Essential Topology

Crossley, Martin D.

doi:10.1007/1-84628-194-6

Cited by 34 publications

(26 citation statements)

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“…Bott then shows [20] that this suspension captures enough of G i−1 /G i that we obtain the same isomorphism. The two approaches are related because of the natural identification [23] Map * (ΣX, Y ) = Map * (X, ΩY )…”

Section: Now We Show That Spacesmentioning

confidence: 99%

“…The full machinery of K-theory is intimidatingly abstract, but the simpler mechanism that underlies the period-two or period-eight pattern of correlations can be understood with relatively unsophisticated mathematical tools -representation theory and the basics of homotopy as described in [22], or perhaps [23]. It aim of this present paper to explain how this mechanism works, and to fill in some of details sketched in [19].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock

Stone¹,

Chiu

Roy

2010

J. Phys. A: Math. Theor.

145

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We provide an account of some of the mathematics of Bott periodicity and the Atiyah, Bott, Shapiro construction. We apply these ideas to understanding the twisted bundles of electron bands that underly the properties of topological insulators, spin Hall systems, and other topologically interesting materials. 74.20.Rp, 74.45.+c, 72.25.Dc Now introduce J 5 and construct M = J 1 J 4 J 5 which has M 2 = I and commutes with K and J 1 . M therefore acts within the V + (or V − ) eigenspace of K and divides it into two mutually DIII ≡ O(16r)/U(8r): The m ∈ m 0 's are real skew symmetric matrices that anticommute with J 1 . We can keep C = ϕ as a particle-hole symmetry and take T = ϕ ⊗ J 1 as a time reversal that commutes with H = −iσ 2 ⊗ M and squares to −I. The product of C and T is J 1 , and this a linear (commutes with −iσ 2 ⊗ I) "P " type symmetry that anticommutes with H. AII≡ U(8r)/Sp(4r): The generators m ∈ m 1 are real skew matrices that commute with J 1 and anticommute with J 2 . They can be regarded as skew-quaternion-hermitian matrices with complex entries. We no longer need set i → −iσ 2 ⊗ I as the matrices no longer have elements coupling between the artificial copies. We instead use J 1 as the surrogate for "i." Now H = J 1 m is real symmetric, and commutes with T = J 2 . This T acts as a time reversal operator squaring to −I. CII≡ Sp(4r)/Sp(2r) × Sp(2r): The matrices m ∈ m 2 commute with J 1 and J 2 but anticommute with J 3 . Again H = J 1 m. We can set T = J 3 as this commutes with with H and squares to −I. P = J 2 J 3 anticommutes with H but commutes with J 1 (and so is a linear map) while C = J 2 anticommutes with H, is antilinear and squares to −I. C≡ {Sp(2r) × Sp(2r)}/Sp(2r) ≃ Sp(2r): The matrices m ∈ m 3 commute with J 1 , J 2 , J 3 , and anticommute with J 4 , and we can restrict ourselves to the subspace in which K = J 1 J 2 J 3 takes a definite value, say +1. The Hamiltonian J 1 m commutes with J 4 -but 15 J 4 does not commute with J 1 J 2 J 3 and so is not allowed as an operator on our subspace.Indeed no product involving J 4 is allowed. But C = J 2 commutes with J 1 J 2 J 3 and still anticommutes with H. Thus we still have a particle-hole symmetry squaring to −1.The old time reversal J 3 now anticommutes with H and looks like another particlehole symmetry, but is not really an independent one as in this subspace J 3 = J 2 J 1 and J 1 is simply multiplication by "i." CI≡ Sp(2r)/U(2r): The m ∈ m 4 anticommute with J 5 . Now J 4 J 5 commutes with J 1 J 2 J 3 , and so is an allowed operator. It anticommutes with H = J 1 m and commutes with J 1 .It is therefore a "P "-type linear map. The map T = J 2 J 4 J 5 is antilinear (anticommutes with J 1 ), commutes with H and T 2 = +I. We can take C = J 2 again. AI ≡ U(2r)/O(2r): The m ∈ m 5 n anticommute with J 6 , and we are to restrict ourselves to the subspace on which K = J 1 J 2 J 3 = +1 and M = J 1 J 4 J 5 = +1. The map T = J 3 J 4 J 6 commutes with K and M and commutes H = J 1 m. We have T 2 = +I.We could equivalently take T = J 2 J 4 J 6 .The m ∈ ...

