Dimension Conjecture Research Articles

Dimension of hyperbolic measures of random diffeomorphisms

We consider dynamics of compositions of stationary random C 2 diffeomorphisms. We will prove that the sample measures of an ergodic hyperbolic invariant measure of the system are exact dimensional. This is an extension to random diffeomorphisms of the main result of Barreira, Pesin and Schmeling (1999), which proves the Eckmann-Ruelle dimension conjecture for a deterministic diffeomorphism.

A geometric approach to the Jacobian conjecture in dimension two

The Jacobian conjecture in dimension two holds true if a conjecture, which we call below the generalized Sard property (GSP), holds true for the affine plane and an A 1 -fibration on A 2 . The observations will be made for affine pseudo-planes with the Jacobian conjecture enlarged to the generalized Jacobian conjecture.

The Divergence-Free Jacobian Conjecture in Dimension Two

A special case, called the divergence-free case, of the Jacobian Conjecture in dimension two is proved. This note outlines an argument for a special case of the Jacobian conjecture in dimension two: Suppose F : C → C is a polynomial so that F (0) = 0, F ′(0) = I, det(F ′(z)) = 1, z ∈ C. (1) where I is the identity transformation on C. Write F (x, y) = ( r(x, y) + x s(x, y) + y ) , (x, y) ∈ C where r, s have no non-zero constant or linear terms and observe that detF ′ = {r, s}+∇ · ( r s ) + 1

Tilting modules and Gorenstein rings

In his pioneering work [30], Miyashita extended classical tilting theory to the setting of finitely presented modules of finite projective dimension over an arbitrary ring. More recently, tilting theory has been generalized to arbitrary modules of finite projective dimension [2], [3], [6], et al. Encouraged by this developement, in the present paper, we apply infinite dimensional tilting theory to the setting of modules over Iwanaga-Gorenstein rings. We start with pointing out the central role played by resolving subcategories of modR for an arbitrary ring R. In Theorem 2.2 we prove that these subcategories correspond bijectively to tilting classes of finite type in ModR, as well as to cotilting classes of cofinite type in RMod. This generalizes a classical result of Auslander-Reiten [9] characterizing cotilting classes in modR over an artin algebra R. Moreover, it provides for a bijection between certain cotilting left modules and tilting right modules which extends the duality between finitely generated cotilting and tilting modules over artin algebras. We then consider Iwanaga-Gorenstein rings, that is (not necessarily commutative) two-sided noetherian rings of finite selfinjective dimension on both sides. We show that the first finitistic dimension conjecture holds true for any Iwanaga-Gorenstein ring (Theorem 3.2). We prove that Gorenstein injectivity, and ∗This work was started while the first author was a Ramon y Cajal Fellow at the Universitat Autonoma de Barcelona. The research of the second author was partially supported by the DGESIC (Spain) through the project PB98-0873, and by the Comissionat per Universitats i Recerca de la Generalitat de Catalunya. Third author supported by grant SAB2001-0092 of Secretaria de Estado de Educacion y Universidades MECD at CRM IEC Barcelona, and by MSM 113200007.

The Weinstein conjecture for planar contact structures in dimension three

In this paper we describe a general strategy for approaching the Weinstein conjecture in dimension three. We apply this approach to prove the Weinstein conjecture for a new class of contact manifolds (planar contact manifolds). We also discuss how the present approach reduces the general Weinstein conjecture in dimension three to a compactness problem for the solution set of a first order elliptic PDE.

Classification of finite-dimensional estimation algebras of maximal rank with arbitrary state–space dimension and mitter conjecture

In the late seventies, the concept of the estimation algebra of a filtering system was introduced. It was proven to be an invaluable tool in the study of non-linear filtering problems. In the early eighties, Brockett proposed to classify finite dimensional estimation algebras and Mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one. Despite the massive effort in understanding the finite dimensional estimation algebras, the 20 year old problem of Brockett and Mitter conjecture remains open. In this paper, we give a classification of finite dimensional estimation algebras of maximal rank and solve the Mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank. In particular, for an estimation algebra E of maximal rank, we give a necessary and sufficient conditions for E to be finite dimensional in terms of the drift fi (x) and observation hj (x). As an important corollary, we show that the number of statistics needed to compute the conditional density of the state given the observation {y(s):0 ≤ s ≤ t} by the algebraic method is n where n is the dimension of the state.

