The Einstein Relation on Metric Measure Spaces

This note is based on F. Burghart's master thesis at Stuttgart university from July 2018, supervised by Prof. Freiberg. We review the Einstein relation, which connects the Hausdorff, local walk and spectral dimensions on a space, in the abstract setting of a metric measure space equipped with a suitable operator. This requires some twists compared to the usual definitions from fractal geometry. The main result establishes the invariance of the three involved notions of fractal dimension under bi-Lipschitz continuous isomorphisms between mm-spaces and explains, more generally, how the transport of the analytic and stochastic structure behind the Einstein relation works. While any homeomorphism suffices for this transport of structure, non-Lipschitz maps distort the Hausdorff and the local walk dimension in different ways. To illustrate this, we take a look at H\"older regular transformations and how they influence the local walk dimension and prove some partial results concerning the Einstein relation on graphs of fractional Brownian motions. We conclude by giving a short list of further questions that may help building a general theory of the Einstein relation.Comment: 28 pages, 3 figure

New Avenues for Einstein's Gravity: from Penrose's Twistors to Hitchin's Three-Forms

In this thesis we take Einstein theory in dimension four seriously, and explore the special aspects of gravity in this number of dimension. Among the many surprising features in dimension four, one of them is the possibility of `Chiral formulations of gravity' - they are surprising as they typically do not rely on a metric. Another is the existence of the Twistor correspondence. The Chiral and Twistor formulations might seems different in nature. In the first part of this thesis we demonstrate that they are in fact closely related. In particular we give a new proof for Penrose's `non-linear graviton theorem' that relies on the geometry of SU(2)-connections only (rather than on metric). In the second part of this thesis we describe partial results towards encoding the full GR in the total space of some fibre bundle over space-time. We indeed show that gravity theory in three and four dimensions can be related to theories of a completely different nature in six and seven dimension respectively. This theories, first advertised by Hitchin, are diffeomorphism invariant theories of differential three-forms. Starting with seven dimensions, we are only partially succesfull: the resulting theory is some deformed version of gravity. We however found that solutions to a particular gravity theory in four dimension have a seven dimensional interpretation as G2 holonomy manifold. On the other hand by going from six to three dimension we do recover three dimensional gravity. As a bonus, we describe new diffeomorphism invariant functionnals for differential forms in six dimension and prove that two of them are topological.Comment: This thesis can also be found on https://tel.archives-ouvertes.fr/tel-0165003

Doctor of Philosophy

dissertationWe study the geometry of higher dimensional algebraic varieties according to the dichotomy of Kodaira dimensions, negative or nonnegative, and the corresponding pictures in the Minimal Model Conjecture. On the one hand, according to the Minimal Model Conjecture, a variety with nonnegative Kodaira dimension is birational to a minimal model, which has semiample canonical class. This has been done if dimension is less than or equal to three and for varieties of general type in any dimension. In general, the Minimal Model Conjecture is still open. As the first result, we show that the Minimal Model Conjecture for varieties with nonnegative Kodaira dimensions follows from the Minimal Model Conjecture for varieties with Kodaira dimension zero. In particular, the Minimal Model Conjecture is reduced to the Minimal Model Conjecture for varieties of Kodaira dimension zero and the Nonvanishing Conjecture. On the other hand, according to the Minimal Model Conjecture, Fano varieties of Picard number one are the building blocks for varieties with negative Kodaira dimension. The set of mildly singular Fano varieties of given dimension is expected to be bounded. As a second result, we exhibit an effective upper bound of the anticanonical volume for the set of e-klt Q-factorial log Q-Fano threefolds with Picard number one. This result is related to a conjecture open in dimension three and higher, the Borisov-Alexeev-Borisov Conjecture, which asserts boundedness of the set of e-klt log Q-Fano varieties. In the end of this dissertation, we include some partial results of the Nonvanishing Conjecture in the minimal model program. The minimal model program is developed to attack the Minimal Model Conjecture. The Nonvanishing Conjecture is one of the most important missing ingredient for completing the minimal model program

A geometric view of cryptographic equation solving

This paper considers the geometric properties of the Relinearisation algorithm and of the XL algorithm used in cryptology for equation solving. We give a formal description of each algorithm in terms of projective geometry, making particular use of the Veronese variety. We establish the fundamental geometrical connection between the two algorithms and show how both algorithms can be viewed as being equivalent to the problem of finding a matrix of low rank in the linear span of a collection of matrices, a problem sometimes known as the MinRank problem. Furthermore, we generalise the XL algorithm to a geometrically invariant algorithm, which we term the GeometricXL algorithm. The GeometricXL algorithm is a technique which can solve certain equation systems that are not easily soluble by the XL algorithm or by Groebner basis methods

A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

The basic methods of constructing the sets of mutually unbiased bases in the Hilbert space of an arbitrary finite dimension are discussed and an emerging link between them is outlined. It is shown that these methods employ a wide range of important mathematical concepts like, e.g., Fourier transforms, Galois fields and rings, finite and related projective geometries, and entanglement, to mention a few. Some applications of the theory to quantum information tasks are also mentioned.Comment: 20 pages, 1 figure to appear in Foundations of Physics, Nov. 2006 two more references adde

Classical evolution of fractal measures on the lattice

We consider the classical evolution of a lattice of non-linear coupled oscillators for a special case of initial conditions resembling the equilibrium state of a macroscopic thermal system at the critical point. The displacements of the oscillators define initially a fractal measure on the lattice associated with the scaling properties of the order parameter fluctuations in the corresponding critical system. Assuming a sudden symmetry breaking (quench), leading to a change in the equilibrium position of each oscillator, we investigate in some detail the deformation of the initial fractal geometry as time evolves. In particular we show that traces of the critical fractal measure can sustain for large times and we extract the properties of the chain which determine the associated time-scales. Our analysis applies generally to critical systems for which, after a slow developing phase where equilibrium conditions are justified, a rapid evolution, induced by a sudden symmetry breaking, emerges in time scales much shorter than the corresponding relaxation or observation time. In particular, it can be used in the fireball evolution in a heavy-ion collision experiment, where the QCD critical point emerges, or in the study of evolving fractals of astrophysical and cosmological scales, and may lead to determination of the initial critical properties of the Universe through observations in the symmetry broken phase.Comment: 15 pages, 15 figures, version publiced at Physical Review