Dr Nelly Villamizar

Associate Professor, Mathematics

n.y.villamizar@swansea.ac.uk

Office - 303
Third Floor
Computational Foundry
Bay Campus

Available For Postgraduate Supervision

Research Links

About

I am an Associate Professor in the Department of Mathematics at Swansea University. My research interests are applied algebraic geometry, and commutative algebra, in particular, their applications for solving problems arising in geometric modeling and Computer Aided Geometric Design (CAGD). Previously, from 2013 to 2017, I was a Postdoctoral Researcher in the Group of Symbolic Computation at RICAM, in Linz, Austria.

Website: https://sites.google.com/site/nvillami/

Areas Of Expertise

Applied Algebraic Geometry
Commutative Algebra
Spline Approximation
Algebraic Spline Geometry
Computer Aided Geometric Design

Career Highlights

Teaching Interests

MA-233 Projective Geometry: This module gives an introduction to various properties of projective geometry. It aims at strengthening the understanding of linear algebra by demonstrating its geometrical significance, while also pointing towards more advanced algebraic geometry.

MA-231 Advanced Geometry: This module covers conics sections, projective geometry as well as problems in non-euclidean geometry.

MA-312 Higher Algebra: This module approaches the theory of algebraic modules, exact sequences, and classification of abelian groups. It also introduces categories as a language and unifying force in modern mathematics.

MA-212 Groups and Rings: This module approaches the theory of groups, rings and modules as abstract algebraic objects.

Research Award Highlights Collaborations

Publications

Research Highlights

Sirvent, M., Sorokina, T., Villamizar, N., & Yuan, B. (2024). Multivariate polynomial splines on generalized oranges. Journal of Approximation Theory, 299, 106016
https://doi.org/10.1016/j.jat.2024.106016, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa65585
DeepeshToshniwal, . & Villamizar, N. (2023). Algebraic methods to study the dimension of supersmooth spline spaces. Advances in Applied Mathematics, 142
https://doi.org/10.1016/j.aam.2022.102412, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa60699
DiPasquale, M. & Villamizar, N. (2023). A lower bound for the dimension of tetrahedral splines in large degree. Constructive Approximation
https://doi.org/10.1007/s00365-023-09625-5, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa63205
http://dx.doi.org/10.1007/s00365-023-09625-5
Villamizar, N., Morales, A., & Reiner, V. (n.d.) Reflection groups and enumeration. In Federico Ardila, Carolina Benedetti, Alejandro Morales (Ed.), Algebraic and Geometric Combinatorics Cambridge University Press
SU Repository: https://cronfa.swan.ac.uk/Record/cronfa56992
Villamizar, N., Mantzaflaris, A., & Jüttler, B. (2022). Completeness Characterization of Type-I Box Splines. In Geometric Challenges in Isogeometric Analysis. Springer INdAM Series (pp. 279-304). Springer International Publishing.
https://doi.org/10.1007/978-3-030-92313-6_12, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa56991
DiPasquale, M. & Villamizar, N. (2021). A Lower Bound for Splines on Tetrahedral Vertex Stars. SIAM Journal on Applied Algebra and Geometry, 5(2), 250-277.
https://doi.org/10.1137/20m1341118, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa56990
Toshniwal, D. & Villamizar, N. (2020). Dimension of polynomial splines of mixed smoothness on T-meshes. Computer Aided Geometric Design, 80, 101880
https://doi.org/10.1016/j.cagd.2020.101880, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa54305
Gallet, M., Ranestad, K., & Villamizar, N. (2018). Varieties of apolar subschemes of toric surfaces. Arkiv för Matematik, 56(1), 73-99.
https://doi.org/10.4310/ARKIV.2018.v56.n1.a6, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa34564
https://arxiv.org/abs/1601.00694
Mourrain, B. & Villamizar, N. (2013). Homological techniques for the analysis of the dimension of triangular spline spaces. Journal of Symbolic Computation, 50, 564-577.
https://doi.org/10.1016/j.jsc.2012.10.002, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32879
Villamizar, N. & Mourrain, B. (2014). On the Dimension of Spline Spaces on Triangulations. In Tor Dokken and Georg Muntingh (Ed.), SAGA – Advances in ShApes, Geometry, and Algebra (pp. 177-197). Springer, Cham
https://doi.org/10.1007/978-3-319-08635-4_10, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32880

