Singularities appear 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 perspective has provided valuable tools in the global study of geometric properties. The group's purpose is to develop Singularity Theory techniques for differentiable and complex analytic applications and to study their interconnections with other areas, especially geometry. The general objectives of this study can be summarized as: a) Obtaining local and global topological invariants for singularities of differentiable and complex analytic applications, flows, and foliations. b) Application to the study of geometric objects (submanifolds and frontals). c) Development of support techniques for problem-solving in Computer Graphics and microwave circuit design in telecommunications. The group's objectives include achieving various advances in the following areas:
1) Study of topological aspects of real singularities.
2) Development of the differential geometry of submanifolds with singularities (frontals and images of stable maps).
3) Finite determination and classification of singularities of rank greater than or equal to 2. Mond's conjecture.
4) Effective computation of Lojasiewicz exponents.
5) Non-degenericity conditions on polynomial maps.
6) Development of a theory of mixed manifolds for analytic manifolds in the context of Bruce-Roberts numbers.
7) Determination of invariants in the global classification of stable maps, flows, and foliations on surfaces and 3-manifolds.
8) Study of Vassiliev-type semilocal invariants and global invariants for Lagrangian maps.
9) Study of topological aspects and determination of complete invariants for the second-order geometry of surfaces and 3-manifolds in Euclidean space. Applications to string theory.
10) Problems in conformal geometry: Rational curves of the Pythagorean hodograph in computer-aided design. Study of conformal invariants through singularities of squared distance functions on a submanifold.
11) Geometric applications to the design of microwave circuits.
12) Design of Bézier surfaces with predefined boundary properties.
The team has extensive experience in the field, supported by more than 20 years of prior work by several of its members. It includes young researchers and students in training. It maintains active collaboration with numerous international specialists. It is worth noting that the "International Workshop on Singularities in Generic Geometry and Applications" was organized in March 2009, aimed at promoting the international interaction of Singularity Theory with Geometry and Applications. 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), which have placed the team in a leading position internationally.
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