Latest News:
• A new preprint on the stability of Néel walls
in ferromagnetic thin films is available. This is a joint work
with L. Morales (IIMAS, UNAM), A. Capella (IM, UNAM) and C.
Melcher (RWTH, Aachen). The link to the preprint is here.
Current research topics
- Stability of shock profiles
I study the asymptotic stability properties of shock profiles, namely, traveling wave solutions to different regularizations of hyperbolic systems of conservation laws, such as viscous, relaxation or radiative profiles, among others.
- Multidimensional stability of planar phase boundaries
in continuum mechanics
I have applied hyperbolic stability techniques to study the dynamics and stability of planar phase boundaries in continuum mechanics, such as planar interfaces between phases of nonlinear elastic materials, or planar boundaries between simple fluids. The techniques I use are based on the Kreiss-Majda-Métivier theory of stability of classical fronts.
- Existence and stability of nonlinear waves
I study the general existence theory and stability properties of nonlinear waves, appearing in different circumstances. Among them you can find: traveling waves for coupled elasto-chemical models, solitons, periodic wavetrains, or diffusion fronts. - Mathematical Biology
I work on chemotaxis and transport phenomena applied to biosciences. My work covers modelling, asymptotic analysis and numerical simulations.
Earlier work
- Pattern formation (morphogenesis)
I have studied the effects of surface curvature and growth on pattern formation, by adapting the celebrated Turing mechanism of morphogenesis to time-dependent, curved domains.
Selected Talks
Here you can find a list of selected downloadable presentations:
- Dissipative structure of nonlinear viscous dispersive
systems. V Workshop on Nonlinear Dispersive Equations
(WNDE). Universidade Federal de Minas Gerais, Belo
Horizonte, Brazil. November 8-11, 2022. [PDF]
- Existence and spectral instability of bounded periodic
waves for viscous balance laws. III Summer School
(online), Universidade Estaduale de Maringá, Brasil. March
29, 2021. [PDF]
- Stabilization effects of chemotaxis on bacterial aggregation patterns. XII Americas Conference on Differential Equations and Nonlinear Analysis. CIMAT, Guanajuato, Mexico. December 9 - 13, 2019 [PDF].
- Orbital stability of standing waves for the nonlinear
Schrödinger equation with attractive delta potential and
double power repulsive nonlinearity. Plenary talk.
"Stability of Nonlinear Waves: Analysis and Applications: A
conference in honor of the 60th birthday of Kevin Zumbrun".
Institute Henri Poincaré, Paris, France. July 1 - 5, 2019 [PDF]
- Orbital stability of periodic traveling waves for
nonlinear Klein-Gordon equations in several space
dimensions. Nonlinear PDE Seminar. Institute of
Mathematics, UNAM. May 22, 2019 [PDF].
- Diffusive limits of stochastic velocity jump processes
for biological agents. Workshop: "Mathematical
Challenges in the Analysis of Continuum Models for Cancer
Growth, Evolution and Therapy (18w5115)". Casa Matemática
Oaxaca. Oaxaca, Mexico. November 25 - 30, 2018. [PDF]. The
video of this talk can be watched here.
- Spectral and nonlinear stability of traveling fronts
for a hyperbolic Allen-Cahn model with relaxation. The
Tenth IMACS International Conference on Nonlinear Evolution
Equations and Wave Phenomena: Computation and Theory.
Georgia Center for Continuing Education, University of
Georgia, Athens, Georgia. March 29 - April 1, 2017. [PDF]
- On the spectral, modulational and orbital stability of perioidc wavetrains for the sine-Gordon equation. Second Workshop on Nonlinear Dispersive Equations: Celebrating the 70th Anniversary of M. Scialom. IMECC, University of Campinas, Brazil. October 6 to 9, 2015. [PDF]
- Lecture series: "Spectral stability of periodic wavetrains". Institute of Mathematics and Statistics, University of Sao Paulo, Brazil. Auditotio Antonio Gilioli, Bloco A, October 13, 14 and 16, 2015.
