Manifolds, patterns and transitions in a creative life

Using sculpture and drawing as my primary methods of investigation, this research explores ways of shifting the emphasis of my creative visual arts practice from object to process whilst still maintaining a primacy of material outcomes. My motivation was to locate ways of developing a sustained practice shaped as much by new works, as by a creative flow between works. I imagined a practice where a logic of structure within discrete forms and a logic of the broader practice might be developed as mutually informed processes. Using basic structural components of multiple wooden curves and linear modes of deployment – in both sculptures and drawings – I have identified both emergence theory and the image of rhizomic growth (Deleuze and Guattari, 1987) as theoretically integral to this imagining of a creative practice, both in terms of critiquing and developing works. Whilst I adopt a formalist approach for this exegesis, the emergence and rhizome models allow it to work as a critique of movement, of becoming and changing, rather than merely a formalism of static structure. In these models, therefore, I have identified a formal approach that can be applied not only to objects, but to practice over time. The thorough reading and application of these ontological models (emergence and rhizome) to visual arts practice, in terms of processes, objects and changes, is the primary contribution of this thesis. The works that form the major component of the research develop, reflect and embody these notions of movement and change.

Kernel analysis on Grassmann manifolds for action recognition

Modelling video sequences by subspaces has recently shown promise for recognising human actions. Subspaces are able to accommodate the effects of various image variations and can capture the dynamic properties of actions. Subspaces form a non-Euclidean and curved Riemannian manifold known as a Grassmann manifold. Inference on manifold spaces usually is achieved by embedding the manifolds in higher dimensional Euclidean spaces. In this paper, we instead propose to embed the Grassmann manifolds into reproducing kernel Hilbert spaces and then tackle the problem of discriminant analysis on such manifolds. To achieve efficient machinery, we propose graph-based local discriminant analysis that utilises within-class and between-class similarity graphs to characterise intra-class compactness and inter-class separability, respectively. Experiments on KTH, UCF Sports, and Ballet datasets show that the proposed approach obtains marked improvements in discrimination accuracy in comparison to several state-of-the-art methods, such as the kernel version of affine hull image-set distance, tensor canonical correlation analysis, spatial-temporal words and hierarchy of discriminative space-time neighbourhood features.

Random projections on manifolds of symmetric positive definite matrices for image classification

Recent advances suggest that encoding images through Symmetric Positive Definite (SPD) matrices and then interpreting such matrices as points on Riemannian manifolds can lead to increased classification performance. Taking into account manifold geometry is typically done via (1) embedding the manifolds in tangent spaces, or (2) embedding into Reproducing Kernel Hilbert Spaces (RKHS). While embedding into tangent spaces allows the use of existing Euclidean-based learning algorithms, manifold shape is only approximated which can cause loss of discriminatory information. The RKHS approach retains more of the manifold structure, but may require non-trivial effort to kernelise Euclidean-based learning algorithms. In contrast to the above approaches, in this paper we offer a novel solution that allows SPD matrices to be used with unmodified Euclidean-based learning algorithms, with the true manifold shape well-preserved. Specifically, we propose to project SPD matrices using a set of random projection hyperplanes over RKHS into a random projection space, which leads to representing each matrix as a vector of projection coefficients. Experiments on face recognition, person re-identification and texture classification show that the proposed approach outperforms several recent methods, such as Tensor Sparse Coding, Histogram Plus Epitome, Riemannian Locality Preserving Projection and Relational Divergence Classification.

Dictionary learning and sparse coding on Grassmann manifolds : an extrinsic solution

Recent advances in computer vision and machine learning suggest that a wide range of problems can be addressed more appropriately by considering non-Euclidean geometry. In this paper we explore sparse dictionary learning over the space of linear subspaces, which form Riemannian structures known as Grassmann manifolds. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping, which enables us to devise a closed-form solution for updating a Grassmann dictionary, atom by atom. Furthermore, to handle non-linearity in data, we propose a kernelised version of the dictionary learning algorithm. Experiments on several classification tasks (face recognition, action recognition, dynamic texture classification) show that the proposed approach achieves considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelised Affine Hull Method and graph-embedding Grassmann discriminant analysis.

The dynamics of cellular automata on 2-manifolds is affected by topology

In this paper, we demonstrate that the distribution of Wolfram classes within a cellular automata rule space in the triangular tessellation is not consistent across different topological general. Using a statistical mechanics approach, cellular automata dynamical classes were approximated for cellular automata defined on genus-0, genus-1 and genus-2 2-manifolds. A distribution-free equality test for empirical distributions was applied to identify cases in which Wolfram classes were distributed differently across topologies. This result implies that global structure and local dynamics contribute to the long term evolution of cellular automata.

