ACADEMIA COLOMBIANA DE CIENCIAS EXACTAS, FÍSICAS Y NATURALES

Miembro del Colegio Máximo de las Academias de Colombia

**Profesión: Matemática**

**Área (s) de especialización: Análisis**

**Categoría en la Academia Colombiana de Ciencias Exactas, Físicas y Naturales**

Miembro Correspondiente

Fecha de posesión, Miembro Correspondiente: 23 agosto 2017

** **

**Datos personales**

Fecha de nacimiento: 05/07/1964

Lugar de nacimiento

**Ciudad**: Bogota

**Departamento/Estado/Provincia**: Cundinamarca

**País**: Colombia

**Dirección institucional: Department of Mathematics,Universityof Washington, Box 354350, Seattle, WA 98195-4350**

Teléfono: 1-206-543-1173

**Dirección electrónica (e-mail): toro@uw.edu**

** **

**Hoja de vida (Curriculum Vitae en línea): https://sites.math.washington.edu/~toro/web-cv.pdf**

** **

**Página (sitio) web**: http://sites.math.washington.edu/~toro/

** **

**Títulos académicos:**

BS Matematicas,Universidad Nacional de Colombia, 1986

MS Mathematics, Stanford University,1989

PhD Mathematics, Stanford University 1992

**Áreas de investigación: Ecuaciones Diferenciales Parciales, Teoria Geometrica de la Medida**

** **

**Reseña Biográfica (Extensa)**

I was born in Bogotá, Colombia, in 1964, to a family to whom education was a very high priorty. Both of my parents were first generation college graduates. At a Young age I discovered my love for Mathematics. In 1981, at the International Mathematics Olympiad (IMO), held in Washington DC, representing Colombia, I realized that being a mathematician could be a profession. I attended a French school in Colombia, where I was scheduled to take the Baccalaureat exam the following year. I received the best posible mention in my Baccalaureat exam and was accepted at Lycée Louis Le Grand in Paris for the Fall of 1982. For personal reasons I had to return to Colombia at the end of that academic year. I entered the Universidad Nacional de Colombia in the Fall of 1983. Unfortunately, eight months after I started my studies, the university was closed after a series of public demonstrations. I did most of my undergraduate work on my own. After nine months of closure I was allowed to return to the university and take exams to show that I had learned the material. I graduated in 1986 after only four semesters of in class instruction. In 1987, I was awarded a Graduate Study Fellowship from the Universidad Nacional de Colombia (1987-1989) which was given the to student graduating with the highest GPA in the university.

In the Fall of 1987, I started graduate school at Stanford University. I graduated in the Spring of 1992. In the Fall of 1992, I became a member at the Institute for Advanced Study in Princeton, one of the most prestigious research institutions in the country (1992-1993). Subsequently I became a C. B. Morrey Jr. Assistant Professor at University of California: Berkeley (1993-1994). In the Fall of 1994, my husband and I became Assistant Professors in the Mathematics department at the University of Chicago (1994-1996). At that time we also accepted positions at the University of Washington to start in the Fall of 1996. I joined the faculty at the University of Washington as an Assistant Professor (1996-1998). I was promoted to the rank of Associate Professor with tenure in the Fall of 1998 (1998-2002). By invitation I spent the academic year 2001-2002 as Visiting Associate Professor and Visiting Scholar at Harvard University and a Fellow at the Radcli_e Institute for Advanced Study. I was promoted to the rank of Full Professor in the Fall of 2002. I was an Invited Professor at the University College London during the Fall of 2005. I held the Robert R. & Elaine F. Phelps Professor in Mathematics from 2012-2016. In the Fall of 2016 I became the Craig McKibben & Sarah Merner Professor in Mathematics.

My research lies at the interface of Geometric Measure Theory (GMT), Harmonic Analysis and Partial Differential Equations (PDE). The cross-pollenization between these three areas has been one of the pillars of my research. Over the past fifteen years my research has primarily focused on two programs. The first one was initiated in collaboration with C. Kenig. Our goal was to show that weak notions of regularity were well adapted to the study of boundary behavior of solutions of elliptic PDEs and to free boundary regularity problems. Over the past 5 years this program has been continually to be developed by E. Milakis, J. Pipher and myself, as well as several other teams of mathematicians.

