Uso de la Realidad Aumentada para facilitar la lectura e interpretación de planos

[ES] Este proyecto genera productos adicionales de trabajos también disponibles en el repositori, en concreto:

Problems when generating virtual models representing real objects: Hondarribia walls

[EN] This paper is based in the following project:

Determinantal hypersurface from a geometric perspective

In this paper, we give a geometric interpretation of determinantal forms, both in the case of general matrices and symmetric matrices. We will prove irreducibility of the determinantal singular loci and state its dimension. We also provide detailed description of the singular locus for small dimensions.

A geometric analysis of convex demixing

Demixing is the task of identifying multiple signals given only their sum and prior information about their structures. Examples of demixing problems include (i) separating a signal that is sparse with respect to one basis from a signal that is sparse with respect to a second basis; (ii) decomposing an observed matrix into low-rank and sparse components; and (iii) identifying a binary codeword with impulsive corruptions. This thesis describes and analyzes a convex optimization framework for solving an array of demixing problems.

Our framework includes a random orientation model for the constituent signals that ensures the structures are incoherent. This work introduces a summary parameter, the statistical dimension, that reflects the intrinsic complexity of a signal. The main result indicates that the difficulty of demixing under this random model depends only on the total complexity of the constituent signals involved: demixing succeeds with high probability when the sum of the complexities is less than the ambient dimension; otherwise, it fails with high probability.

The fact that a phase transition between success and failure occurs in demixing is a consequence of a new inequality in conic integral geometry. Roughly speaking, this inequality asserts that a convex cone behaves like a subspace whose dimension is equal to the statistical dimension of the cone. When combined with a geometric optimality condition for demixing, this inequality provides precise quantitative information about the phase transition, including the location and width of the transition region.

Geometric quantization and foliation reduction

A standard question in the study of geometric quantization is whether symplectic reduction interacts nicely with the quantized theory, and in particular whether “quantization commutes with reduction.” Guillemin and Sternberg first proposed this question, and answered it in the affirmative for the case of a free action of a compact Lie group on a compact Kähler manifold. Subsequent work has focused mainly on extending their proof to non-free actions and non-Kähler manifolds. For realistic physical examples, however, it is desirable to have a proof which also applies to non-compact symplectic manifolds.

In this thesis we give a proof of the quantization-reduction problem for general symplectic manifolds. This is accomplished by working in a particular wavefunction representation, associated with a polarization that is in some sense compatible with reduction. While the polarized sections described by Guillemin and Sternberg are nonzero on a dense subset of the Kähler manifold, the ones considered here are distributional, having support only on regions of the phase space associated with certain quantized, or “admissible”, values of momentum.

We first propose a reduction procedure for the prequantum geometric structures that “covers” symplectic reduction, and demonstrate how both symplectic and prequantum reduction can be viewed as examples of foliation reduction. Consistency of prequantum reduction imposes the above-mentioned admissibility conditions on the quantized momenta, which can be seen as analogues of the Bohr-Wilson-Sommerfeld conditions for completely integrable systems.

We then describe our reduction-compatible polarization, and demonstrate a one-to-one correspondence between polarized sections on the unreduced and reduced spaces.

Finally, we describe a factorization of the reduced prequantum bundle, suggested by the structure of the underlying reduced symplectic manifold. This in turn induces a factorization of the space of polarized sections that agrees with its usual decomposition by irreducible representations, and so proves that quantization and reduction do indeed commute in this context.

A significant omission from the proof is the construction of an inner product on the space of polarized sections, and a discussion of its behavior under reduction. In the concluding chapter of the thesis, we suggest some ideas for future work in this direction.