Polo Sport  Polo Club  Polo ponies  Players Equipment  The field  Notable players  Handicap players A Theory Of Generalized Donaldsonthomas Invariants RRP $281.99 This book studies generalized DonaldsonThomas invariants $bar{DT}{}^alpha(tau)$. They are rational numbers which 'count' both $tau$stable and $tau$semistable coherent sheaves with Chern character $alpha$ on $X$; strictly $tau$semistable sheaves must be counted with complicated rational weights. The $bar{DT}{}^alpha(tau)$ are defined for all classes $alpha$, and are equal to $DT^alpha(tau)$ when it is defined. They are unchanged under deformations of $X$, and transform by a wallcrossing formula under change of stability condition $tau$. To prove all this, the authors study the local structure of the moduli stack $mathfrak M$ of coherent sheaves on $X$. They show that an atlas for $mathfrak M$ may be written locally as $mathrm{Crit}(f)$ for $f:Uto{mathbb C}$ holomorphic and $U$ smooth, and use this to deduce identities on the Behrend function $nu_mathfrak M$. They compute the invariants $bar{DT}{}^alpha(tau)$ in examples, and make a conjecture about their integrality properties. They also extend the theory to abelian categories $mathrm{mod}$$mathbb{C}Qbackslash I$ of representations of a quiver $Q$ with relations $I$ coming from a superpotential $W$ on $Q$. RRP $372.99 This book provides an extensive and selfcontained presentation of quantum and related invariants of knots and 3manifolds. Polynomial invariants of knots, such as the Jones and Alexander polynomials, are constructed as quantum invariants, i.e. invariants derived from representations of quantum groups and from the monodromy of solutions to the KnizhnikZamolodchikov equation. With the introduction of the Kontsevich invariant and the theory of Vassiliev invariants, the quantum invariants become wellorganized. Quantum and perturbative invariants, the LMO invariant, and finite type invariants of 3manifolds are discussed. The ChernSimons field theory and the WessZuminoWitten model are described as the physical background of the invariants. RRP $354.99 In algebraic topology some classical invariants  such as Betti numbers and Reidemeister torsion  are defined for compact spaces and finite group actions. They can be generalized using von Neumann algebras and their traces, and applied also to noncompact spaces and infinite groups. These new L2invariants contain very interesting and novel information and can be applied to problems arising in topology, KTheory, differential geometry, noncommutative geometry and spectral theory. It is particularly these interactions with different fields that make L2invariants very powerful and exciting. The book presents a comprehensive introduction to this area of research, as well as its most recent results and developments. It is written in a way which enables the reader to pick out a favourite topic and to find the result she or he is interested in quickly and without being forced to go through other material. Search
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