My name is Tania Pernas Castaño, I am a postdoctoral researcher at the Hausdorff Center in Bonn, Germany.
My research experience began during my Bachelor in Mathematics focusing in Differential and Difference Equations, due to the collaboration grant that I received in 2010. After that I continued with my research throughout my Master. In particular, I studied Impulsive Dynamics Equations on time scales which resulted in two scientific articles.
Once I finished my studies with an excellency in 2012, I obtained an FPI scholarship that allowed me to begin my PhD studies. My research line changed and I started to center in Partial Differential Equations arising from Fluid Mechanics. I have been focused in the local existence of solutions and in the study of finite-time singularities for free boundary problems. During my first two years with de FPI grant I published one paper with title "Non-splat singularities for the one-phase Muskat problem" in the reputable journal Transactions of the American Mathematical Society (IF:1,426; 24/311; Q1,Database WoS). In 2015, I obtained a movility grant offered by the Ministry which gave me the opportunity to do a research stay in Princeton University under the supervision of Charles Fefferman (Fields Medal). During the stay I moved forward in to my research studing a more physically meanfull problems, which introduce several technical complications. The bulk of these results have appeared in a single-author paper in Nonlinearity (IF:1,767; 33/255; Q1, Database WoS).
In the last year of my PhD I continued my research line obtaining two new results about finite-time singularities for the more general case of the problem. Both results appeared in the top journal Journal of Nonlinear Science (IF: 1,904; 28/255; Q1, Database WoS). The above mentioned articles conformed my thesis with the title "On the splat singularity for the Muskat problem". I defended it on 27 November 2017 obtaining Suma Cum Laude and the international certificate, in a program with quality mention.
In addition to the research activity, I have been attending other courses, congresses, seminars and workshops. I have also given talks in several national and international congresses. Furthermore, I organized the "Seminario Junior UAM-ICMAT" for three consecutive years. Moreover, I was teaching in different degrees: Mathematics, Enviromental science, Computer engineering, etc. I have also been involved in numerous outreach activities. I organized and designed different activities for the science week and participated in programs with highschool students in order to show them the connections between mathematics and real life. I am a referee for the journal Boundary Value Problems and member of the American Mathematical Society.
2010–2011 Beca de Colaboración, Diferential and difference equations, Instituto de Matemáticas, Universidad de Santiago de Compostela. Santiago
2010–2011 Monitor de clases prácticas, Universidad de Santiago de Compostela. Santiago
2012–2016 FPI Ministerio Economía, Industria y Competitividad, Predoc Fellowship, Universidad de Santiago de Compostela. Santiago
2016–present Profesora Ayudante, Teaching Assistant, Universidad Autónoma de Madrid. Madrid
September 2018 Postdoctoral position, Mentor: Juan J.L. Velázquez, Hausdorff Center. Bonn
2012 Positive solution of singular BVPs for a system of dynamic equations on time scales, Advances in difference equations, 2012:185, Work with: Ariadna Lago, Victoria Otero-Espinar, Available at: www.advancesindifferenceequations.com/content/2012/1/185.
2013 Variational approach to second-order impulsive dynamic equations on time scales, Boundary Value Problems, 2013:119, Work with: Victoria Otero-Espinar, Available at: www.boundaryvalueproblems.com/content/2013/1/119.
2017 Non-splat singularity for the one-phase Muskat problem, Transactions of the AMS, 369(1):711–754, Work with: Diego Córdoba, Available at: http://www.ams.org/journals/tran/2017-369-01/S0002-9947-2016-06688-X/ S0002-9947-2016-6688-X.pdf.
2017 Local-existence for the Inhomogeneous Muskat problem, Nonlinearity, 30(5):2063, Available at: http://iopscience.iop.org/article/10.1088/1361-6544/aa6691/meta.
2017 On the splash and splat singularities for the One-phase Inhomogeneous Muskat Problem, Journal of Nonlinear Science, pages 1–50, Work with: Diego Córdoba, http://rdcu.be/xHKd.
2019 Analysis of a thin film approximation for two-fluid Taylor-Couette Flows. Journal of Differential Equations, Volume 269, Issue 1, pages 377-419, 2020. Work with: Juan J.L. Velázquez, https://arxiv.org/pdf/1905.13606.pdf
01/03/15 to 29/05/15 Princeton University, Mentor: Charles Fefferman.
