Mariel Sáez Trumper

Associate Professor, Pontificia Universidad Católica de Chile

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> About

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, I also held a visiting professor positions in 2013 at Klaus Ecker's group, in 2017 in Monash University in Melbourne Australia (funded by the Bartnik fellowship), in 2018 in University College London (funded by the Leverhulme Visiting Professor program and International Exchanges program of the Royal Society) and I was a member of MSRI (currently SLMath) in 2008, 2016 and 2019.

I served as the director department between 2014 and 2017, I was in charge of the mathematics committee of becas Conicyt and I was part of the study group of Fondecyt grant.

I was the director of the interdisciplinary project Anillo de Matemáticas y Género and I participate of the Colectivo de mujeres matemáticas de Chile

• Curriculum Vitae (short)

• Curriculum Vitae (long)

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> Research

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

• On the existence and classification of k-Yamabe solitons, joint with M.F. Espinal. Preprint.
• The k-Yamabe Flow and its solitons. Joint with M.F. Espinal. Accepted for publication in Women in Mathematics Latin America}, Trends in Mathematics Series, Springer-Birkhäuser.
• Classification of Semigraphical Translators., joint with F. Mart\'{\i}n, R.Tsiamis and B. White. Preprint
• Uniqueness of semigraphical translators., joint with F. Mart\'{\i}n and R.Tsiamis. Preprint.
• The Hadamard formula for nonlocal eigenvalue problems, joint with R. Benguria and M. Pereira. Math. Ann. (2023). \url{https://doi.org/10.1007/s00208-023-02673-z}.
• Uniqueness of entire graphs evolving by Mean Curvature flow., Joint with P. Daskalopoulos. Journal f\"ur die reine und angewandte Mathematik (Crelle's Journal) (2023) \url{https://doi.org/10.1515/crelle-2022-0085.}
• Eigenvalue bounds for the Paneitz operator and its associated third-order boundary operator on locally conformally flat manifolds., Joint with M.d. M. Gonzalez. Ann. Sc. Norm. Super. Pisa Cl. Sci.(2022). \url{ https://doi.org/10.2422/2036-2145.202109_018}.
• Short time existence for the network flow, Joint with J. Lira, R. Mazzeo and A. Pluda. Comm. Pure Appl. Math., 76: 3968-4021. https://doi.org/10.1002/cpa.22111
• Fractional Laplacians and extension problems: the higher rank case, Joint with M.d. M. Gonzalez. Trans. Amer. Math. Soc. 370 (2018), no. 11, 8171--8213. .
• On the evolution by Fractional Mean Curvature, Joint with E. Valdinoci. Comm. Anal. Geom. 27(1), 211--249 (2019).
• 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 .

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> Seminars

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

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> Teaching

Current Courses:




Some previous Courses:

• Second Semester 2024: . MAT1630: Cálculo 3

• First Semester 2023: MAT2515 Análisis Real .

• First Semester 2021, 2022, 2023:Introducción al álgebra
• 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.
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• First Semester 2009: MPG31030 Ecuaciones en derivadas parciales (graduate level),
• First Semester 2008: MAT1600 Introducción al cálculo.





Other teaching:

• At Summer School `` Geometric Flows and Relativity'', Minicourse `` Mean Curvature Flow'', March 2024, Montevideo, Uruguay.
• At Workshop "Nonlocal and Local Coupled Equations", Minicourse on ``Nonlocal operators in geometry and nonlocal geometry'', February 2023, Montevideo, Uruguay.
• At TYAN Workshop in Mathematics 2022, Minicourse on ``Eigenvalues of operators in conformal geometry'' , November 2022, Sao Paulo, Brazil.
• At AMSI Winter School 2018, Minicourse on the ``Network Flow'', July 2018, University of Queensland, Australia.
• At LSGNT mini course on the Allen-Cahn equation and its connections with geometry,mJune 2018, University College of London, London, UK.
• At Mini course on non-local equations and conformal geometry, January 2019 University College of London, London, UK.
• At School of Differential Geometry, Minicourse '' Geometric flows ", July 2016.<\li>
• At Berlin Mathematical School (BMS), Partial Differential Equations II, Wintersemester 2013.