinitesimal situation: let F = G/C(T) be a flag manifold and let o ∈ F. We have a splitting of the Lie algebra of G

g^C = h ⊕ m

with m ≅ T_oF and h the Lie algebra of the stabiliser of o in G. An invariant complex structure on F induces an ad h-invariant splitting of m^C into (1, 0) and (0, 1) spaces m^C = m⁺ ⊕ m⁻ with [m⁺, m⁺] ⊂ m⁺ by integrability. One can show that m⁺ and m⁻ are nilpotent subalgebras of g^C and in fact h^C ⊕ m⁻ is a parabolic subalgebra of g^C with nilradical m⁻. If P is the corresponding parabolic subgroup of G^C then P is the stabiliser of o and we obtain a biholomorphism between the complex coset space G^C/P and the flag manifold F.

Conversely, let P ⊂ G^C be a parabolic subgroup with Lie algebra p and let n be the conjugate of the nilradical of p (with respect to the real form g). Then H = G ∩ P is the centraliser of a torus and we have orthogonal decompositions (with respect to the Killing inner product)

p = h^C ⊕ n^–, g^C = h^C ⊕ n ⊕ n^–

which define an invariant complex structure on G/H realising the biholomorphism with G^C/P.

The relationship between a flag manifold F = G^C/P and an inner symmetric space comes from an examination of the central descending series of n. This is a filtration 0 = n_k+1 ⊂ n_k ⊂…⊂ n₁ = n of n defined by n_i = [n, n_i−1].

We orthogonalise this filtration using the Killing inner product by setting

g_i = n^⊥_i+1 ∩ n_i

for i ≥ 1 and extend this to a decomposition of g^C by setting g₀ = h^C = (g ∩ p)^C and g_−i = g^–_i for i ≥ 1. Then

g^C = ∑g_i

is an orthogonal decomposition with

p = ∑_i≤0 g_i, n = ∑_i>0 g_i 

The crucial property of this decomposition is that

[g_i, g_j] ⊂ g_i+j

which can be proved by demonstrating the existence of an element ξ ∈ h with the property that, for each i, adξ has eigenvalue √−1i on g_i. This element ξ (necessarily unique since g is semi-simple) is the canonical element of p. Since ad ξ has eigenvalues in √−1Z, ad exp πξ is an involution of g which we exponentiate to obtain an inner involution τ_ξ of G and thus an inner symmetric space G/K where K = (G^τ_ξ)₀. Clearly, K has Lie algebra given by

k = g ∩ ∑_i g_2i

and so contains H whence we obtain a homogeneous fibration G/H → G/K of our flag manifold over our inner symmetric space. Moreover, this fibration is essentially unique: the only ambiguity in the prescription is that several points in the symmetric space might have the same stabiliser K (e.g. antipodal points on a sphere). However, the number of such points is finite and so we only get a finite number of such fibrations.

Let F = G^C/P be a flag manifold. Then there is a unique inner symmetric space G-space N associated to F together with a finite number of homogeneous fibrations F → N.

Let us emphasise that this construction depends on nothing but the conjugacy class of p ⊂ g^C and the choice of compact real form g. Equivalently, it depends solely on the choice of invariant complex structure on F.

Every flag manifold fibres over an inner symmetric space. Conversely, it is straightforward to show that every inner symmetric space is the target of the canonical fibrations of at least one flag manifold. Let us see in the next post how this story relates to the geometry of J(N)……

Micropolitics, an Aesthetic. Thought of the Day 28.0

The Situationists, mutating from their Futurist, Surrealist and Dadaist predecessors, were pivotal in introducing a variegated political aesthetic that traversed polemics and play: taking the détournement to the streets.

