This volume is devoted specifically to the mathematical aspects of Clifford algebras and their applications in physics. Algebraic geometry, cohomology, non-communicative spaces, "q"-deformations and the related quantum groups, and projective geometry provide the basis for algebraic topics covered.

## Geometric Algebra and Applications to Physics

Physical applications and extensions of physical theories such as the theory of quaternionic spin, a projective theory of hadron transformation laws, and electron scattering are also presented, showing the broad applicability of Clifford geometric algebras in solving physical problems. Treatment of the structure theory of quantum Clifford algebras, the connection to logic, group representations, and computational techniques including symbolic calculations and theorem proving rounds out the presentation.

Have doubts regarding this product? Whereas the algebras of Clifford and Grassmann are well known in advanced mathematics and physics, they have never made an impact in elementary textbooks where the vector algebra of Gibbs-Heaviside still predominates. Sobczyk first learned about the power of geometric algebra in classes in electrodynamics and relativity taught by Hestenes at Arizona State University from to He still vividly remembers a feeling of disbelief that the fundamental geometric product of vectors could have been left out of his undergraduate mathematics education.

This book is addressed to a broad audience of applied mathematicians, physicists, computer scientists, and engineers. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Free Preview. Buy eBook. Buy Hardcover. Buy Softcover.

In this talk, I will begin by introducing the quantum computational model, and describing the famous quantum algorithm due to Grover that solves unstructured search problems in approximately the square root of the time required classically. I will then go on to describe more recent work on a quantum algorithm to speed up classical search algorithms based on the technique known as backtracking "trial and error" , and very recent work on calculating the level of quantum speedup anticipated when applying this algorithm to practically relevant problems.

The talk will aim to give a flavour of the mathematics involved in quantum algorithm design, rather than going into the full details. Quantum walk speedup of backtracking algorithms, Theory of Computing to appear ; arXiv Recently two approaches to twisting of the real structure of spectral triples were introduced.

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In this talk we present and compare these two approaches. The Yang-Baxter equation YBE is a cubic matrix equation which plays a prominent in several fields such as quantum groups, braid groups, knot theory, quantum field theory, and statistical mechanics. Its invertible normal solutions "R-matrices" define representations and extremal characters of the infinite braid group. These characters define a natural equivalence relation on the family of all R-matrices, and I will describe a research programme aiming at classifying all solutions of the YBE up to this equivalence.

I will then describe the current state of this programme.

In the special case of normal involutive R-matrices, the classification is complete joint work with Simon and Ulrich. The more general case of R-matrices with two arbitrary eigenvalues is currently work in progress, and I will present some partial results, including a classification of all R-matrices defining representations of the Temperley-Lieb algebra and a deformation theorem for involutive R-matrices.

Algebraic structures that are usually referred to as cohomological field theories arise from geometry of Deligne-Mumford compactifications of moduli spaces of curves with marked points.

### 1st Edition

I shall talk about some new rather remarkable algebraic varieties that have a lot in common with [genus 0] Deligne-Mumford spaces, and several new algebraic structures that naturally arise from studying those varieties. K-theory of some AF-algebras from braided categories In the s, Renault, Wassermann, Handelman and Rossmann explicitly described the K-theory of the fixed point algebras of certain actions of compact groups on AF-algebras as polynomial rings.

Similarly, Evans and Gould in explicitly described the K-theory of certain AF-algebras related to SU 2 as quotients of polynomial rings. I will also give new explicit descriptions of the K-theory of certain AF-algebras related to SU 3 , Sp 4 and G 2 2 as quotients of polynomial rings.

Finally, I will attempt to explain how this work is motivated by the Freed-Hopkins-Teleman formula for the fusion rings of WZW-models in conformal field theory. This is all based on joint work with David E. Of course, in some cases, for example, mathematical physics, the concern is with unbounded operators such as position and momentum in quantum mechanics. In particular, our axioms are such that an analogue of the Gelfand-Naimark theorem holds. In this formulation it admits a direct description in terms of KK-theory of certain section algebras and thus has tight connections for instance to the geometry of scalar curvature.

Modern homotopy theory on the other hand provides a universally twisted companion for every coherently multiplicative cohomology theory by means of parametrised spectra. This construction has very appealing formal properties and, indeed, applied to K-theory allows for much more general twists than those afforded by the operator algebraic one. Necessarily then, such twisted companions are defined in a much more formal manner and thus in general not easily tied to geometry.

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## Algebraic Geometry and Mathematical Physics

The goal of my talk is to briefly explain the category of the title, that naturally houses both constructions and then sketch that, indeed, a suitable restriction of the universal one reproduces the operator theoretic version of twisted K-theory. GAPT Seminars Our researchers are working across disciplines to tackle major challenges facing society, the economy and our environment.

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## [] Geometric Algebra

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Geometry, Algebra, Mathematical Physics and Topology Research Group Overview Research People Events Our interests sweep a broad range of topics, from algebra, geometry, topology, including operator algebras, and non-commutative geometry in pure mathematics, to algebraic and conformal quantum field theory and integrable statistical mechanics in mathematical physics.

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The main areas of research within the current group are: Pure mathematics Algebraic Geometry DG categories and derived categories associated to algebraic varieties Operator algebras and non-commutative geometry Subfactors and planar algebras Orbifolds and the McKay correspondence in Algebraic Geometry and Subfactor Theory Categorification problems, Mirror symmetry, Moduli spaces Quiver representations in Algebraic Geometry and Subfactor Theory K-theory - including twisted and equivariant versions Quantum symmetries: subfactors, tensor categories, Hopf algebras, quantum groups; Enumerative Combinatorics.