- A chemotactic model of bacterial aggregation with nonlinear cross diffusion. Applied Mathematics Seminar, Department of Information Engineering, Computer Science and Mathematics, University of L'Aquila, Italy. November 27, 2013 [PDF]
- Modulational and spectral (in)stability of periodic traveling wave solutions to the nonlinear Klein-Gordon equation. Applied Mathematics Seminar, Department of Pure and Applied Mathematics, University of L'Aquila, Italy. November 15, 2012 [PDF]
- On the stability of degenerate shock profiles. XIV Conference on Hyperbolic Problems (HYP2012). University of Padova, Italy. June 29, 2012 [PDF]
- A reaction-diffusion-chemotactic biocontrol model: Asymptotics and numerical simulations. Applied Mathematics Seminar, Department of Mathematics, Texas A&M University. April 2, 2012 [PDF]
Funding
My research has been partially supported by the following sources:
Current:
Research grants:
- Principal investigator: Homogenization and nonlinear
dynamics of coherent structures in material sciences.
Co-PI: A. Capella. CONAHCyT, "Ciencia de Frontera 2023"
program, project no. CF-2023-G-122. August, 2023 - December,
2025. $97,000.00 USD (approx.)
- Principal investigator: Analysis and simulation of PDEs of evolution, DGAPA-UNAM, Program PAPIIT, Grant IN-104922. January 2022 - December 2024. $19,000.00 USD (approx.)
- Member of the National Researchers Network (SNI-CONAHCyT), Level III. January 2024 - December 2028.
Past:
Research grants:
- Principal investigator: Hyperbolic-parabolic systems of balance laws, DGAPA-UNAM, Program PAPIIT, Grant IN-100318. January 2018 - December 2020. $17,000.00 USD (approx.)
- Principal investigator: Analytical methods in the study of nonlinear stability of periodic wavetrains, DGAPA-UNAM, Program PAPIIT, Grant IN-104814. January 2014 - December 2016. $15,000.00 USD (approx.)
- Principal investigator: Analytical and numerical study of partial differential equations in the modeling of reaction and transport phenomena, characterized by the formation of complex structures. (Co-director: Dra. Chiara Simeoni.) CONACyT - Italian Ministry of Research (MIUR), MAE Program, Grant No. 146529. January 2011 - December 2013. $30,000.00 USD (approx.)
- Principal investigator: Non-standard semigroups and stability of traveling waves. DGAPA-UNAM, Program PAPIIT, Grant IN-109008. January 2008 - December 2009. $15,000.00 USD (approx.)
- Member of the National Researchers Network (SNI-CONACyT), Level II. January 2019 - December 2023.
- Member of the National Researchers Network (SNI-CONACyT), Level II. January 2015 - December 2018.
- Member of the National Researchers Network (SNI-CONACyT),
Level I. January 2011 - December 2014.
- Member of the National Researchers Network (SNI-CONACyT),
Level I. January 2008 - December 2010.
- Main researcher: Research grant for doctoral studies, Ministry of Education (SEP), México. Grant: 204-6-07106. January 2000 - December 2002. $10,800.00 USD
Research Visits
This is a list of recent academic departments I have visited:
- Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Universitá degli Studi dell'Aquila, Italia. September 11-15, 2023. Host: C. Lattanzio.
- Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Universitá degli Studi dell'Aquila, Italia. October 21 - December 21, 2017. Hosts: D. Donatelli, C. Lattanzio.
- Dipartimento di Matematica 'Guido Castelnuovo', University
of Rome, La Sapienza, Rome, Italy. July 17 - 30, 2017. Host:
C. Mascia.
- Department of Mathematics and Statistics. University of Sao Paulo, Brazil. October 1 - 20, 2015. Host: J. Angulo Pava.
- Department of Mathematical Sciences. University of Bath, UK. September 13 - 20, 2014. Host: A. Kyprianou.
- Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Universitá degli Studi dell'Aquila, Italia. November 5 - 30, 2013. Host: C. Lattanzio.
- Dipartimento di Matematica Pura ed Applicata, Universitá degli Studi dell'Aquila, Italia. September 20 - December 20, 2012. Host: C. Lattanzio.
- Department of Mathematics, Texas A&M University. April 2-7, 2012. Host: P. Howard.
- Dipartimento di Matematica Pura ed Applicata, Universitá degli Studi dell'Aquila, Italia. October 31 - November 5, 2011. Host: C. Simeoni.
- Departamento de Matemática Aplicada, Universidad Complutense de Madrid, Spain. April 10-24, 2010. Hosts: G. Oleaga, J.J.L. Velázquez.