Emerging and fading in the Outside ‘of uncertain doubles and partial deaths’ is a micropolitics, the molecular shapings of perceptions, attitudes, representational systems, etc.; transgressive lines of resistance (Foucault), lines of flight (Deleuze), ‘desiring-productions’ that interface a creative in-between of extreme macropolitical forces such as fascism and capitalism. Micropolitics produces an ethical aesthetic of the affective kind. It resonates with artistic practice in addressing Foucault’s insight that ‘resistance comes first’. Micropolitics is a multiplicity, disavowing identifiable unities. Both micropolitics and microperceptions actualize from the immanent cause of the unformed-unthought, from lines of resistance, to perceive, think, act and distribute processually. Points of deterriorialization are cutting edges, the avant-garde as it were, of the  distinguishing and convolving of matter and function. As the abstract machine becomes formalized through its conjunctive lines of resistance/flight and cutting edges, it effects a biopolitical aesthetic.

Cartographies of Disjunction’s Relational Dust. Thought of the Day 27.0

The biogrammatic interface, generates a political aesthetic in which action is felt through the affective modulations and/or tonalities it incites. A doubling occurs in the moving towards realization, the rearticulation, of the becoming-thing/gesture. This doubling divides in a central differentiation, referencing a voluminous vocabulary of the interstitial – fissure, gap, disjunction, in-between, crack, interface, fold, non-place – descriptors of a bifurcating rift between content and expression, necessary for realization. Deleuze sums up the crux of this Foucauldian argument:

Things can be realized only through doubling or dissociation, creating divergent forms among which they can be distributed. It is here that we see the great dualities: between different classes, or the governing and the governed, or the public and the private. But more than this, it is here that the two forms of realization diverge or become differentiated: a form of expression and a form of content, a discursive and a non- discursive form, the form of the visible and the form of the articulable. It is precisely because the immanent cause, in both its matter and its functions, disregards form, that it is realized on the basis of a central differentiation which, on the one hand will form visible matter, and on the other will formalize articulable functions.’ (Gilles Deleuze, Sean Hand-Foucault)

It can be argued that this central differentiation or interface distinguishes between the movements of two diagrammatic registers: outside from inside and the forms of realization. Transductive processes between these registers mark portals of entry through which all points of the diagram are in superposition, in passage as intensities of non-localizable relations from one point to another. The diagram distributes affective intensities within the context it maps.

Deleuze elasticizes Foucault’s reach by translating his oeuvre within the folding/unfolding of a knowledge-power-subjectivity continuum, mapping Foucault’s relays between the bifurcating polarities of content/expression, visibilities/statements as they differentiate and integrate through the folding ‘zone of subjectification’. The biogramming interface. The ‘event’ of rearticulation, of knowledge-capture and distribution, takes place through the perceptual filter of differential relations becoming-actual as a perception or thought. This is a topological dynamic mapped by the diagram, affected through the central differentiation (biogram) ‘or the ‘non-place’, as Foucault puts it, where the informal diagram is swallowed up and becomes embodied instead in two different directions that are necessarily divergent and irreducible. The concrete assemblages are therefore opened up by a crack that determines how the abstract machine performs’. It’s the process of swallowing up the relational intensities of a milieu and spitting back out certain selected somethings to be swallowed again that’s of particular interest to political aesthetics of the performative event. Foucault imagined a cartographic container of forces, affects, attractions and repulsions that modulate the diagram, excite the disjunction that separates forms of realization. The abstract machine begins to actualize its virtual potential as it distributes its relational dust.

…….

Kenneth Knoespel notes that diagramma in the original Greek does ‘not simply mean something that is marked out by lines, a figure, a form or a plan, but also carries a second connotation of marking or crossing out,’ suggesting not only ephemerality but also an incompleteness that carries an expectation of potential. ‘What is interesting is that the diagram participates in a geneology of figures that moves from the wax tablet to the computer screen […] the Greek setting of diagram suggests that any figure that is drawn is accompanied by an expectancy that it will be redrawn […] Here a diagram may be thought of as a relay. While a diagram may have been used visually to reinforce an idea one moment, the next it may provide a means of seeing something never seen before. Diagrams As Piloting Devices…

 

Optimal Hedging…..

Risk management is important in the practices of financial institutions and other corporations. Derivatives are popular instruments to hedge exposures due to currency, interest rate and other market risks. An important step of risk management is to use these derivatives in an optimal way. The most popular derivatives are forwards, options and swaps. They are basic blocks for all sorts of other more complicated derivatives, and should be used prudently. Several parameters need to be determined in the processes of risk management, and it is necessary to investigate the influence of these parameters on the aims of the hedging policies and the possibility of achieving these goals.

