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L2-invariants

RRP $354.99

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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 non-compact spaces and infinite groups. These new L2-invariants contain very interesting and novel information and can be applied to problems arising in topology, K-Theory, differential geometry, non-commutative geometry and spectral theory. It is particularly these interactions with different fields that make L2-invariants 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.


A Theory Of Generalized Donaldson-thomas Invariants

RRP $281.99

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This book studies generalized Donaldson-Thomas 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 wall-crossing 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 $Q$.


Classical And Involutive Invariants Of Krull Domains

RRP $546.99

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This monograph is devoted to Krull domains and its invariants. The book shows how a serious study of invariants of Krull domains necessitates input from various fields of mathematics, including rings and module theory, commutative algebra, K-theory, cohomology theory, localization theory and algebraic geometry. About half of the book is dedicated to so-called involutive invariants, such as the involutive Brauer group, and is essentially the first to cover these topics. In a structured and methodical way, the work presents a large quantity of results previously scattered throughout the literature. Audience: This volume is recommended as a first introduction to this rapidly developing subject, but will also be useful as a state-of-the-art reference work, both to students at graduate and postgraduate levels and to researchers in commutative rings and algebra, algebraic K-theory, algebraic geometry, and associative rings.



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