Research

Ph. D in Mathematics, Departamento de Análisis Matemático. Universidad Complutense de Madrid. 2011.

Thesis title: Some topics in lipschitz analyisis on metric spaces.

Thesis Advisor: Prof. Jesús A. Jaramillo Aguado (Universidad Complutense de Madrid).

My main research interests are analysis on metric spaces, geometric analysis, non linear potential theory and functional analysis.

PUBLICATIONS

The following papers are earlier versions of the final articles and therefore may differ from the journal article.

• Metric differentiability of Lipschitz maps defined on Wiener spaces with L. Ambrosio. Rend. Circ. Mat. Palermo (2) 58 (2009) 1-10.
• Pointwise Lipschitz functions on metric spaces  with J. A. Jaramillo. J. Math. Anal. Appl. 363 (2010) 525-548.
• Some stability results under domain variation for Neumann problems in metric spaces with A. Lemenant. Ann. Ac. Sci. Fen. Math. Vol. 35, (2010), 1-27.
• Lipschitz-type Sobolev spaces in metric measure spaces. J. Analysis. Volume 18 (2010), 129-144. Proceedings of the ICM2010. Satellite Conference International Workshop on Harmonic and Quasiconformal Mappings.
• Rectifiable curves in Sierpinski carpets, with J. Tyson. Indiana Univ. Math. J. Vol 60 (2011), 285-310. See also Erratum. 
• The $\infty$-Poincaré inequality in metric measure spaces with J. A. Jaramillo and N. Shanmugalingam. Michigan Math. J. Vol. 61, (2012), 63-85.
• p-Poincaré inequality vs. $\infty$-Poincaré inequality: some counterexamples with N. Shanmugalingam and A. Williams. Math. Z. Vol 271, Issue 1 (2012), 447-467.
• The effect of projections on dimension in the Heisenberg group with Z.M. Balogh, K. Fässler, P. Mattila, J. Tyson. Rev. Mat. Iberoamericana 29 (2013), no.2, 381-432.
• First order Poincaré inequalities in metric measure spaces with J. A. Jaramillo and N. Shanmugalingam. Ann. Ac. Sci. Fen. Math.Vol 38 (2013) 1-22.
• An elementary proof of Cheeger’s theorem on reflexivity of Newton-Sobolev spaces of functions in metric measure spaces, with N. Shanmugalingam. J. Analysis 21 (2013), 73-83. Proceedings of the International Symposium on Complex Analysis and Conformal Geometry.
•   On Whitney-type characterization of approximate differentiability on metric measure spaces with L. Ihnatsyeva, R. Korte and M. Szumanska.  Canadian Journal of Mathematics 66 (2014), no. 4, 721-742.
• Rectifiability of Self-contracted curves in the euclidean space and applications with A. Daniilidis, G. David and A. Lemenant. Journal of Geometric Analysis 25 (2015), 1211-1239.
• Preservation of p-Poincaré inequality for large p under sphericalization and flattening, with Xining Li. Illinois Journal of Mathematics 59 (4) (2015), 1043-1069.
• Geometric characterizations of p-Poincaré inequalities with J. A. Jaramillo and N. Shanmugalingam. Publicacions Matemàtiques 60 (2016), 81-111.
• Preservation of bounded geometry under sphericalization and flattening: quasiconvexity and $\infty$-Poincaré inequality, with Xining Li. Available at CVGMT preprint server. Ann. Ac. Sci. Fen. Math 42 (2017), 1-22.
• Self contracted curves in Riemannian manifolds, with A. Daniilidis, R. Deville and L. Rifford.  J. Math. Anal. Appl. 457 (2018), 1333-1352.

PREPRINTS

• Sierpinski type fractals are differentiably trivial, with J. Gong, and J.A. Jaramillo. Preprint 2018 in CVGMT preprint server of Scuola Normale Superiore, Pisa, Italy.
• Metric and geometric relaxations of self-contracted curves, with  A. Daniilidis, and   R. Deville. Preprint 2018 in CVGMT preprint server of Scuola Normale Superiore, Pisa, Italy.
• Self-contracted curves are gradient flows of convex functions, with A. Lemenant. Preprint 2018 in CVGMT preprint server of Scuola Normale Superiore, Pisa, Italy.
• Existence and uniqueness of $\infty$-harmonic functions under assumption of $\infty$-Poincaré inequality, with J. A. Jaramillo and N. Shanmugalingam. Preprint 2016 in CVGMT preprint server of Scuola Normale Superiore, Pisa, Italy.
• Stepanov’s theorem in Wiener spaces with L. Ambrosio. Preprint 2010 in CVGMT preprint server of Scuola Normale Superiore, Pisa, Italy.

RESEARCH STAYS

1. Scuola Normale Superiore di Pisa. Pisa, Italy. September 2008-December 2009. Supervisor: Prof. Luigi Ambrosio.
2. Mathematisches Institut, Universität Bern. Bern, Switzerland. September 2009-December 2009. Supervisor: Prof. Zoltán Balogh.
3. Department of Mathematical Sciences, University of Cincinnati. Cincinnati, USA. June 2010-July 2010. Supervisor: Prof. Nageswari Shanmugalingam.
4. Department of Mathematics, School of Science and Technology. Aalto University. Helsinki, Finland. April 2011.
5. Instituto de Investigación de la Universidad de Sevilla (IMUS). Departamento de Análisis Matemático, Universidad de Sevilla. Sevilla. May-June 2011
6. Laboratoire Jacques Louis Lions. Paris, France. March 2012-June 2012.
7. Institute for Pure and Applied Mathematics, UCLA. Los Angeles, USA. March 2013-May 2013. Program Interactions between Analysis and Geometry.
8. Department of Mathematical Sciences, University of Cincinnati. Cincinnati, USA. May 2014.
9. Laboratoire Imath, Université de Toulon. France. July 2014.
10. Departamento de Ingeniería Matemática de la Universidad de Chile. Santiago de Chile, Chile. March 2015.
11. IMPAM, Warsaw. Poland. October 2015
12. Department of Mathematical Sciences, University of Cincinnati. Cincinnati, USA. April 2016.
13. Departamento de Ingeniería Matemática de la Universidad de Chile. Santiago de Chile, Chile. June 2016.
14. Laboratoire Jacques Louis Lions. Paris, France. September 2017-November 2017.

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