The success of this program has significantly expanded the collection of domains where the boundary regularity of solutions of elliptic divergence form operators can be established. It has also enlarged the class of free boundary regularity problems which are well understood. In particular, in joint work with Kenig and Preiss, we provide a complete characterization of the boundary of a domain in Euclidean space in terms of the behavior of the exterior and the interior harmonic measures. My former student Matthew Badger (currently an Assistant Professor at the University of Conneticut) has re_ned these ideas and has found novel applications of Preiss’s work, providing another indication the synergy between GMT and free boundary regularity problems yields interesting new mathematics. My work, as well as that of my collaborators and graduate students, has enhanced the idea that weak notions of regularity are suitable to study problems that thus far had only been considered in terms of classical notions of regularity.

The second program concerns non-smooth geometry and addresses several classical questions in Geometric Measure Theory. One of the main themes of this research concerns the relationship between the doubling properties of a measure and the regularity of its support. This family of questions highlights the synergy between Harmonic Analysis and GMT. The other main theme, whose origin goes back to Reifenberg and his _rst attempt to understand the regularity of solutions to the Plateau problem in higher dimensions, concerns the existence of good parameterizations for sets which are well approximated by nice sets, that is planes, cones or Lipschitz graphs.

Since 1991 I have been partially funded by both NSF and private foundations. During the last year of graduate school I received an Alfred P. Sloan Doctoral Dissertation Fellowship (1991-1992). In 1994 I was awarded a National Science Foundation Mathematical Sciences Postdoctoral Research Fellowship (1994-1998). In 1996 I was awarded an Alfred P. Sloan Research Fellowship (1996-2000). I held a Simons Foundation Fellowship during the academic year 2012-2013. I received a Guggenheim Foundation Fellowship in 2015. I became a Fellow of the AMS in 2016. My research has been continuously supported since 1994 by grants from the National Science Foundation. From 2009 to 2014 I was a co-PI in an NSF Research Training Grant in Inverse Problems and Partial Di_erential Equations at the University of Washington. This grant supported four postdoctoral fellows, twenty three graduate student and many undergraduates. Thirteen graduate students who finished their PhD during this period of time are currently employed in either academia or industry.

At least half of undergraduate students supported by the grant are currently enrolled in graduate school in the mathematical sciences.

I have also been professionally recognized by distinguished invitations. In 1999 I gave a one hour invited address at the American Mathematical Society (AMS) meeting at the University of Texas, Austin. In 2007 I was an invited speaker to the Lars Ahlfors Centennial Celebration in Helsinki, Finland. In January 2011 I delivered a one hour invited address at the Joint Mathematics Meeting in New Orleans. This meeting brought together the Mathematics associations of the US: AMS, MAA, AWM, SIAM and NAM. I was an invited speaker at the Analysis session of the International Congress of Mathematicians (ICM 2010) in Hyderabad India. I became a Corresponding Member of the Academia Colombiana de Ciencias Exactas, Fsicas y Naturales in 2017.

My professional service includes several activities which I summarize below. I currently serve as an editor for the Proceedings of the AMS, the AMS University Lecture Series, Analysis and Geometry in Metric Spaces, and, starting in February 2015, I will become an editor of the Transactions of the AMS. I served as Chair of the Scienti_c Review Panel of the Pacific Institute for the Mathematical Sciences (PIMS) from 2009 to 2013. I am a member of the Pacific Rim Mathematical Association (PRIMA) steering committee. I joined the Board of Trustees at the Institute for Pure and Applied Mathematics (IPAM) in 2009. In addition, I was a member of the ICM 2014 sectional panel for Section 8: Analysis and Applications. The charge of this panel was speaker selection. I am a member of the Scientific Advisory Committee for the Mathematical Sciences Research Institute (MSRI) since 2016.

My professional activities also include organizing conferences and thematic programs. Among others, I co-organized programs and workshops at MSRI (Mathematical Sciences Research Institute) in 1997 and 2011. With the support of IPAM and PIMS I co-organizes Lat-Math, at IPAM in April 2015. Thanks to the success of this conference we are currently co-organizing the second edition to take place at IPAM in March 2018. LatMath’s main goal is to promote diversity in the mathematical sciences. This three day conference will showcase latinos in the mathematical sciences. The events will include community lectures and activities geared to attract a broad audience, from undergraduate students to senior researchers in academia, research laboratories and industry. This is a truly unique program designed to increase the representation in the mathematical sciences of a minority group whose needs have not been su_ciently addressed by any of the existing programs. The expectation is that LatMath will become a recurring meeting which will have a signi_cant impact on the mathematical and latino communities.

