A Wandering Mathematician personal notes

About me

My name is Felipe Eduardo Ponce Vanegas, and I am a postdoc at the Basque Center for Applied Mathematics (BCAM). I am interested in industrial applications of mathematics, in particular, to advanced manufacturing. My office is located at the Aeronautics Advanced Manufacturing Center (CFAA) in the Basque Country Technology Park.

BCAM signed a collaboration agreement with CFAA, IDEKO, and IMH Campus with the objective of promoting new projects on digitalization and AI in the context of Industry 4.0 and sustainability, and my work is framed within this collaboration.

First stages of my career

I finished my BSc. in Chemistry, and during my thesis — supervised by Édgar Delgado at the National University of Colombia, sede Bogotá — I developed a material to remove fluoride from water. The objective of the project was to provide safe drinking water to poor communities with limited access to other sources of water. The material has a high efficiency of removal, it is affordable, easy to manufacture, and harmless to the human body, so we decided to patent it.

Late during my BSc., I noticed that I enjoyed more spending my time doing math, even though I felt the lab was a great place to discover and create new materials for the benefit of people. Finally, I decided to pursue a Master in applied mathematics, working on problems of existence and uniqueness of solutions for equations describing elastic bodies. I was excited when I found that the Calderón–Zygmund theory allows us to prove very general theorems of existence and uniqueness, which was the content of my thesis Minimization of energy functionals in elasticity, supervised by Leonid Lebedev; see also my first paper.

Continuing at the National University of Colombia, I did my PhD thesis on Harmonic Analysis, under the supervision of Javier Ramos and Germán Fonseca. My thesis was on the problem of restriction of the Fourier transform, and it is available in this link. The main result of my thesis was a sharp estimation for a trilinear restriction inequality for the saddle; see here.

My initial work at BCAM

I was initially hired by BCAM to work with Pedro Caro, and I started my job applying restriction theory to inverse problems. Later, I got interested in the problem of pointwise convergence of solutions of dispersive equations to the initial data; see, for example, this paper. In general, I worked on analysis and PDEs, but gradually my interests began to shift towards applied mathematics. My last event within the Analysis community was at Oberwolfach (3 July – 9 July, 2022).

Current interests

Nowadays, I work in modelling and monitoring of manufacturing processes. In general, I try to pay attention to all the problems engineers pose to me, or I notice. For example, how to avoid undesirable tool vibrations during cutting processes.

Also, check out this git course I gave at BCAM.

List of papers

  1. with Irastorza, M. (2026). The Tustin Method for approximating eigenvalues of delay systems. J. Comput. Appl. Math., 472, 116772. https://doi.org/10.1016/j.cam.2025.116772. Repo.

  2. with Bartfai, A., Hogan, J., Kuske, R., & Dombovari, Z. (2025). Semi-analytical framework for the study of finite-time stability of forced dynamical systems with slowly varying parameters. J. Sound Vib., 618, 119359. https://doi.org/10.1016/j.jsv.2025.119359.

  3. with Bartfai, A., & Dombovari, Z. (2025). Semi-analytical Estimation for the Escape of Solutions of Linear Differential Equations with Slowly Varying Coefficients. SIAM J. Appl. Math., 85(4), 1519–1549. https://doi.org/10.1137/24M1685481. Repo.

  4. with Bizzarri, M., & Barton, M. (2023). On $C^0$ and $C^1$ continuity of envelopes of rotational solids and its application to 5-axis CNC machining. CAGD, 107, 102245. https://doi.org/10.1016/j.cagd.2023.102245. Preprint and repo.

  5. with Eceizabarrena, D., (2022). Counterexamples for the fractal Schrödinger convergence problem with an Intermediate Space Trick. Commun. Pure Appl., 21(11), 3777–3812. https://doi.org/10.3934/cpaa.2022122. Preprint.

  6. with Eceizabarrena, D., (2022). Pointwise convergence over fractals for dispersive equations with homogeneous symbol. J. Math. Anal. Appl., 515, 126385. https://doi.org/10.1016/j.jmaa.2022.126385.

  7. with Lucà, R., (2022). Convergence over fractals for the Schrödinger equation. Indiana Univ. Math. J., 71(6), 2283–2307. http://dx.doi.org/10.1512/iumj.2022.71.9302. Preprint.

  8. with Kumar, S., & Vega, L., (2022). Static and dynamical, fractional uncertainty principles. Trans. Am. Math. Soc., 375(8), 5691–5725. https://doi.org/10.1090/tran/8655.

  9. (2021). The Bilinear Strategy for Calderón’s Problem. Rev. Mat. Iberoam., 37(6), 2119–2160. https://doi.org/10.4171/rmi/1257. Preprint.

  10. (2020). Reconstruction of the derivative of the conductivity at the boundary. Inverse Problems and Imaging, 14(4), 701–718. https://doi.org/10.3934/ipi.2020032. Preprint.

  11. (2020). A Trilinear Restriction Estimate for the Hyperbolic Paraboloid with Sharp Dependence on Transversality. Int. Math. Res. Not., 2020(18), 5723–5753. https://doi.org/10.1093/imrn/rny178.

  12. (2018). Examples of measures with slow decay of the spherical means of the Fourier transform. Proc. Am. Math. Soc., 146(6), 2617–2621. http://dx.doi.org/10.1090/proc/13999.

  13. with Galvis, J., & Mantilla, L.M. (2014). Some numerical studies of oscillating chemical reactions using discontinuous finite elements. Revista Facultad de Ciencias Universidad Nacional de Colombia, Sede Medellín, 3(2), 81–93. Article.

  14. with Cloud, M., & Lebedev, L. (2014). Finite element method in equilibrium problems for a nonlinear shallow shell with an obstacle. Int. J. Eng. Sci., 80, 43–52. https://doi.org/10.1016/j.ijengsci.2014.02.024.

  15. with Lebedev, L., & Rendón, L. (2013). On Weak Solvability of Boundary Value Problems for Elliptic Systems. Rev. Colomb. Mat., 47(2), 191–204. Article.

Conference papers

  1. (2026). On complex mode shapes and their identification. In: Vehovszky, B. (eds), Journal of Physics: Conference Series: vol 3190 International Conference for Acoustic and Vibration Engineers (pp 012009). IOP Publishing. https://doi.org/10.1088/1742-6596/3190/1/012009.

  2. with Kumar, S., Roncal, L., & Vega, L., (2024). The Frisch–Parisi Formalism for Fluctuations of the Schrödinger Equation. In: Machihara, S. (eds), Springer Proceedings in Mathematics & Statistics: vol 451. Mathematical Physics and Its Interactions (pp 199–223). Springer. https://doi.org/10.1007/978-981-97-0364-7_7. Preprint.

Former students

  1. García Sanabria, D. A. (2026). Collision Detection, Robotics, and Path Planning [Bachelor’s Thesis, DigiPen Institute of Technology Europe-Bilbao]. Co-directed with Michael Barton.

  2. Irastorza, M. (2024). Numerical methods for delay differential equations [Bachelor’s Thesis, University of the Basque Country]. Thesis