Associate Professor, Pontificia Universidad Católica de Chile

P. Universidad Catolica de Chile

Facultad de Matemáticas

Avda. Vicuña Mackenna 4860

Macul, Santiago, 6904441,Chile

+56-2-23544523

mariel at mat (dot) puc (dot) cl

I am currently an Associate Professor at the mathematics department of Pontificia Universidad Católica de Chile.

I received my Ph.D. in 2005 from Stanford University, under the supervision of Rafe Mazzeo. Between 2005 and 2008, I held a postdoctoral position at Max Plack instute for Gravitational Physics (Albert Eintein's institute) in Gerhardt Huisken's group. I also held a visiting professor position in 2013 at Klaus Ecker's group and I was a member of MSRI in 2008 and 2016.

Since 2014,
I am also serving as the director of the department and, until recently, I was in charge of the mathematics committee of becas Conicyt. Currently, I am part of the study group of Fondecyt grant and I participate of the

My primary field of interest is geometric analysis. I am currently interested in geometric flows, minimal surfaces and fractional equations arising from geometry.

- Fractional Laplacians and extension problems: the higher rank case, Joint with M.d. M. Gonzalez. To appear in Trans. Amer. Math.
- On the evolution by Fractional Mean Curvature, Joint with E. Valdinoci. To appear in Comm.Anal.Geom
- Some constructions for the fractional Laplacian on noncompact manifolds, Joint with M.M. Gonzalez and V. Banica, Rev. Mat. Iberoam. 31 (2015), no. 2, 681--712
- Mean curvature flow without singularities, Joint with Oliver Schnürer, J. Differential Geom. 97 (2014), no.~3, 545--570
- Layer solutions for the fractional Laplacian on hyperbolic space: existence, uniqueness and qualitative properties, Joint with M.M. Gonzalez and Y. Sire, Annali di Matematica Pura ed Applicata (1923 -) December 2014, Vo$
- Multiple-layer solutions to the Allen-Cahn equation on hyperbolic space , Joint with R. Mazzeo, Proc. Amer. Math. Soc. 142 (2014), no. 8, 2859--2869
- Semi-local inversion of the geodesic ray transform in the hyperbolic plane., Joint with M. Courdurier, Inverse Problems, Volume 29, Issue 6, 2013. doi:10.1088/0266-5611/29/6/065010
- Uniqueness of self-similar solutions to the network flow in a given topological class Comm. Partial Differential Equations, Volume 36, Issue 2, 185-204, 2011.
- Evolution of convex lens-shaped networks under curve shortening flow with Schnürer, Oliver C., Azouani Abderrahim, Georgi Marc, Hell Juliette, Jangle Nihar, Koeller Amos, Marxen Tobias, Ritthaler Sandra, Sáez Mariel, Schulze Felix, Smith Brian). Trans. Amer. Math. Soc. 363 (2011), 2265--2294.
- Relaxation of the flow of triods by curve shortening flow via the vector-valued parabolic Allen-Cahn equation. J. Reine Angew. Math. (Crelle's). Volume 2009, number 634, 143--168.
- Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential. Indiana Univ. Math. J. Volume 58, Number 1, 213--267, 2009.
- Self-similar expanding solutions for the planar network flow. with R. Mazzeo, S\'eminaires \& Congr\`es 19, 2008, p. 159--173.
- Relaxation of the curve shortening flow via the parabolic Ginzburg-Landau equation . Calc. of Var. Volume 31, Number 3, 359-386, March, 2008.
- Optimal Regularity for the Pseudo Infinity Laplacian. Joint with J. Rossi, ESAIM - Control, Optimisation and Calculus of Variations 13(2), 294--304, 2007.
- Asymptotic description in a fast-diffusion equation with absortion, with M. Del Pino, Differential Integral Equations 15 (2002), no. 8, 1009-1023.
- On the extiction profile for solutions of $u_t = \Delta u^{N-2\over N+2},$ joint with with M. Del Pino, Indiana University Mathematical Journal 50 (2001) no1, 611-628 .
- A min-max construction of embedded hypersurfaces with prescribed constant mean curvature in Riemannian manifolds. Joint with M. Li.
*Second Semester 2016: MPG3970: Geometria Diferencial.*- First Semester 2015: MAT1206 Introducción al Algebra.
- Second Semester 2014: MAT2515\MAT251I Análisis Real.
- First Semester 2013: MAT1206 Introducción al Algebra.
- First Semester 2012: MPG3950 Variable Compleja (graduate level).
- First Semester 2012: MAT1630 Cálculo III.
- First Semester 2011: MPG 3100 Análisis Real (graduate level)
- First Semester 2010: MAT1620 Cálculo II. .
- First Semester 2009: MPG31030 Ecuaciones en derivadas parciales (graduate level),
- First Semester 2008: MAT1600 Introducción al cálculo.
*At School of Differential Geometry, Minicourse '' Geometric flows ", July 2016.<\li>**At Berlin Mathematical School (BMS), Partial Differential Equations II, Wintersemester 2013.*

**In preparation: **

I currently participate of the Analysis and Geometry Seminar and the Colloquium at PUC .

Current Courses:

Some previous Courses:

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Other teaching:

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