- Mathematics Institute, University of Leipzig, Germany. November 3-15, 2009. Hosts: W. König, S. Luckhaus.
- Department of Mathematics, Indiana University, Bloomington, U.S.A. April 5-18, 2009. Host: K. Zumbrun.
- Department of Mathematics, Indiana University, Bloomington, U.S.A. April 23 - May 4, 2008. Host: K. Zumbrun.
- Department of Mathematics, Indiana University, Bloomington, U.S.A. April 9-18, 2007. Host: K. Zumbrun.
Mentorship
Post-doctoral associates:
- Luis Fernando López Ríos (Ph.D.: U. de Chile/U. de Aix-Marseille), 2017 - 2019.
- César A. Hernández Melo (Ph.D.: U. de Sao Paulo), 2018 - 2019.
- Delyan Zhelyazov (Ph.D.: Gran Sasso Science Institute, Italy), 2021-2023.
- Lauro Morales Montesinos (Ph.D.: UNAM), 2022-2025.
- Felipe Angeles García (Ph.D.: UNAM), 2023.
Theses under supervision:
Ph.D.
- José Manuel Valdovinos
- Anna Naumkina
M.Sc.
- Abraham Quiles
Past students
- (2023) Luis Ibáñez (M.Sc. in Mathematics, UNAM)
Currently a Ph.D. student at GSSI, Italy. - (2023) Fabio Vallejo (Ph.D. in Mathematics, UNAM)
- (2021) Enrique Álvarez (Ph.D. in Mathematics, UNAM).
Currently a Post-doctoral Associate at IIMAS, UNAM. - (2020) José Manuel Valdovinos (M.Sc. in Mathematics, UNAM)
Currently a Ph.D. candidate in Mathematics at UNAM. - (2020) Luis Alejandro Rosas (M.Sc. in Mathematics, UNAM)
Currently a Ph.D. candidate at INRIA, Paris (France). - (2020) Isai Padilla (M.Sc. in Mathematics, UNAM)
- (2018) Eduardo Parra (B.Sc. in Mathematics, UNAM).
- (2010, 2017) Juan Francisco Leyva (M.Sc. and Ph. D. in
Mathematics).
Currently at the Benemérita Universidad Autónoma de Puebla. - (2017) Edgar Ávalos (B.Sc. in Mathematics, UNAM).
Currently a M.B.A. student at the Centro de Investigación y Docencia Económicas (CIDE). - (2016) Alejandro Butanda (M.Sc. in Mathematics, UNAM)
Currently a Ph. D. candidate at CIMAT. - (2015) Karina Islas (B.Sc. in Mathematics, UNAM)
Ph.D. graduate from the University of Monash (Australia). Currently at InterVenn BioSciences.
- (2014, 2016, 2021) Felipe Angeles (B. Sc. in Physics; M.
Sc. and Ph. D. in Mathematics, UNAM).
Currently a Post-doctoral Associate at Institute of Mathematics, UNAM. - (2012) Ignacio Pérez (M.Sc. in Mathematics, UNAM).
- (2010) Mónica Romero (B.Sc. in Mathematics, UNAM).
Ph.D. graduate from UCLA, California. Currently at Genentech.
Collaborators
- Kevin Zumbrun (Indiana)
- Heinrich Freistühler (Konstanz)
- Gilberto Flores (IIMAS-UNAM)
- Toan Nguyen (Brown)
- Corrado Lattanzio (L'Aquila)
- Chiara Simeoni (U. Cote D'Azur)
- Corrado Mascia (Roma)
- Pablo Padilla (IIMAS-UNAM)
- Rafael Barrio (IFUNAM)
- Philip K. Maini (Oxford)
- Faustino Sánchez (Ciencias, UNAM)
- Carlos Málaga (Ciencias, UNAM)
- Peter D. Miller (Michigan)
- Chris Jones (North Carolina)
- Robert Marangell (Sydney)
- Jaime Angulo Pava (Sao Paulo)
- Víctor Breña (ITAM)
- Pablo Castañeda (ITAM)
- César Hernández-Melo (Maringá)
- Luis F. López Ríos (IIMAS-UNAM)
- Donato Pera (L'Aquila)
- Raffaele Folino (IIMAS-UNAM)
- Marta Strani (U. Ca' Foscari Venezia Mestre)
- J. Francisco Leyva (BUAP)
- C. Melcher (RWTH, Aachen)