All Research

Journal Articles

Sirvent, M., Sorokina, T., Villamizar, N., & Yuan, B. (2024). Multivariate polynomial splines on generalized oranges. Journal of Approximation Theory, 299, 106016
https://doi.org/10.1016/j.jat.2024.106016, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa65585
DiPasquale, M. & Villamizar, N. (2023). A lower bound for the dimension of tetrahedral splines in large degree. Constructive Approximation
https://doi.org/10.1007/s00365-023-09625-5, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa63205
http://dx.doi.org/10.1007/s00365-023-09625-5
DeepeshToshniwal, . & Villamizar, N. (2023). Algebraic methods to study the dimension of supersmooth spline spaces. Advances in Applied Mathematics, 142
https://doi.org/10.1016/j.aam.2022.102412, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa60699
DiPasquale, M. & Villamizar, N. (2021). A Lower Bound for Splines on Tetrahedral Vertex Stars. SIAM Journal on Applied Algebra and Geometry, 5(2), 250-277.
https://doi.org/10.1137/20m1341118, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa56990
Toshniwal, D. & Villamizar, N. (2020). Dimension of polynomial splines of mixed smoothness on T-meshes. Computer Aided Geometric Design, 80, 101880
https://doi.org/10.1016/j.cagd.2020.101880, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa54305
Gallet, M., Ranestad, K., & Villamizar, N. (2018). Varieties of apolar subschemes of toric surfaces. Arkiv för Matematik, 56(1), 73-99.
https://doi.org/10.4310/ARKIV.2018.v56.n1.a6, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa34564
https://arxiv.org/abs/1601.00694
Gallet, M., Koutschan, C., Li, Z., Regensburger, G., Schicho, J., & Villamizar, N. (2017). Planar linkages following a prescribed motion. Mathematics of Computation, 86(303), 473-506.
https://doi.org/10.1090/mcom/3120, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32864
Blidia, A., Mourrain, B., & Villamizar, N. (2017). G 1 -smooth splines on quad meshes with 4-split macro-patch elements. Computer Aided Geometric Design, 52-53, 106-125.
https://doi.org/10.1016/j.cagd.2017.03.003, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32862
Mourrain, B., Vidunas, R., & Villamizar, N. (2016). Dimension and bases for geometrically continuous splines on surfaces of arbitrary topology. Computer Aided Geometric Design, 45, 108-133.
https://doi.org/10.1016/j.cagd.2016.03.003, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32863
http://www.sciencedirect.com/science/article/pii/S0167839616300309
Villamizar, N., Mantzaflaris, A., & Jüttler, B. (2016). Characterization of bivariate hierarchical quartic box splines on a three-directional grid. Computer Aided Geometric Design, 41, 47-61.
https://doi.org/10.1016/j.cagd.2015.11.004, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32865
Postinghel, E., Sottile, F., Villamizar, N., & Villamizar, N. (2015). Degenerations of real irrational toric varieties. Journal of the London Mathematical Society, 92(2), 223-241.
https://doi.org/10.1112/jlms/jdv024, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32867
Mourrain, B. & Villamizar, N. (2014). Bounds on the Dimension of Trivariate Spline Spaces: A Homological Approach. Mathematics in Computer Science, 8(2), 157-174.
https://doi.org/10.1007/s11786-014-0187-8, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32869
Mourrain, B. & Villamizar, N. (2013). Homological techniques for the analysis of the dimension of triangular spline spaces. Journal of Symbolic Computation, 50, 564-577.
https://doi.org/10.1016/j.jsc.2012.10.002, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32879