I am fortunate to have been able to do some outreach activities in the Seattle area. Between 2005 and 2009, I organized and co-organized several Math Fairs in different elementary schools with the Seattle Public School District. The general format included training undergraduate students to work with elementary school students for four weeks; teaching them different games that enhanced mathematical concepts, making them fun and appealing to a broad audience. The culminating event was an evening \Math Fair” where the kids the opportunity to teach their parents and guardians the games. This was an empowering event for the children involved. During the academic year 2007-2008 the Seattle Public School system adopted a new curriculum, Everyday Math, to be supplemented by the Singapore Math curriculum. I was involved in the joint implementation of these two very different curricula for 4th grade at Wedgwood Elementary. My involvement included setting the guidelines for each lecture and teaching one lecture a week. Patricia Hagen, a local artist, and I jointly ran a project for all the 4th grade classes at Wedgwood Elementary combining art and symmetry.

** **

**Premios y reconocimientos recibidos:**

2017 Miembro Correspondiente de la Academia Colombiana de Ciencias Exactas, Fisicas y Naturales

2016 Fellow of the AMS

2016 Craig McKibben & Sarah Merner Professor in Mathematics

2015 Guggenheim Foundation Fellowship

2012-2016 Robert R. & Elaine F. Phelps Professorship in Mathematics Simons Foundation Fellowship

1996-2000 Alfred P. Sloan Research Fellowship

1994-1998 NSF Mathematical Sciences Postdoctoral Research Fellowship

1991- 1992 Alfred P. Sloan Doctoral Dissertation Fellowship

ICM 2010 Invited speaker to the Analysis session at ICM 2010 in Hyderabad, India, August 2010.

AMS 2011 Invited one hour address at the Joint Mathematics Meeting in New Orleans, Louisiana, January 2011.

2016 NAM Clayton-Woodard Lecture at the Joint Mathematics Meeting in Seattle, Washington, January 2016.

2017 23rd Nevanlinna Colloquium, June 2017, ETH, Zurich.

**Publicaciones (lista completa)**

** **

- Surfaces with generalized second fundamental form in L2 are Lipschitz manifolds, J. Differential Geometry, 39, 1994, 65-101.
- Geometric conditions and existence of bilipschitz parameterizations, Duke Mathematical Journal, 77, 1995, 193-227.
- Compactness Properties of Weakly p-Harmonic Maps into Homogeneous Spaces, (with C.Y. Wang), Indiana Univ. Math. J., 44, 1995, 87-113.
- Harmonic Measure on Locally Flat Domains, (with C. Kenig), Duke Mathematical Journal, 87, 1997, 509-551.
- Doubling and Flatness: Geometry of Measures, Notices of the AMS, 44, 1997, 1087-1094.
- Free Boundary Regularity for Harmonic Measures and Poisson Kernels (with C. Kenig), Annals of Math. 150, 1999, 369-454.
- Reifenberg Flat Metric Spaces, Snowballs and Embeddings, (with G. David), Math. Ann., 315, 1999, 641-720.
- Asympotically Optimally Doubling Measures and Reifenberg Flat Sets with Vanishing Constant, (with G. David and C. Kenig), Comm. Pure Appl. Math, 54, 2001, 385-449.
- A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, (with C. Kenig, H. Koch and J. Pipher), Adv. Math. 153, 2000, no. 2, 231–298.
- Free Boundary Regularity for the Poisson Kernel, Proceedings of the Eleventh Tokyo Conference in Nonlinear PDE 2001. 11. Free Boundary Regularity for the Poisson Kernel below the Continuous Threshold, (joint C. Kenig), Math. Research Letters, 9, 2002, 247-254.
- Poisson Kernel Characterization of Reifenberg Flat Chord Arc Domains, (with C. Kenig), Ann. Scient. Ec. Norm. Sup. 36, 2003, 323-401.
- On the free boundary regularity theorem of Alt and Caffarelli, (with C. Kenig), Discrete and Continuous Dynamical Systems 10, 2004, 397-422.
- Regularidad por debajo del umbral de continuidad, Boletin de Matematicas, 8, 2001, 60-67.
- On the free boundary regularity theorem of Alt and Caffarelli, Errata, (with C. Kenig), Discrete and Continuous Dynamical Systems 14, 2006, 857-859.
- Free boundary regularity below the continuous threshold: 2-phase problems (with C. Kenig), J. Reine Angew. Math. 596, 2006, 1-44.
- When do good parameterizations exit?, (extended abstract) Proceedings of the AWM and MSRI workshop Women in mathematics: the legacy of Ladyzhenskaya and Oleinik , May 2006.
- Stability of Lewis and Vogel’s result (with D. Preiss), Revista Matematica Iberoamericana 23, 2007, 17-55.
- Two phase free boundary regularity problem for Harmonic measure and Poisson kernel Report No 33/2005, Mathematisches Forschungsinstitut Oberwolfach, 1898-1900.
- Regularity of measures with H ̈older density ratio Report No 35/2007, Mathematisches Forschungsin- stitut Oberwolfach, 2046-2048.
- A generalization of Reifenberg’s theorem in R3 (with G. David and T. DePauw), GAFA 18 2008, 1168-1235.
- Geometry of Measures: Harmonic Analysis meets Geometric Measure Theory Handbook of geometric analysis. No. 1, 449465, Adv. Lect. Math. (ALM), 7, Int. Press, Somerville, MA, 2008
- Boundary Structure and size in terms of interior and exterior harmonic measures in higher dimensions (with C. Kenig and D. Preiss), J. Amer. Math. Soc. 22 (2009), 771-796.

