Algebraic geometry is one of the leading fields of the mathematics, and in the last decades its subfield, the theory of singularities became a mainstream research area. Its recent connections with algebraic and differential topology, number theory, arithme tics and combinatorics are really fascinating. All the connections can be seen very clearly in the theory of complex surface singularities. This theory incorporates the rich structure of algebraic/analytic invariants (like the geometric genus, multiplicity ), new topological invariants (e.g. the Casson-Walker or Seiberg-Witten invariants of their links), combinatorics of their resolution graphs involving e.g. Newton diagrams and generalized Dedekind-Fourier sums. This is the area in which the researcher of t he present proposal, András Nemethi, is actively working. The main goal of the proposal is to create a mathematically rich and supporting environment for him by strongly qualified local team of the Renyi Institute of Mathematics, Budapest, Hungary, in orde r to develop the theory and attack the recent important open problems and conjectures in the field. The efforts would be enforced by the complementary skills of the experts at the host in combinatorics and in the Seiberg-Witten theory. In addition the rese archer would receive advanced training in various topics that are important for his future carrier. As a result, we expect a substantial impact in the field by clarification of the hierarchy of numerical surface singularity invariants. Since the researche r originates from Rumania and has been working in the USA for more than a decade, the project contributes towards reversing brain drain. In summary the project fits nicely with the objectives of scientific excellence of the European Research Area and with the objectives of the host institute.
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