Symmetries of Real Hypersurfaces in Complex 3-Space

The main result of the paper consists in the proof of the fact that for any germ of a real analytic hypersurface in complex 3-space the following alternative (dimension conjecture) takes place: either the dimension of the group of holomorphic symmetries of the germ is at most the dimension of that of a nondegenerate hyperquadric (the latter equals 15), or the group is infinite-dimensional. We also discuss mistakes found in A. Ershova’s paper.

Classification des variété approximativement kähleriennes homogénes

A Riemannian homogeneous manifold admitting a strict nearly-Kahler structure is 3-symmetric. We actually classify them in dimension 6 and use previous results of Swann, Cleyton and Nagy to prove the conjecture in higher dimensions. The six-dimensional homogeneous spaces, S3 × S3, S6, CP(3) and the flag manifold F(1, 2) have a unique (after a change of scale) nearly-Kahler, invariant structure. For the first one we solve a differential equation on the SU(3)-structure given by Reyes Carrion. For the last two it is obtained by canonical variation of the Kahler structure of the twistor space over a four-dimensional manifold. Finally, from Bar, a nearly-Kahler structure on the sphere S6 corresponds to a constant 3-form on the Riemannian cone R7.

Two theorems on harmonic manifolds

A Riemannian manifold is called harmonic , if for any point x it admits a nonconstant harmonic function depending only on the distance to x . A.Lichnerowicz conjectured that any harmonic manifold is two-point homogeneous. This conjecture is proved in dimension n \le 4 and also for some classes of manifolds, but disproved in general, with the first counterexample of dimension 7 . We prove the Lichnerowicz Conjecture in dimension 5 : a five-dimensional harmonic manifold has constant sectional curvature. We also obtain a functional equation for the volume density function \theta(r) of a harmonic manifold and show that \theta(r) is an exponential polynomial, a finite linear combination of the terms of the form \mathrm{Re} (c e^{\lambda r} r^m) , with c, \lambda complex constants.

The second law of thermodynamics and multifractal distribution functions: Bin counting, pair correlations, and the Kaplan–Yorke conjecture

We explore and compare numerical methods for the determination of multifractal dimensions for a doubly-thermostatted harmonic oscillator. The equations of motion are continuous and time-reversible. At equilibrium the distribution is a four-dimensional Gaussian, so that all the dimension calculations can be carried out analytically. Away from equilibrium the distribution is a surprisingly isotropic multifractal strange attractor, with the various fractal dimensionalities in the range 1 < D < 4. The attractor is relatively homogeneous, with projected two-dimensional information and correlation dimensions which are nearly independent of direction. Our data indicate that the Kaplan–Yorke conjecture (for the information dimension) fails in the full four-dimensional phase space. We also find no plausible extension of this conjecture to the projected fractal dimensions of the oscillator. The projected growth rate associated with the largest Lyapunov exponent is negative in the one-dimensional coordinate space.

On the finitistic dimension conjecture I: related to representation-finite algebras

We use the class of representation-finite algebras to investigate the finitistic dimension conjecture. In this way we obtain a large class of algebras for which the finitistic dimension conjecture holds. The main results in this paper are: (1) Let A be an artin algebra and let I j ,1⩽ j⩽ n be a family of ideals in A with I 1 I 2⋯ I n =0, such that proj.dim( A I j )<∞ and proj.dim( I j ) A =0 for all j⩾3. If A/ I 1 and A/ I 2 are representation-finite and if A/ I j has finite finitistic dimension for j⩾3, then the finitistic dimension of A is finite. In particular, the finitistic dimension conjecture is true for algebras obtained from representation-finite algebras by forming dual extensions, trivially twisted extensions, Hochschild extensions, matrix algebras and tensor products with algebras of radical-square-zero. (2) Let A, B and C be three artin algebras with the same identity such that (i) C⊆ B⊆ A, and (ii) the Jacobson radical of C is a left ideal of B and the Jacobson radical of B is a left ideal of A. If A is representation-finite, then C has finite finitistic dimension. We also provide a way to construct algebras satisfying all conditions in (2), and this leads to a new reformulation of the finitistic dimension conjecture.

Nef dimension of minimal models

We reduce the Abundance Conjecture in dimension 4 to the following numerical statement: if the canonical divisor K is nef and has maximal nef dimension, then K is big. From this point of view, we classify in dimension 2 nef divisors which have maximal nef dimension, but which are not big.