- On the smoothness of Holder-doubling measures (with D. Preiss and X. Tolsa), Calculus of Variations and PDE’s 35 (2009), 339-363.
- Divergence form operators in Reifenberg flat domains (with E. Milakis), Mathematische Zeitschrift 264 (2010), 15-41.
- The Cauchy problem for Schr ̈odinger flows into K ̈ahler manifolds (joint Kenig, Lamm, Pollack, Staffilani) Discrete and Continuous Dynamical Systems 27, 2010, 389-439.
- Potential Analysis meets Geometric Measure Theory, Proceedings of ICM 2010, III. 28. Reifenberg parameterizations for sets with holes (joint G. David), Memoirs of the AMS, 215

2012. 29. Harmonic Analysis on Chord Arc Domains (joint with E. Milakis and J. Pipher), J. Geom. Anal. 23 (2013), 2091–2157.

- Quasisymmetry and rectifiability of quasispheres (joint with M. Badger, J. Gill and S. Rohde), Trans. Amer. Math. Soc.366, (2014), 1413-1431.
- Perturbations of elliptic operators in chord arc domains (joint with E. Milakis and J. Pipher), Contemporary Mathematics, 612 (2014) Amer. Math. Soc., 143-161.
- Rectifiability via a square function and Preiss’ theorem (joint with X. Tolsa), to appear in International Mathematics Research Notices 2014; (DOI) 10.1093/imrn/rnu082
- Regularity of Almost Minimizers with Free Boundary (joint with G. David), Calculus of Variations and PDEs, 54 (2015), 455-524.
- Square Functions and the A∞ Property of Elliptic Measures (joint C. Kenig, B. Kirchheim, and J. Pipher) The Journal of Geometric Analysis, 26 (2016), 2383-2410. (DOI) 10.1007/s12220- 015-9630-6
- Quasiconformal planes with bi-Lipschitz pieces and extensions of almost affine maps (joint with J. Azzam and M. Badger), Adv. Math., 275 (2015), 195-259.
- Wasserstein Distance and the Rectifiability of Doubling Measures: Part I (joint with J. Azzam and G. David), Mathematische Annalen, 364 (2016), 151-224.
- Wasserstein Distance and the Rectifiability of Doubling Measures: Part II (joint with J. Azzam and G. David) Mathematische Zeitschrift, 286 (2017), 861-891.
- Structure of sets which are well approximated by zero sets of harmonic polynomials (joint M. Badger and M. Engelstein) Analysis & PDE 10-6 (2017), 1455–1495. DOI 10.2140/apde.2017.10.1455 (arXiv:1509.03211)
- A new characterization of chord-arc domains (joint J. Azzam, S. Hofmann, J.M. Martell, and K. Nystr ̈om) to appear in JEMS.
- A∞ implies NTA for a class of variable coefficient elliptic operators (joint S. Hofmann and J.M. Martell) (arXiv:1611.09561) to appear in Journal of Differential Equations.
- Boundary rectifiability and elliptic operators with W1,1 coefficients (joint with Z. Zhao) to appear in Advances in Calculus of Variations.
- Free boundary regularity for almost-minimizers (joint G. David and M. Engelstein)(arXiv:1702.06580)
- Uniform rectifiability and elliptic operators with small Carleson norm(joint S. Hofmann, J.M. Martell, S. Mayboroda, and Z. Zhao) arXiv:1710.06157