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Singularity Theory and Algebraic Geometry

Research Directions

The development of singularity theory has always been influenced by nearby areas in algebra, geometry, and topology. A typical example is the study of topological aspects of singularities and degenerations which ran parallel with the development of differential topology: the algebraic and analytic study of degenerations and deformations, including the variation of complex and Hodge structures, D-modules, and Hodge theory, was largely motivated and related with concurrent developments of  arithmetic nature around the Decomposition Theorem. Interesting connections have been found with representation theory and combinatorics.

In the same way, recent major developments in geometry are bringing new methods and ideas. They put long-standing questions in a new light, create new connections and are ultimately reshaping the theory. Several techniques arising from various fields are now becoming central in Singularity Theory: tropical and non-Archimedean methods, log-geometry methods (with the origin in arithmetic geometry, but playing a role in the SYZ problem via Kato-Nakayama spaces), O-minimal and model theoretic aspects, which are very related to the metric study of singularities. The perspective coming from Mori’s Minimal Model Program further connects singularity theory with algebraic geometry.

Our team aims to incorporate all the variety of methods from different directions in the common framework of Singularity Theory: to explore relations between them, their impact in classical questions, and to formulate new questions motivated by them. The following will be at focus:

 

Local algebra

Local algebra is primarily understood as the study of singularities using the methods of commutative algebra. Our work mostly follows in two primary directions therein, but we are interested in and worked on many connected topics. 

  • Positive characteristic methods colloquially mean the study of singularities defined over a field of positive characteristic by exploiting the existence of the Frobenius homomorphism. There are many invariants and classes of singularities defined via various properties of the Frobenius. These methods are currently used towards to possible extension of Minimal Model Program to positive characteristic, and even to answer questions in characteristic 0 via reduction modulo a prime.
  • Multiplicity theory concerns the study of multiplicity, the oldest and, perhaps, the most fundamental singularity invariant. Multiplicity has algebraic, analytic, and topological interpretations presenting the magnitude of methods that singularity theory employs.

These two directions are not disjoint. Theorems about multiplicity can be proved using positive characteristic methods and extended to characteristic 0, while multiplicity theory may suggests what properties to search in positive characteristic invariants.

 

Singularities from the viewpoint of geometry and topology.

While the topological aspects of singularities and degenerations are widely studied and are not yet completely understood, their symplectic and metric aspects are in a much more incipient stage.

  • On the symplectic side, there are methods such as the invariants arising from the Floer theory, the symplectic study of vanishing cycles and cell and handle decompositions, contact structures and open books. We are interested in Floer-theoretic invariants associated with singularities and degenerations, and in finding out possible algebraic counterparts, like the Lattice Cohomology program pursued by A. Nemethi and collaborators or conjectural descriptions of Floer-theoretic invariants in terms of arc spaces.
  • On the metric side, the study of singularities from the Lipschitz geometry view point (a development that taking place mainly within singularity theory) is one of the most flourishing trends, which probably will transcend the borders of the singularity theory and influence other areas in the future. In contrast with the well developed topological theory of singularities, the metric structure is still at an initial stage and is at focus nowadays.
  • The Mirror Symmetry also provides a new motivation and new questions: the study of Strominger-Yau-Zaslow problem and the Gross-Siebert program provided a completely new viewpoint, motivation and set of new goals and tools in the study of 1-parameter degenerations of algebraic varieties. Simultaneously, Kontsevich’s categorical mirror symmetry also deeply influences the theory, for instance it brings a new light in the classical study of matrix factorizations and Cohen-Macaulay modules via the connections with singularity categories and derived categories of coherent sheaves.

 

Opportunities

There are several opportunities for stays in our group, please contact us.

There are also several sources of external funding. These fellowships are competitive and we advise to consult with us well in advance. 

 

Past & Future activities

Eusko Jaurlaritza - Gobierno Vasco ikerbasque - Basque Foundation for Science Bizkaia xede. Bizkaiko Foru Aldundia innobasque - Agencia vasca de la innovación Universidad del Pa├îs Vasco (UPV/EHU)