2006–2011 Bachelor in Mathematics (BSc), Universidad de Santiago de Compostela, Santiago de Compostela, Degree: 8.5581.
2011–2012 Master in Mathematics (MSc), Universidad de Santiago de Compostela, Degree: 9.3250, Thesis: Variational Methods on impulsive Dynamics equations, Degree: 10. Advisor: MaVictoria Otero Espinar
2012–2017 PhD on Partial Differential Equation arising from Fluids Dynamics, Instituto de Ciencias Matemáticas (ICMAT)-CSIC, UAM, UC3M, UCM, Departament of mathematics- Universidad Autónoma de Madrid, Degree: Suma Cum Laude, Thesis: On the splat singularity for the Muskat problem. Advisor: Diego Córdoba Gazolaz
200 hours Modeling in fluids and structures, Master of Mathematics and Applications, Universidad Autónoma de Madrid. Imparted by: Juan Luis Vázquez.
10 hours Harmonic Analysis, Working Group, Instituto de Ciencias Matemáticas (ICMAT).
12 hours Calderón-Zygmund Theory and beyond: Fourier Analysis, Uncertainty and Modelling, PhD Courses, Universidad Autónoma de Madrid. Imparted by: Antonio Córdoba.
131 hours Working Pizza Seminar on PDE’s and Fluids Mechanics, Working Group, Instituto de Ciencias Matemáticas (ICMAT).
3/06/13 to 7/06/13 “Journées EDP 2013” Congress, Biarritz. Organizers: DMA, Ecole Normale Supérieure y CNRS UMR 8553.
17/06/13 to 21/06/13 Fluids Mechanics and Singular integrals, Sevilla. Organizers: Instituto de matemáticas de la Universidad de Sevilla (IMUS) y Instituto de Ciencias Matemáticas (ICMAT).
4/11/13 to 8/11/13 Mathematics and Geosciences: Global and Local Perspectives, Madrid. Organizado por Instituto de Ciencias Matemáticas (ICMAT) CSIC, UAM, UC3M, UCM.
16/12/13 to 19/12/13 deLeónfest Congress, Madrid. Organizers: Instituto de Ciencias Matemáticas (ICMAT) CSIC, UAM, UC3M, UCM.
3/06/14 to 5/06/14 Three days on analysis and PDEs, Madrid. Organizers: Instituto de Ciencias Matemáticas (ICMAT) CSIC, UAM, UC3M, UCM.
19/09/16 to 21/09/16 Spanish-French Workshop on Analysis of PDE’s from Fluid Mechanics, Madrid. Organizers: Instituto de Ciencias Matemáticas (ICMAT) CSIC, UAM, UC3M, UCM.
13/03/17 to 14/03/17 ERC and Mathematics in Spain, Madrid. Organizers: Instituto de Ciencias Matemáticas (ICMAT) CSIC, UAM, UC3M, UCM.
30/05/12 Variational Approach to Impulsive Dynamic Equations on Time Scale, Workshop on Differential Equations and Dynamical Systems, Santiago de Compostela. Organizers: Departament of Analysis of USC and Instituto de Matemáticas de la Universidad de Santiago de Compostela.
13/03/13 Las ecuaciones de Navier-Stokes, Seminario de Iniciación a la Investigación (SII), Santiago de Compostela. Organizers: Instituto de Matemáticas de la Universidad de Santiago de Compostela.
09/04/14 The Muskat problem, Seminario Junior, Madrid. Organizers: Departament of mathematics, Universidad Autónoma de Madrid; Instituto de Ciencias Matemáticas (ICMAT).
9/10/14 The Muskat problem, Seminario de Doctorandos de la UCM, Madrid. Organizers: Doctorandos Matemáticas UCM; Universidad Complutense Madrid (UCM).
12/11/14 Non-splat singularity for the one-phase Muskat problem, Working Pizza Seminar on PDE’s and Fluid Mechanics, Madrid. Organizers: Instituto de Ciencias Matemáticas (ICMAT) CSIC, UAM, UC3M, UCM.
17/11/14 to 21/11/14 Non-splat singularity for the one-phase Muskat problem, ICMAT- China Exploratory Workshop, Madrid. Organizers: Instituto de Ciencias Matemáticas (ICMAT) CSIC, UAM, UC3M, UCM.
19/07/15 to 24/07/15 Poster: Non-splat singularity for the one-phase Muskat problem, Summer School on Transport,Fluids and Mixing, Levico Terme (Trento). Organizers: Centro Internazionale per la Ricerca Matematica (C.I.R.M.).
