PDEmas

PEOPLE

MARÍA JOSÉ CÁCERES GRANADOS

LINES OF RESEARCH

My research has a double aspect: analytical and numerical. Currently one of my lines of research is focused on the analysis of mesoscopic and microscopic models applied to Computational Neuroscience. In this field I am working on numerical solvers for already known models, on the study of new mathematical models and on the analysis of known models but many of their mathematical properties were unknown. Among my collaborators, in this direction, there are experimentalist researchers.

In another of my lines of research, I work on the development of efficient numerical solvers to describe the behavior of semiconductor devices in collaboration with electronic engineers, computer scientists and mathematicians.

Finally, another of the lines of work in which I am interested is the analysis of asymptotic behavior in mathematical models that describe coagulation-fragmentation phenomena and in models for diffusion reaction processes.

ARTICLES

  1. M. J. Cáceres, J. A. Cañizo, N. Torres. Comparison principles and asymptotic behavior of delayed age-structured neuron models. Preprint , 2025. Journal · arXiv
  2. M. J. Cáceres, J. A. Cañizo, A. Ramos Lora. Sequence of pseudo-equilibria describes the long-time behaviour of the NNLIF model with large delay. Physical Review E. 110(6),064308, 2024. Journal · arXiv
  3. M. J. Cáceres, J. A. Cañizo, A. Ramos Lora. On the asymptotic behavior of the NNLIF neuron model for general connectivity strength.  To appear in Communications in Mathematical Physics, 2024. Journal · arXiv
  4. Torres, N., Cáceres, M. J., Perthame, B., Salort, D. An elapsed time model for strongly coupled inhibitory and excitatory neural networks. Physica D: Nonlinear Phenomena: 132977, 2021. Journal · arXiv
  5. Cáceres, M. J., Ramos Lora, A. An understanding of the physical solutions and the blow-up phenomenon for Nonlinear Noisy Leaky Integrate and Fire neuronal models. Commun. Comput. Phys. 30, 820-850, 2021. Journal · arXiv
  6. M. J. Cáceres. Rivista di Matematica della Università di Parma. Rivista di Matematica della Università di Parma 10 (2), 199-446, 2019. Journal · arXiv
  7. Cáceres, M. J., Roux, P., Schneider, R., Salort D. Global-in-time solutions and qualitative properties for the NNLIF neuron model with synaptic delay. Commun. in Partial Diff. Eqs. vol. 44, no 12, 1358-1386, 2019. Journal · arXiv
  8. M. J. Cáceres, J. A. Cañizo. Close-to-equilibrium behaviour of quadratic reaction–diffusion systems with detailed balance. Nonlinear Analysis 159, 62-84, 2017. Journal · arXiv
  9. Cáceres, M. J. and Schneider, R. Blow-up, steady states and long time behaviour of excitatory-inhibitory nonlinear neuron models. Kinetic and Related Models 10(3):587–612, 2017. Journal · arXiv
  10. M. J. Cáceres, R. Schneider. Towards a realistic NNLIF model: Analysis and numerical solver for excitatory-inhibitory networks with delay and refractory periods. arXiv preprint arXiv:1705.02205, 2017. Journal · arXiv
  11. Bisi, Marzia, Cáceres, María J. A BGK relaxation model for polyatomic gas mixtures. Communications in Mathematical Sciences 14:297-325, 2016. Journal · arXiv
  12. Chevallier, J., Cáceres, M. J., Doumic, M. and Reynaud Bouret, P. Microscopic approach of a time elapsed neural model. Mathematical Models and Methods in Applied Sciences 25(14):2669–2719, 2015. Journal · arXiv
  13. Vecil, F., Mantas, J. M., Cáceres, M. J., Sampedro, C., Godoy, A. and Gámiz, F. A parallel deterministic solver for the schrödinger–poisson–boltzmann system in ultra-short DG-MOSFETs: Comparison with Monte-Carlo. Computers & Mathematics with Applications 67(9):1703–1721, 2014. Journal · arXiv
  14. Cáceres, María J., Perthame, Benoît. Beyond blow-up in excitatory integrate and fire neuronal networks: Refractory period and spontaneous activity. Journal of Theoretical Biology, 81-89, 2014. Journal · arXiv
  15. M. J. Cáceres, P. G. Rodelas. Numerical analysis of the BGK model with external confining potential. Universidad de Granada, 2013.
  16. Cáceres, M. J., Carrillo, J. A. and Perthame, B. Analysis of nonlinear noisy integrate & fire neuron models: blow-up and steady states. The Journal of Mathematical Neuroscience 1(1):1–33, 2011. Journal · arXiv
  17. Cáceres, M. J., Carrillo, J. A. and Tao, L. A numerical solver for a nonlinear Fokker-Planck equation representation of neuronal network dynamics. J. Comp. Phys. 230:1084–1099, 2011. Journal · arXiv
  18. M. J. Cáceres, J. A. Cañizo and S. Mischler. Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations. Journal de Mathémathiques Pures et Appliquées Vol. 96, No. 4, pp. 334-362, 2011. Journal · arXiv
  19. M. J. Cáceres, J. A. Cañizo, S. Mischler. Rate of convergence to self-similarity for the fragmentation equation in L^ 1 spaces. Communications in Applied and Industrial Mathematics 1 (2), pp.299-308, 2011. Journal · arXiv
  20. M. Bisi, M. J. Cáceres, G. Spiga. A Bhatnagar–Gross–Krook kinetic approach to fast reactive mixtures: Relaxation problems. Physica A: Statistical Mechanics and its Applications 389 (21), 4528-4544, 2010. Journal · arXiv
  21. R. Bosi, M. J. Cáceres. The BGK model with external confining potential: existence, long-time behaviour and time-periodic Maxwellian equilibria. Journal of Statistical Physics 136, 297-330, 2009. Journal · arXiv
  22. Abdallah, N. B., Cáceres, M. J., Carrillo, J. A., & Vecil, F. A deterministic solver for a hybrid quantum-classical transport model in nanoMOSFETs. Journal of Computational Physics 228(17), 6553-6571, 2009. Journal · arXiv
  23. Mantas, J. M., Cáceres, M. J. Efficient deterministic parallel simulation of 2D semiconductor devices based on WENO-Boltzmann schemes. Computer Methods in Applied Mechanics and Engineering 198 (5-8), 693-704, 2009. Journal · arXiv
  24. M. J. Cáceres, G. Toscani. Kinetic approach to long time behavior of linearized fast diffusion equations. Journal of Statistical Physics 128, 883-925, 2007. Journal · arXiv
  25. M. J. Cáceres, J. A. Carrillo, A. Majorana. Deterministic Simulation of the Boltzmann–Poisson System in GaAs-Based Semiconductors. SIAM Journal on Scientific Computing 27 (6), 1981-2009, 2006. Journal · arXiv
  26. M. J. Cáceres, J. A. Carrillo, I. M. Gamba, A. Majorana, C. W. Shu. Deterministic kinetic solvers for charged particle transport in semiconductor devices. Transport Phenomena and Kinetic Theory: Applications to Gases, Semiconductors, Photons, and Biological Systems, 151-171, 2006. Journal · arXiv
  27. M. J. Cáceres, J. A. Carrillo, I. Gamba, A. Majorana, C. W. Shu. Dsmc versus weno-bte: A double gate mosfet example. Journal of Computational Electronics 5, 471-474, 2006. Journal · arXiv
  28. M. Cáceres, J. Carrillo, G. Toscani. Long-time behavior for a nonlinear fourth-order parabolic equation. Transactions of the American Mathematical Society 357 (3), 1161-1175, 2005. Journal · arXiv
  29. M. J. Cáceres, J. A. Carrillo, T. Goudon. Equilibration rate for the linear inhomogeneous relaxation-time Boltzmann equation for charged particles. Taylor & Francis Group 28 (5-6), 969-989, 2003. Journal · arXiv
  30. M. J. Cáceres, J. A. Carrillo, J. Dolbeault. Nonlinear Stability in Lp for a Confined System of Charged Particles. SIAM journal on mathematical analysis 34 (2), 478-494, 2002. Journal · arXiv
  31. M. J. Cáceres, J. A. Carrillo, P. Degond. The Child–Langmuir limit for semiconductors: a numerical validation. ESAIM: Mathematical Modelling and Numerical Analysis 36 (6), 1161-1176, 2002. Journal · arXiv
  32. F. Vecil, J. M. Mantas, M. J. Cáceres, C. Sampedro, A. Godoy, F. Gámiz. Implementation on a high-performance computing platform of a deterministic solver for Double-Gate MOSFETs. 19th European Conference on Mathematics for Industry, 325. Journal · arXiv

