Singularities occur naturally in various branches of science. This places Singularity Theory as a focus of interest, both in Mathematics and in the context of applications. An important application within Mathematics is Generic Geometry, in which geometric phenomena are translated and manipulated in terms of singularities. This viewpoint has provided valuable tools in the global study of geometric properties. The purpose of the group is the development of Singularity Theory techniques of complex differentiable and analytic applications and the study of their interconnections with other areas, in particular with Geometry. The general objectives of this study can be summarised as follows: 

• Obtaining local and global topological invariants for singularities of differentiable applications, complex analytic applications, flows and foliations. 
• Application to the study of geometric objects (subvarieties and fronts). 
• Development of techniques to support the resolution of problems in Computer Graphics and in the design of microwave circuits in Telecommunications.

The objectives of the group are to make progress on the following topics: 

1. Study of topological aspects of real singularities. 
2. Development of the differential geometry of subvarieties with singularities (fronts and images of stable applications). 
3. Finite determination and classification of singularities of range greater than or equal to 2. Mond's conjecture. 
4. Effective computation of Lojasiewicz exponents. 
5. Non-degenericity conditions on polynomial applications. 
6. Development of a theory of mixed multiplicities for analytic varieties in the context of Bruce-Roberts numbers. 
7. Determination of invariants in the global classification of stable applications, flows and foliations on surfaces and 3-manifolds. 
8. Study of semilocal invariants of Vassiliev type and global invariants for Lagrangian applications. 
9. Study of topological aspects and determination of complete invariants for second-order geometry of surfaces and 3-manifolds in Euclidean space. Applications to string theory. 
10. Problems in Conformal Geometry: Rational curves of Pythagorean hodograph in computer-aided design. Study of conformal invariants through singularities of squared distance functions on a subvariety. 
11. Geometric applications to the design of microwave circuits. 
12. Design of Bézier surfaces with prefixed boundary properties. 

The team has extensive experience in the field, backed by more than 20 years of previous work by several of its members. It has young researchers and students undergoing research training. It maintains an active collaboration with numerous international specialists. In March 2009, it organised the "International Workshop on Singularities in Generic Geometry and Applications" aimed at boosting the interactions between Singularity Theory, Geometry and Applications at an international level. This has led to a biennial series of conferences: Bedlewo, Poland 2011 (Valencia II), Edinburgh, UK 2013 (Valencia III) and Kobe-Kyoto, Japan 2015 (Valencia IV) that place the team in a position of reference in the international arena.

Partners

• Ana María Arnal Pons - Universitat Jaume I de Castelló 
• Washington Luiz Marar - Universidade de Sao Paulo (Brasil) 
• Catarina Mendes De Jesus - Universidade Federal do Viçosa (Brasil) 
• Bruna Oréfice Okamoto - Universidade Federal de São Carlos (Brasil) 
• Federico Sánchez Bringas - Universidad Nacional Autónoma de México 
• João Nivaldo Tomazella - Universidade Federal de São Carlos (Brasil) 

Investigation activities