On The Topology Of Manifolds With Positive Isotropic Curvature

We show that a closed orientable Riemannian n-manifold, n >= 5, with positive isotropic curvature and free fundamental group is homeomorphic to the connected sum of copies of Sn-1 x S-1.

On the Geometry of Infinite-Dimensional Grassmannian Manifolds and Gauge Theory

This PhD Thesis is about certain infinite-dimensional Grassmannian manifolds that arise naturally in geometry, representation theory and mathematical physics. From the physics point of view one encounters these infinite-dimensional manifolds when trying to understand the second quantization of fermions. The many particle Hilbert space of the second quantized fermions is called the fermionic Fock space. A typical element of the fermionic Fock space can be thought to be a linear combination of the configurations m particles and n anti-particles . Geometrically the fermionic Fock space can be constructed as holomorphic sections of a certain (dual)determinant line bundle lying over the so called restricted Grassmannian manifold, which is a typical example of an infinite-dimensional Grassmannian manifold one encounters in QFT. The construction should be compared with its well-known finite-dimensional analogue, where one realizes an exterior power of a finite-dimensional vector space as the space of holomorphic sections of a determinant line bundle lying over a finite-dimensional Grassmannian manifold. The connection with infinite-dimensional representation theory stems from the fact that the restricted Grassmannian manifold is an infinite-dimensional homogeneous (Kähler) manifold, i.e. it is of the form G/H where G is a certain infinite-dimensional Lie group and H its subgroup. A central extension of G acts on the total space of the dual determinant line bundle and also on the space its holomorphic sections; thus G admits a (projective) representation on the fermionic Fock space. This construction also induces the so called basic representation for loop groups (of compact groups), which in turn are vitally important in string theory / conformal field theory. The Thesis consists of three chapters: the first chapter is an introduction to the backround material and the other two chapters are individually written research articles. The first article deals in a new way with the well-known question in Yang-Mills theory, when can one lift the action of the gauge transformation group on the space of connection one forms to the total space of the Fock bundle in a compatible way with the second quantized Dirac operator. In general there is an obstruction to this (called the Mickelsson-Faddeev anomaly) and various geometric interpretations for this anomaly, using such things as group extensions and bundle gerbes, have been given earlier. In this work we give a new geometric interpretation for the Faddeev-Mickelsson anomaly in terms of differentiable gerbes (certain sheaves of categories) and central extensions of Lie groupoids. The second research article deals with the question how to define a Dirac-like operator on the restricted Grassmannian manifold, which is an infinite-dimensional space and hence not in the landscape of standard Dirac operator theory. The construction relies heavily on infinite-dimensional representation theory and one of the most technically demanding challenges is to be able to introduce proper normal orderings for certain infinite sums of operators in such a way that all divergences will disappear and the infinite sum will make sense as a well-defined operator acting on a suitable Hilbert space of spinors. This research article was motivated by a more extensive ongoing project to construct twisted K-theory classes in Yang-Mills theory via a Dirac-like operator on the restricted Grassmannian manifold.

Manifolds with nonnegative isotropic curvature

We prove that if (M-n, g), n >= 4, is a compact, orientable, locally irreducible Riemannian manifold with nonnegative isotropic curvature,then one of the following possibilities hold: (i) M admits a metric with positive isotropic curvature. (ii) (M, g) is isometric to a locally symmetric space. (iii) (M, g) is Kahler and biholomorphic to CPn/2. (iv) (M, g) is quaternionic-Kahler. This is implied by the following result: Let (M-2n, g) be a compact, locally irreducible Kahler manifold with nonnegative isotropic curvature. Then either M is biholomorphic to CPn or isometric to a compact Hermitian symmetric space. This answers a question of Micallef and Wang in the affirmative. The proof is based on the recent work of Brendle and Schoen on the Ricci flow.

Open manifolds, Ozsvath-Szabo invariants and exotic R-4 ` S

We construct an invariant of certain open four-manifolds using the Heegaard Floer theory of Ozsvath and Szabo. We show that there is a manifold X homeomorphic to R-4 for which the invariant is non-trivial,showing that X is an exotic R-4. This is the first invariant that detects exotic R-4' s. (C) 2009 Published by Elsevier GmbH.

