Handbook of Geometry and Topology of Singularities VIII by Cisneros-Molina, José Luis; Dũng Tráng, Lê; Seade, José;

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This is the eight volume of the Handbook of Geometry and Topology of Singularities, a series that provides an accessible account of the state of the art of the subject, its frontiers and its interactions with other areas of research.

This volume consists of twelve chapters with reader-friendly introductions to several important topics and aspects of singularity theory, such as:

• Plane curve singularities studied by means of divides, which capture a lot of their topology.
• Viro’s method to study the topology of real algebraic varieties, providing a wide range of possible combinations of topological and combinatorial invariants.
• Local tropicalization, a technique for attaching a combinatorial object to germs of subvarieties of algebraic tori and toric varieties.
• The theory of Zariski pairs and superisolated singularities.
• The McKay correspondence, a deep connection that links group theory, algebraic geometry, and representation theory.
• Calculations with Characteristic Cycles, a deep concept in the interplay between algebraic geometry, representation theory and microlocal analysis.
• The monodromy zeta functions in singularity theory.
• The singularities of the minimal model program of complex quasi-projective varieties.
• A general theory of Thom polynomials associated to the classification of map-germs.
• A discussion on indices and residues, intertwining the theories of complex analytic singular varieties and singular holomorphic foliations.
• The Monodromy in Integral Geometry and PDE.
• The topological theory of Hyperplane Arrangements.

The book is addressed to graduate students and newcomers to the theory, as well as to specialists who can use it as a guidebook.

Tartalomjegyzék:

1 Norbert A’Campo and Pablo Portilla Cuadrado, Plane curve singularities via divides.- 2 Ilia Itenberg, Viro method in topology of real algebraic varieties.- 3 Patrick Popescu-Pampu and Dmitry Stepanov, An introduction to local tropicalization.- 4 Enrique Artal Bartolo, Superisolated singularities and friends.- 5 José Luis Cisneros-Molina and Agustín Romano-Velázquez, A review on the McKay correspondence.- 6 David B. Massey, Calculations with Characteristic Cycles.- 7 Sabir M. Gusein-Zade, Monodromy zeta functions in singularity theory.- 8 Christopher Hacon, Singularities of the minimal model program.- 9 Toru Ohmoto, Thom polynomials for singularities of maps.- 10 Maurício Corrêa and José Seade, Indices and residues: from Poincaré-Hopf to Baum-Bott, and Marco Brunella.- 11 V. A. Vassiliev, Monodromy in Integral Geometry and PDE.- 12 Masahiko Yoshinaga, Topology of hyperplane arrangements via real structure.