The problem of determining the optimal strike price and optimal hedging ratio is considered, where a put option is used to hedge market risk under a constraint of budget. The chosen option is supposed to finish in-the-money at maturity in the, such that the predicted loss of the hedged portfolio is different from the realized loss. The aim of hedging is to minimize the potential loss of investment under a specified level of confidence. In other words, the optimal hedging strategy is to minimize the Value-at-Risk (VaR) under a specified level of risk.

A stock is supposed to be bought at time zero with price S₀, and to be sold at time T with uncertain price S_T. In order to hedge the market risk of the stock, the company decides to choose one of the available put options written on the same stock with maturity at time τ, where τ is prior and close to T, and the n available put options are specified by their strike prices K_i (i = 1, 2,··· , n). As the prices of different put options are also different, the company needs to determine an optimal hedge ratio h (0 ≤ h ≤ 1) with respect to the chosen strike price. The cost of hedging should be less than or equal to the predetermined hedging budget C. In other words, the company needs to determine the optimal strike price and hedging ratio under the constraint of hedging budget.

Suppose the market price of the stock is S₀ at time zero, the hedge ratio is h, the price of the put option is P₀, and the riskless interest rate is r. At time T, the time value of the hedging portfolio is

S₀e^rT + hP₀e^rT —– (1)

and the market price of the portfolio is

S_T + h(K − S_τ)⁺ e^r(T−τ) —– (2)

therefore the loss of the portfolio is

L = (S₀e^rT + hP₀e^rT) − (S_T +h(K−S_τ)⁺ e^r(T−τ)) —– (3)

where x⁺ = max(x, 0), which is the payoff function of put option at maturity.

For a given threshold v, the probability that the amount of loss exceeds v is denoted as

α = Prob{L ≥ v} —– (4)

in other words, v is the Value-at-Risk (VaR) at α percentage level. There are several alternative measures of risk, such as CVaR (Conditional Value-at-Risk), ESF (Expected Shortfall), CTE (Conditional Tail Expectation), and other coherent risk measures. The criterion of optimality is to minimize the VaR of the hedging strategy.

The mathematical model of stock price is chosen to be a geometric Brownian motion, i.e.

dS_t/S_t = μdt + σdB_t —– (5)

where S_t is the stock price at time t (0 < t ≤ T), μ and σ are the drift and the volatility of stock price, and B_t is a standard Brownian motion. The solution of the stochastic differential equation is

S_t = S₀ e^{σB_t + (μ−1/2σ2)t} —– (6)

where B₀ = 0, and S_t is lognormally distributed.

Proposition:

For a given threshold of loss v, the probability that the loss exceeds v is

Prob {L ≥ v} = E [I_{{X ≤ c1}} F_Y (g(X) − X)] + E [I_{{X ≥ c1}} F_Y (c₂ − X)] —– (7)

where E[X] is the expectation of random variable X. I_{{X < c}} is the index function of X such that I_{{X < c}} = 1 when {X < c} is true, otherwise I_{{X < c}} = 0. F_Y (y) is the cumulative distribution function of random variable Y , and

c₁ = 1/σ [ln(K/S₀) − (μ−1/2σ²)τ] ,

g(X) = 1/σ [(ln (S₀ + hP₀)e^rT − h (K − f(X)) e^r(T−τ) −v)/S₀ − (μ − 1/2σ²) T],

f(X) = S₀ e^σX + (μ−1/2σ²)τ,

c₂ = 1/σ [(ln (S₀ + hP₀) e^rT − v)/S₀ − (μ− 1/2σ²) T

X and Y are both normally distributed, where X ∼ N(0,√τ), Y ∼ N(0,√(T−τ).

For a specified hedging strategy, Q(v) = Prob {L ≥ v} is a decreasing function of v. The VaR under α level can be obtained from equation

Q(v) = α —– (8)

The expectations in Proposition can be calculated with Monte Carlo simulation methods, and the optimal hedging strategy which has the smallest VaR can be obtained from equation (8) by numerical searching methods….