Combinatorial manifolds with complementarity

A simplicial complex is said to satisfy complementarity if exactly one of each complementary pair of nonempty vertex-sets constitutes a face of the complex.

Surfaces of Bounded Mean Curvature In Riemannian Manifolds

Consider a sequence of closed, orientable surfaces of fixed genus g in a Riemannian manifold M with uniform upper bounds on the norm of mean curvature and area. We show that on passing to a subsequence, we can choose parametrisations of the surfaces by inclusion maps from a fixed surface of the same genus so that the distance functions corresponding to the pullback metrics converge to a pseudo-metric and the inclusion maps converge to a Lipschitz map. We show further that the limiting pseudo-metric has fractal dimension two. As a corollary, we obtain a purely geometric result. Namely, we show that bounds on the mean curvature, area and genus of a surface F subset of M, together with bounds on the geometry of M, give an upper bound on the diameter of F. Our proof is modelled on Gromov's compactness theorem for J-holomorphic curves.

Minimal Triangulations of Manifolds

Finding vertex-minimal triangulations of closed manifolds is a very difficult problem. Except for spheres and two series of manifolds, vertex-minimal triangulations are known for only few manifolds of dimension more than 2 (see the table given at the end of Section 5). In this article, we present a brief survey on the works done in last 30 years on the following:(i) Finding the minimal number of vertices required to triangulate a given pl manifold. (ii) Given positive integers n and d, construction of n-vertex triangulations of different d-dimensional pl manifolds. (iii) Classifications of all the triangulations of a given pl manifold with same number of vertices.In Section 1, we have given all the definitions which are required for the remaining part of this article. A reader can start from Section 2 and come back to Section 1 as and when required. In Section 2, we have presented a very brief history of triangulations of manifolds. In Section 3,we have presented examples of several vertex-minimal triangulations. In Section 4, we have presented some interesting results on triangulations of manifolds. In particular, we have stated the Lower Bound Theorem and the Upper Bound Theorem. In Section 5, we have stated several results on minimal triangulations without proofs. Proofs are available in the references mentioned there. We have also presented some open problems/conjectures in Sections 3 and 5.

An infinite family of tight triangulations of manifolds

We give explicit construction of vertex-transitive tight triangulations of d-manifolds for d >= 2. More explicitly, for each d >= 2, we construct two (d(2) + 5d + 5)-vertex neighborly triangulated d-manifolds whose vertex-links are stacked spheres. The only other non-trivial series of such tight triangulated manifolds currently known is the series of non-simply connected triangulated d-manifolds with 2d + 3 vertices constructed by Kuhnel. The manifolds we construct are strongly minimal. For d >= 3, they are also tight neighborly as defined by Lutz, Sulanke and Swartz. Like Kuhnel complexes, our manifolds are orientable in even dimensions and non-orientable in odd dimensions. (c) 2013 Elsevier Inc. All rights reserved.

On stellated spheres and a tightness criterion for combinatorial manifolds

We introduce k-stellated spheres and consider the class W-k(d) of triangulated d-manifolds, all of whose vertex links are k-stellated, and its subclass W-k*; (d), consisting of the (k + 1)-neighbourly members of W-k(d). We introduce the mu-vector of any simplicial complex and show that, in the case of 2-neighbourly simplicial complexes, the mu-vector dominates the vector of Betti numbers componentwise; the two vectors are equal precisely for tight simplicial complexes. We are able to estimate/compute certain alternating sums of the components of the mu-vector of any 2-neighbourly member of W-k(d) for d >= 2k. As a consequence of this theory, we prove a lower bound theorem for such triangulated manifolds, and we determine the integral homology type of members of W-k*(d) for d >= 2k + 2. As another application, we prove that, when d not equal 2k + 1, all members of W-k*(d) are tight. We also characterize the tight members of W-k*(2k + 1) in terms of their kth Betti numbers. These results more or less answer a recent question of Effenberger, and also provide a uniform and conceptual tightness proof for all except two of the known tight triangulated manifolds. We also prove a lower bound theorem for homology manifolds in which the members of W-1(d) provide the equality case. This generalizes a result (the d = 4 case) due to Walkup and Kuhnel. As a consequence, it is shown that every tight member of W-1 (d) is strongly minimal, thus providing substantial evidence in favour of a conjecture of Kuhnel and Lutz asserting that tight homology manifolds should be strongly minimal. (C) 2013 Elsevier Ltd. All rights reserved.