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Section: Now We Show That Spacesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock

Stone¹,

Chiu

Roy

2010

J. Phys. A: Math. Theor.

145

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“…Topology describes the properties of special mathematical spaces that are unaltered under continuous deformation (Crossley, 2006).…”

Section: Lithological Topologymentioning

confidence: 99%

Topological Analysis in Monte Carlo Simulation for Uncertainty Estimation

Pakyuz-Charrier¹,

Jessell²,

Giraud³

et al. 2019

Preprint

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Abstract. This paper proposes and demonstrates improvements for the Monte Carlo simulation for Uncertainty Estimation (MCUE) method. MCUE is a type of Bayesian Monte Carlo aimed at input data uncertainty propagation in implicit 3D geological modeling. In the Monte Carlo process, a series of statistically plausible models are built from the input data set which uncertainty is to be propagated to a final probabilistic geological model (PGM) or uncertainty index model (UIM). Significant differences in terms of topology are observed in the plausible model suite that is generated as an intermediary step in MCUE. These differences are interpreted as analogous to population heterogeneity. The source of this heterogeneity is traced to be the non-linear relationship between plausible datasets’ variability and plausible model’s variability. Non-linearity is shown to arise from the effect of the geometrical ruleset on model building which transforms lithological continuous interfaces into discontinuous piecewise ones. Plausible model heterogeneity induces geological incompatibility and challenges the underlying assumption of homogeneity which global uncertainty estimates rely on. To address this issue, a method for topological analysis applied to the plausible model suite in MCUE is introduced. Boolean topological signatures recording lithological units’ adjacency are used as n-dimensional points to be considered individually or clustered using the Density-Based Spatial Clustering of Applications with Noise (DBSCAN) algorithm. The proposed method is tested on two challenging synthetic examples with varying levels of confidence in the structural input data. Results indicate that topological signatures constitute a powerful discriminant to address plausible model heterogeneity. Basic topological signatures appear to be a reliable indicator of the structural behavior of the plausible models and provide useful geological insights. Moreover, ignoring heterogeneity was found to be detrimental to the accuracy and relevance of the PGMs and UIMs.

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“…We explain these ideas in greater detail for the remainder of this section. Our discussions in the first two subsections below recapitulate presentations in texts such as [33,34].…”

Section: Topological Data Analysis and Persistent Homologymentioning

confidence: 99%

Topological Data Analysis of Biological Aggregation Models

2015

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We apply tools from topological data analysis to two mathematical models inspired by biological aggregations such as bird flocks, fish schools, and insect swarms. Our data consists of numerical simulation output from the models of Vicsek and D'Orsogna. These models are dynamical systems describing the movement of agents who interact via alignment, attraction, and/or repulsion. Each simulation time frame is a point cloud in position-velocity space. We analyze the topological structure of these point clouds, interpreting the persistent homology by calculating the first few Betti numbers. These Betti numbers count connected components, topological circles, and trapped volumes present in the data. To interpret our results, we introduce a visualization that displays Betti numbers over simulation time and topological persistence scale. We compare our topological results to order parameters typically used to quantify the global behavior of aggregations, such as polarization and angular momentum. The topological calculations reveal events and structure not captured by the order parameters.

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Essential Topology

Cited by 34 publications

References 0 publications

Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock

Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock

Topological Analysis in Monte Carlo Simulation for Uncertainty Estimation

Topological Data Analysis of Biological Aggregation Models

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