Linear connections on matrix geometries

A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection.Comment: 14p, LPTHE-ORSAY 94/9

Linear Connections on Fuzzy Manifolds

Linear connections are introduced on a series of noncommutative geometries which have commutative limits. Quasicommutative corrections are calculated.Comment: 10 pages PlainTex; LPTHE Orsay 95/42; ESI Vienna 23

Linear Connections in Non-Commutative Geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of $\Omega^1$. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of $\Omega^1$. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois- Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of $\Omega^1$. These constructions are illustrated with the example of the algebra of $n \times n$ matrices.Comment: 15 pages, LMPM ../94 (uses phyzzx

On Curvature in Noncommutative Geometry

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared.Comment: 16 pages, PlainTe

On the first order operators in bimodules

We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.Comment: 13 pages, AMSLaTe

Risks and return of banking activities related to hedge funds.

There are approximately 10,000 hedge funds worldwide, managing assets of over USD 1.5 trillion. Investment banking activities are more and more intertwined with hedge funds, as hedge funds obtain financing from banks through prime brokerage and are clients or counterparties of banks for all sorts of products. The development of hedge funds has therefore created many opportunities for investment banks. Bank benefit from hedge funds activities directly to the extent that hedge funds are their clients. All capital market activities benefit from it, from brokerage and research to derivatives. Prime brokerage has become a growing source of income. Banks have a very important business of providing derivatives and products, from vanilla products to more complex, customized and exotic products. Hedge funds are also possible underlyings for derivatives. Many banks, including Société Générale, have developed a business of writing options on hedge funds as well as providing leverage to funds of funds. Investment banks are not only making profits by transacting with hedge funds. They also benefi t indirectly through more trading: on certain specifi c specialized market, like structured complex derivatives, there would be no market at all without the availability of hedge funds that are willing to take the risks. Together, as two intertwined partners, hedge funds and investment banks have extended the reach and effi ciency of capital markets. The benefi ts that this system brings to the economy as a whole is widely recognized. Not only do hedge funds provide important benefi ts for the economy in general but their risks are manageable. The risks for investors are overplayed. Whatever the risk measure, hedge funds are clearly less risky than equities. As regards operational risks, the market itself is able to generate protection solutions. Academic research has shown that operational risks can be dealt in the most extensive way by using managed account platforms, such as the Lyxor platform. The risks for banks are under control and the move toward “risk-based margining” has improved very much their risk management. Banks in general invest a lot of resources in monitoring hedge funds qualitatively through due-diligences. They also put different types of limits in order to cover different aspects of risks: nominal limits, stress test limits, limits on delta, limits on vega, expected tail loss limits. Moreover, they regulate their capital requirements using not only Value at Risk, the usual tool used by banks to allocate capital to market risks, but also stress tests losses based on the worst possible scenarios. These very sophisticated models are quite convincing. There is no reason to believe that they will not work in practice under stress conditions. There are also general consideration about a systemic risk that would be something else than banking risks, but it has no real argument to back it up. Hedge funds are fi rst of all the result of a signifi cant improvement of asset management techniques. These improvements are here to stay, whatever the regulatory environment will become, since these techniques will be more and more part of the mainstream asset management world. Hedge funds are more and more institutionalized. They will eventually merge with “classical” asset management, while some forms of compromises between hedge funds and classical asset management, such as absolute return funds or 130-30 funds, are becoming more common. Hedge funds are just a nice new development of capital markets that, like all past capital market developments, will be irreversible and will contribute to a more effi cient fi nancial system.

Examples of derivation-based differential calculi related to noncommutative gauge theories

Some derivation-based differential calculi which have been used to construct models of noncommutative gauge theories are presented and commented. Some comparisons between them are made.Comment: 22 pages, conference given at the "International Workshop in honour of Michel Dubois-Violette, Differential Geometry, Noncommutative Geometry, Homology and Fundamental Interactions". To appear in a special issue of International Journal of Geometric Methods in Modern Physic

Curvature and geometric modules of noncommutative spheres and tori

When considered as submanifolds of Euclidean space, the Riemannian geometry of the round sphere and the Clifford torus may be formulated in terms of Poisson algebraic expressions involving the embedding coordinates, and a central object is the projection operator, projecting tangent vectors in the ambient space onto the tangent space of the submanifold. In this note, we point out that there exist noncommutative analogues of these projection operators, which implies a very natural definition of noncommutative tangent spaces as particular projective modules. These modules carry an induced connection from Euclidean space, and we compute its scalar curvature