Deformation of Weyl Modules and Generalized Parking Functions

Local Weyl modules over two-dimensional currents with values in Glr are deformed into spaces with bases related to parking functions. Using this construction, we (1) propose a simple proof that dimension of the space of diagonal coinvariants is not less than the number of parking functions; (2) describe the limits of Weyl modules in terms of semi-infinite forms and find the limits of characters; and (3) propose a lower bound and state a conjecture for dimensions of Weyl modules with arbitrary highest weights. Also we express characters of deformed Weyl modules in terms of ρ-parking functions and the Frobenius characteristic map.

Compactness results in Symplectic Field Theory

This is one in a series of papers devoted to the foundations of Symplectic Field Theory sketched in [Y Eliashberg, A Givental and H Hofer, Introduction to Symplectic Field Theory, Geom. Funct. Anal. Special Volume, Part II (2000) 560--673]. We prove compactness results for moduli spaces of holomorphic curves arising in Symplectic Field Theory. The theorems generalize Gromov's compactness theorem in [M Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985) 307--347] as well as compactness theorems in Floer homology theory, [A Floer, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988) 775--813 and Morse theory for Lagrangian intersections, J. Diff. Geom. 28 (1988) 513--547], and in contact geometry, [H Hofer, Pseudo-holomorphic curves and Weinstein conjecture in dimension three, Invent. Math. 114 (1993) 307--347 and H Hofer, K Wysocki and E Zehnder, Foliations of the Tight Three Sphere, Annals of Mathematics, 157 (2003) 125--255].

Non-negative cd-coefficients of [formula omitted] posets

We give a convolution formula for cd-index coefficients. The convolution formula, together with the proof by Davis and Okun of the Charney–Davis Conjecture in dimension 3, imply that certain cd-coefficients are non-negative for all Gorenstein ∗ posets. Additional coefficients are shown to be non-negative by interpreting them in terms of the top homology of certain Cohen–Macaulay complexes. In particular we verify, up to rank 6, Stanley's conjecture that the coefficients in the cd-index of a Gorenstein ∗ ranked poset are non-negative.

On the Isomorphism Conjecture in algebraic K-theory

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and arbitrary coefficient rings R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RΓ in terms of the K-theory of R and the homology of the group.

Three-dimensional surgery theory, UNil-groups and the Borel conjecture

We present almost complete computations of the surgery obstruction for 3-manifolds, as well as consequences for a suitable version of the structure set in dimension three. Applications leading to new results and conjectures concerning the Borel conjecture in dimension ⩾5 and UNil-groups are also discussed.

The commuting derivations conjecture

This paper proves the Commuting Derivations Conjecture in dimension three: if D 1 and D 2 are two locally nilpotent derivations which are linearly independent and satisfy [ D 1, D 2]=0 then the intersection of the kernels, A D 1 ∩ A D 2 equals C[f] where f is a coordinate. As a consequence, it is shown that p( X) Y+ Q( X, Z, T) is a coordinate if and only if Q( a, Z, T) is a coordinate for every zero a of p( X). Next to that, it is shown that if the Commuting Derivations Conjecture in dimension n, and the Cancellation Problem and Abhyankar–Sataye Conjecture in dimension n−1, all have an affirmative answer, then we can similarly describe all coordinates of the form p( X) Y+ q( X, Z 1,…, Z n−1 ). Also, conjectures about possible generalisations of the concept of “coordinate” for elements of general rings are made.

The Direct Summand Conjecture in Dimension Three

The direct summand conjecture asserts that if R is a regular local ring and S is a module-finite R-algebra containing R, then R is a direct summand of S as an R-module. It was previously known to be true if R contains a field or if dim R is at most two. In this article, the result is demonstrated for mixed characteristic rings of dimension three. The proof is accomplished by showing that an extension of plus closure has the colon-capturing property in dimension three.

Tilting theory and the finitistic dimension conjectures

Let R R be a right noetherian ring and let P &gt; ∞ \mathcal {P}^{&gt;\infty } be the class of all finitely presented modules of finite projective dimension. We prove that findim R = n &gt; ∞ R = n &gt; \infty iff there is an (infinitely generated) tilting module T T such that pd T = n T = n and T ⊥ = ( P &gt; ∞ ) ⊥ T ^\perp = (\mathcal P^{&gt;\infty })^\perp . If R R is an artin algebra, then T T can be taken to be finitely generated iff P &gt; ∞ \mathcal P^{&gt;\infty } is contravariantly finite. We also obtain a sufficient condition for validity of the First Finitistic Dimension Conjecture that extends the well-known result of Huisgen-Zimmermann and Smalø.