07/09/15 to 11/09/15 Non-splat singularity for the one-phase Muskat problem, III Congreso de Jóvenes Investigadores RSME, Murcia. Organizers: Real Sociedad Matemática Española (RSME); Universidad de Murcia.
18/07/16 to 20/07/16 Finite-time singularities for the Muskat problem, Summer school on Elliptic PDE’s at Tenerife, La Laguna. Organizers: Universidad de la Laguna; Departamento de Análisis Matemático.
04/09/17 to 08/09/17 The inhomogeneous Muskat problem, IV Congreso de Jóvenes Investigadores RSME, Valencia. Organizers: Real Sociedad Matemática Española (RSME); Universitat de València.
24/10/17 O problema de Muskat non-homogéneo, Seminario de Ecuacións Diferenciáis e Análise Funcional, Santiago de Compostela. Organizers: Universidad de Santiago de Compostela; Instituto de Matemáticas
05/12/17 to 08/12/17 On the splat singularity for the Muskat problem, Oberseminar, Bonn. Organizers: Department of Functional Analysis; Institut für Angewandte Mathematik.
7/07/14 a 11/07/14 Voluntary:The 10th AIMS Conference on Dynamical Systems,Differential Equations and Applications, Madrid. Organizers: The American Institute of Mathematical Sciences, Instituto de Ciencias Matemáticas (ICMAT), Universidad Autónoma de Madrid (UAM) y University of North Carolina at Wilmington
Year 2014/2015 Seminario Junior UAM-ICMAT, Madrid.
Year 2015/2016 Seminario Junior UAM-ICMAT, Madrid.
Year 2016/2017 Seminario Junior UAM-ICMAT, Madrid.
Year 2010/2011 2º Mathematics Degree, Assistant, Topology of Euclidean Spaces. Universidad de Santiago de Compostela (USC)
July–13 VI Edition of the Escuela JAE de matemáticas 2013, Course, Fluids Mechanics, Instituto de Ciencias Matemáticas (ICMAT). CSIC, UAM, UC3M, UCM.
February–14 Máster en Matemáticas P1141105, Course, Temas actuais na Investigación Matemática, The Muskat problem. Universidad de Santiago de Compostela (USC).
Year 2015/2016 2º Joint Degree in Computer Engineering and Mathematics, Lineal Algebra and Geometry. Universidad Autónoma de Madrid (UAM).
Year 2016/2017 2º Joint Degree in Computer Engineering and Mathematics, Lineal Algebra and Geometry. Universidad Autónoma de Madrid (UAM).
Year 2016/2017 1º Environmental Science Degree, Statistics. Universidad Autónoma de Madrid (UAM).
Year 2017/2018 2º Joint Degree in Computer Engineering and Mathematics, Analysis. Universidad Autónoma de Madrid (UAM).
Year 2017/2018 2º Mathematics Degree, Analysis. Universidad Autónoma de Madrid (UAM).
2013 Tutor, Program "4o ESO + Empresa", Instituto de Ciencias Matemáticas(ICMAT), CSIC, UAM, UC3M, UCM.
2014 Moderator at a roundtable: “Olimpiada Matemática”, Program "4o ESO + Empresa", Instituto de Ciencias Matemáticas(ICMAT), CSIC, UAM, UC3M, UCM.
2014 Speaker at a roundtable: “Jóvenes Matemáticos”, Science and technology week, Instituto de Ciencias Matemáticas(ICMAT), CSIC, UAM, UC3M, UCM.
2016 Member of a Jury, "Ciencia en Acción 17", Modality: Math laboratory.
2016 Design and Organization, Mathematical Conversations, “The night of the (very young) researchers at the UAM”, VII edition of the European Night of researchers from Madrid.
2016 Design and Organization, “Math ICMAT Gymkhana”, Science and technology week, Instituto de Ciencias Matemáticas(ICMAT), CSIC, UAM, UC3M, UCM. Ayuntamiento de Alcobendas
2017 Design and Organization, “Math ICMAT Gymkhana”, Science and technology week, Instituto de Ciencias Matemáticas(ICMAT), CSIC, UAM, UC3M, UCM. Ayuntamiento de Alcobendas
Referee, Boundary Value Problems, SpringerOpen Journal.
Member, American Mathematical Society, (AMS).