PARTICIPATION IN CONFERENCES

These are the most recent conferences i have participated in during the last 10 years:

  1. 9th European Congress of Mathematics. MS: Nonlinear and nonlocal PDE in biology and collective behaviour. Sevilla (España), Julio 2024.
  2. Alhambra PDE days. 9ECM Satellite Conference, Granada (España), Julio 2024.
  3. Topics on Neuroscience, Collective Migration and Parameter Estimation Workshop. Oxford (Reino Unido), Julio 2023.
  4. Kinetic Equations: Recent Developments and Novel Applications. Oaxaca (México), Octubre 2022.
  5. Frontiers in kinetic equations for plasmas and collective behavior. Isaac Newton Institute for Mathematical Sciencies, Cambridge (Reino Unido), Abril 2022.
  6. Multi-scale modeling for pattern formation in biological systems. Institut Mittag-Leffler (IML), Estocolmo (Suecia), Julio 2021.
  7. Conference of the Euro-Maghreb International Research Network in Mathematics and Applications. Madrid (España), Noviembre 2019.
  8. Differential Equations arising form Organising Principles in Biology. Mathematisches Forschungsinstitut, Oberwolfach (Alemania), Septiembre 2018.
  9. Minicurso en la escuela de verano: Summer School on “Methods and Models of Kinetic Theory” (M&MKT2018). Porto Ercole (Italia), Junio 2018.
  10. Cross diffusion and kinetic equations for biology. Viena (Austria), Mayo 2017.