Book Chapters

Villamizar, N. & Mourrain, B. (2014). On the Dimension of Spline Spaces on Triangulations. In Tor Dokken and Georg Muntingh (Ed.), SAGA – Advances in ShApes, Geometry, and Algebra (pp. 177-197). Springer, Cham
https://doi.org/10.1007/978-3-319-08635-4_10, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa32880
Villamizar, N., Morales, A., & Reiner, V. (n.d.) Reflection groups and enumeration. In Federico Ardila, Carolina Benedetti, Alejandro Morales (Ed.), Algebraic and Geometric Combinatorics Cambridge University Press
SU Repository: https://cronfa.swan.ac.uk/Record/cronfa56992

Conference Contributions

Villamizar, N., Mantzaflaris, A., & Jüttler, B. (2022). Completeness Characterization of Type-I Box Splines. In Geometric Challenges in Isogeometric Analysis. Springer INdAM Series (pp. 279-304). Springer International Publishing.
https://doi.org/10.1007/978-3-030-92313-6_12, SU Repository: https://cronfa.swan.ac.uk/Record/cronfa56991

Supervision

Ore extensions of trusses (current)

PhD

Other supervisor: Prof Tomasz Brzezinski
Solving Polynomial Systems for Feedback Control (current)

PhD

Other supervisor: Prof Chenggui Yuan
Higher rank tropicalisation (current)

PhD

Other supervisor: Prof Tomasz Brzezinski

Other supervisor: Prof Jeffrey Giansiracusa

Other supervisor: Dr Edwin Beggs

Taught Modules

MA-233 Projective Geometry: theory and applications

This module gives an introduction to various properties of projective geometry. It aims at strengthening the understanding of linear algebra by demonstrating its geometrical significance, while also pointing towards more advanced algebraic geometry. The module starts by discussing affine transformations and isometries in Euclidean geometry. This is followed by the study of some properties of the projective plane, the connection of the subject to projections, and the idea of points at infinity where parallel lines meet. Subsequently, the focus will be in the axiomatic presentation of projective geometry, and the basic properties of projective spaces over arbitrary fields. In the last part of the module, the study of conics in projective space guides the introduction of homogeneous coordinates and the study of easy cases of Bézout's theorem, one of the fundamental results in both algebraic geometry and computer algebra. The module is developed via explicit examples and emphasizes those parts of projective geometry that are important in areas of applications such as Information Security, Statistics, Computer Graphics and Computer Vision.
MA-312 Higher Algebra

This course approaches the theory of groups, rings and modules as abstract algebraic objects. The course also introduces categories as a language and unifying force in modern mathematics.
MA-M07 Further Mathematics for Data Science

The module develops mathematical foundations for data science and machine learning. The module covers basic problems of Vector Calculus, Linear Algebra and Optimization including concepts of partial derivatives, gradient and Hessian, eigenvalues and eigenvectors of matrices and their use in optimisation problems, method of gradient descent and its application in Machine Learning. It also highlights the use of Python in the numerical solution of these problems, as well as in symbolic computation.

Key Grants and Projects

High order methods for computational engineering and data analysis. Alliance Hubert Curien Grant 2021-2022: Swansea University & Inria Sophia Antipolis Méditerranée 2021 - 2023

Grant awarded to strengthen collaborations between UK and France, with Dr. Angelos Mantzaflaris, Inria Sophia Antipolis Méditerranée., British Council & French Embassy in UK – Alliance Hubert Curien programme
Algebraic Spline Geometry: towards algorithmic shape representation. 2021 - 2024

This project is funded by the EPSRC New Investigator Award: EP/V012835/1., EPSRC - Engineering and Physical Sciences Research Council

Affiliated Positions

Research Scientist, RICAM

2013 - 2016

Dr Nelly Villamizar

Email address

Research Links

About

Areas Of Expertise

Career Highlights

Research Highlights

All Research

Ore extensions of trusses (current)

Solving Polynomial Systems for Feedback Control (current)

Higher rank tropicalisation (current)

MA-233 Projective Geometry: theory and applications

MA-312 Higher Algebra

MA-M07 Further Mathematics for Data Science

Research Scientist, RICAM