There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

Symmetry-protected topological (SPT) phases of matter have been interpreted in terms of anomalies, and it has been expected that a similar picture should hold for SPT phases with fermions. Here, we describe in detail what this picture means for phases of quantum matter that can be understood via band theory and free fermions. The main examples we consider are time-reversal invariant topological insulators and superconductors in 2 or 3 space dimensions. Along the way, we clarify the precise meaning of the statement that in the bulk of a 3d topological insulator, the electromagnetic -angle is equal to .

Fermion Path Integrals And Topological Phases

Harmonic analysis on semi‐simple lie groups—i

Compact Lie groups are ‘perfect’ entities in modern mathematics because: 

(a)

They are differentiable (and even analytical) manifolds of finite dimension and therefore can be analysed with the help of the most powerful tool of mathematics: ordinary and partial differential operators (equations);




(b)

Denoting by £(G) or L(G) = T e (G) the tangent space to G in unity e, we find that L(G) can be identified with the set of left-invariant vector fields on G (or with the set of appropriate differential operators of order one). Thus L(G) naturally becomes a Lie algebra: on L(G) there exists a structure [·, ·] which satisfies the axioms of Lie algebra (see below);




(c)

This rich algebraic structure, in turn, makes it possible to apply the powerful algebraic tools (theory of Lie algebras);




(d)

An abstract (finite dimensional) Lie algebra g defines (up to an isomorphism) a simply connected Lie group G such that £(G) = g. Therefore investigations of the Lie group G can be to large extent replaced by investigations of its Lie algebra g;




(e)

Compactness of a Lie group G makes it possible to use the theory of representations of compact groups developed by Weyl and Peter. This theory is extraordinarily beautiful and simple: it reduces to large extend to the spectral theory of compact operators;




(f)

And finally every complex, compact (connected) commutative Lie group is a torus. Moreover, as it was shown by E. Cartan, every compact Lie group contains a maximal torus T (any other such a torus is conjugated T 1 = gTg −1 ); and any point g 0 of the group G belongs to some maximal torus: G = U g gTg −1. This fundamental Cartan theorem makes it possible to apply to T (and thus to the whole of G) the theory of representations of abelian groups: every representation of the torus T decomposes into the finite (and orthogonal) sum of unit representations (over ℂ), the so called weights. Thus the knowledge of these weights is the fundamental element of the theory of representations of compact Lie groups.

Representations of Compact Lie Groups

We prove a relative index theorem for Dirac operators acting on sections of an A ? C-Cliiord bundle, where A is some C-algebra. We develop the index theory of complex and real Callias-type Dirac operators.

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A K-theoretic relative index theorem and Callias-type Dirac operators

We study a topological version of the T-duality relation between pairs consisting of a principal U(1)-bundle equipped with a degree-three integral cohomology class. We describe the homotopy type of a classifying space for such pairs and show that it admits a selfmap which implements a T-duality transformation. We give a simple derivation of a T-duality isomorphism for certain twisted cohomology theories. We conclude with some explicit computations of twisted K-theory groups and discuss an example of iterated T-duality for higher-dimensional torus bundles.

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On the Topology of T-Duality

In this paper we consider smooth extensions of cohomology theo- ries. In particular we construct an analytic multiplicative model of smooth K-theory. We further introduce the notion of a smooth K-orientation of a proper submersion p: W → B and define the associated push-forward ˆ p! : ˆ K(W) → ˆ K(B). We show that the push-forward has the expected properties as functoriality, compatibility with pull-back diagrams, projection formula and a bordism formula. We construct a multiplicative lift of the Chern character ˆ ch : ˆ K(B) → ˆ H(B, Q),

Smooth k-theory

In string theory, the concept of T-duality between two principal U(1)-bundles E_1 and E_2 over the same base space B, together with cohomology classes $h_1\in H^3(E_1)$ and $h_2\in H^3(E_2)$, has been introduced. One of the main virtues of T-duality is that $h_1$-twisted K-theory of $E_1$ is isomorphic to $h_2$-twisted K-theory of $E_2$. In this paper, a new, very topological concept of T-duality is introduced. The study pairs (E,h) as above from a topological point of view and construct a classifying space of such pairs. Using this, we construct a universal dual pair to a given pair. Our construction immediately gives a number of known and new properties of the dual. In particular it implies existence of a dual of any pair (E,h), and it also describes the ambiguity upto which the dual is well defined. In order to deal with twisted K-theory, some care is needed, in particular when dealing with naturality questions, because the twisted K-theory depends on the explicit model for the twists and the twisted theory --care which is missing in some of the existing literature. We illustrate the use of T-duality by some explicit calculations of twisted K-groups.

On the topology of T-duality

We provide an axiomatic framework for the study of smooth extensions of generalized cohomology theories. Our main results are about the uniqueness of smooth extensions, and the identification of the flat theory with the associated cohomology theory with ℝ/ℤ-coefficients. In particular, we show that there is a unique smooth extension of K-theory and of MU-cobordism with a unique multiplication, and that the flat theory in these cases is naturally isomorphic to the homotopy theorist's version of the cohomology theory with ℝ/ℤ-coefficients. For this we only require a small set of natural compatibility conditions.

Ulrich Bunke

Papers

A K-theoretic relative index theorem and Callias-type Dirac operators

On the Topology of T-Duality

Smooth k-theory

On the topology of T-duality

Uniqueness of smooth extensions of generalized cohomology theories