Geometry

My interest in geometry are Discrete Differential Geometry, Applied geometry, Algebraic Geometry and Tropical Geometry.

Geometric Processing

Animation.mp4

Under the supervision of Helmut Potman as his Master/Ph. D student, I participated in the course of Geometric processing where I was able to replicate part of the results presented in the paper: Quad-mesh based isometric mappings and developable surfaces.

Discrete Differential Geometry

Discrete Differential Geometry is a recent area in mathematics that has put its efforts into providing a formal process of discretization within Erlangen program, which studies geometric objects based on group theory and projective geometry [2]. The relevance of discrete differential geometry is related to the interesting relations with integrability theory and, most recently, algebraic geometry, without mentioning the number of applications in computer-aided geometric design, computer graphics, animation, and architecture.

I had the pleasure to participate in an internship supervised by Helmut Potmann at KAUST (King Abdullah University of Science and Technology), where I met for the first time Discrete Differential Geometry. As a result, it has become my new focus of interest. However, I am still in a learning process about this topic. I have explored the following concepts in this area: nets, nets on quadrics, discretization principles, Lie Sphere Geometry, Möbius geometry (basic concepts), and Laguerre geometry (in the context of Lie Sphere Geometry).

This image is related to one of the objects that I studied during my internship. It was extracted from https://www.dmg.tuwien.ac.at/geom/ig/publications/snets/snets.pdf

Wu's Algorithm and Voronoi Diagrams of Quadrics

During my undergraduate, I participated in a project in the field of Applied Geometry. The project was a combination of functional programming, classical algebraic geometry, and computational geometry. The project aimed to implement Wu's Algorithm [1] on Haskell to obtain an analytic expression that represents the Voronoi vertices of the corresponding Voronoi Diagram of the quadrics (i.e consider the quadrics as generators instead of points). The project was quite ambitious, but due to the lack of time, we only achieved an analytic expression for the bisector surface of two quadrics. This project was presented at the Sixth International Conference on Analytic Number Theory and Spatial Tessellations. Unfortunately, due to some inconvenience with the organization and some servers, the paper was not published. You can find the code here.

Tropical Geometry

Tropical Geometry, also known as the combinatorial shadow of algebraic geometry, is a relatively novel field in mathematics that has caught the attention of researchers. Mathematically speaking is the study of tropical polynomials constructed over the tropical semiring, where the sum corresponds to the min or max operation, and the multiplication operation is the classical sum. Its relevance is due to its relations with combinatorics and algebraic geometry, with many applications in several branches of science [3-15]. With this motivation, I collaborate in the creation of a package implemented on Haskell to deal with tropical calculations of polynomials. We had the opportunity to present the first stage of the package at the CASC 2020 (Computer Algebra and Scientific Computing Conference, 2020) and hold it online (video). The code can be found here.

[1] Wu, W. T. Basic principles of mechanical theorem proving in elementary geometries. (1984). J. Syst. Sci. Math. Sci., 4, 207–35

[2]Sullivan, J. M., Schröder, P., & Ziegler, G. (2008). Discrete differential geometry (Vol. 38). A. I. Bobenko (Ed.). Basel: Birkhäuser.

[3] Brugallé, E., Cueto, M. A., Dickenstein, A., Feichtner, E. M., & Itenberg, I. (2013). Algebraic and Combinatorial Aspects of Tropical Geometry (Vol. 589). American Mathematical Society.

[4] Maclagan, D, Sturmfels, B.: Introduction to Tropical Geometry (Graduate Studies in Mathematics). American Mathematical Society (2015)

[5] Ardila, F., Develin, M.: Tropical hyperplane arrangements and oriented matroids. Mathematische Zeitschrift 262(4), 795–816 (Aug 2008). https://doi.org/10.1007/s00209-008-0400-z

[6] Katz, E.: Matroid theory for algebraic geometers. In: Nonarchimedean and tropical geometry, pp. 435–517. Springer (2016)

[7] Kovacsics, P.C., Edmundo, M., Ye, J.: Cohomology of algebraic varieties over non-archimedean fields (2020)

[8] Mikhalkin, G.: Enumerative tropical algebraic geometry in R 2 . Journal of the American Mathematical Society 18(2), 313–377 (2005)

[9] Cerbu, A., Marcus, S., Peilen, L., Ranganathan, D., Salmon, A.: Topology of tropical moduli of weighted stable curves. Advances in Geometry 0(0) (Jan 2020). https://doi.org/10.1515/advgeom-2019-0034

[10] Gross, M.: Tropical geometry and mirror symmetry. Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society with support from the National Science Foundation, Providence, R.I (2011)

[11] Allamigeon, X., Benchimol, P., Gaubert, S., Joswig, M.: Log-barrier interior point methods are not strongly polynomial. SIAM Journal on Applied Algebra and Geometry 2(1), 140–178 (2018)

[12] Monod, A., Lin, B., Yoshida, R., Kang, Q.: Tropical Geometry of Phylogenetic Tree Space: A Statistical Perspective (2018)

[13] Zhang, L., Naitzat, G., Lim, L.H.: Tropical Geometry of Deep Neural Networks (2018)

[14] Hampton, M., Moeckel, R.: Finiteness of relative equilibria of the fourbody problem. Inventiones Mathematicae 163(2), 289–312 (Feb 2006). https://doi.org/10.1007/s00222-005-0461-0

[15] Kim, Y., Lee, J.H.: Non-diplaceable toric fibers on compact toric manifolds via tropicalizations. arXiv: Symplectic Geometry (2016)

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