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Óscar Maciá has a degree in Physical Sciences from the University of Valencia (2002) and a Doctor in Toric Physics from the University of Valencia (2007). He currently works as a university professor (2023), assigned to the Department of Mathematics (Area of Geometry and Topology) of the Faculty of Mathematical Sciences of the University of Valencia. He is a member of the UV Geometric Analysis research group (GAGUV).
His research deals primarily with differential geometry, often inspired by problems arising in theoretical physics. Generally, it focuses on the study of varieties endowed with some additional structure (Riemannian, complex, etc.) defined through the action of a Lie group. Topics covered include: quaternionic and exceptional geometry, harmonic maps, manifolds with special holonomy, spinors, differential operators, and complex geometry.
In these areas he has participated in multiple national and international research projects.
 

I was born in Mislata (Valencia) in 1954. Thanks to the help of my two brothers, I was able to complete my undergraduate studies in Mathematics in 1975 and in Physics in 1976 at the University of Valencia.

In 1979, I earned my doctorate in Mathematics from the same university, under the supervision of Antonio Martínez Naveira, to whom I owe almost everything I have been able to achieve in the scientific field. Other mathematicians who have influenced my work include Lieven Vanhecke (University of Leuven) and, especially, Alfred Gray (University of Maryland).

My attendance at the Besse seminar (University of Paris VI) has had a decisive influence on my work. My last three-month stay in Paris in 1985 radically changed my line of research. Another short four-month stay at the University of Maryland in 1992 has also had significant repercussions.

I began supervising my first doctoral student in 1986. I have supervised six more doctoral students since then. Collaborating with them has made my work both the most challenging and the most rewarding. In 1997, I invited A. Borisenko (University of Kharkiv) to Valencia and experienced firsthand how fruitful the Latin-Russian symbiosis is for mathematical research. With him, I began my work on negative curvature, and with him (and my penultimate doctoral student, E. Cabezas-Rivas) I initiated my current research topic on geometric flows, also taking advantage of my sabbatical year in 2004.

Since 1982, I have participated in various research projects funded by the Ministry and the Generalitat (the Catalan government), some of them as Principal Investigator. My last three-month stay away from my university was in 2008 at the CRM (Research Centre for Mathematics) in Barcelona, ​​where I collaborated on the organization of a thematic research term.

I have also collaborated on the organization of six scientific meetings and the editing of three books. I have published 51 original research articles, 5 review articles, and 4 popular science articles.

In 1984, I was appointed Associate Professor of Geometry and Topology at the University of Valencia, and in 1999, Full Professor in the same field and at the same university, where I have been teaching various mathematics courses since 1976.

I am part of the group of mathematicians who are striving to better understand the curvature of space. Two types of curvature are distinguished: intrinsic (that which an observer perceives from within the space itself, as humans observe the curvature of the universe) and extrinsic (that which is observed from the outside, as we can observe the curvature of a road). How are these curvatures related to the properties (distances, angles, volumes, energies, sounds, motions, etc.) of space? That is the question we are trying to answer.

For example, in my most cited work (with E. Cabezas-Rivas), we studied the motion of a surface moving at a velocity determined by an extrinsic curvature, with the condition that the volume enclosed by the surface remains constant. This is a problem called "geometric flow," and it is used as a model for the motion of surfaces separating two immiscible fluids, one of which is incompressible. In 1987, it was determined that if the ambient space in which the surface moves is not curved and the surface is initially convex, it evolves, with exponential speed, toward a spherical shape. At the same time, it was indicated that it seemed impossible for the same to occur in curved spaces. Twenty years later, we demonstrated that this was not the case, that the same phenomenon also occurred in ambient spaces with negative curvature, provided the appropriate concept of convexity is used.

The length of time a problem remains unsolved is one of the criteria used to determine the importance of research in mathematics; hence the reference in the previous example to the 20 years that passed between the problem's formulation and its solution. In fact, our interest in geometric flows stems from their use by Perelman (the Russian mathematician who, on June 6, 2010, failed to accept a million-dollar prize) to prove the Poincaré conjecture, which had remained unsolved for a century. To prove our aforementioned result, we also used another (article 23, with A. Borisenko) on convex bodies in negative curvature, which resolved a problem that had been open for 27 years.

The work mentioned  is the first I have written on geometric flows, and that has been my line of research ever since. I was introduced to the study of flows by A. Borisenko. My collaboration with A. Borisenko, E. Cabezas-Rivas, and F. Viñado has contributed to popularizing the study of flows with densities and flow in non-Euclidean environments. I am interested in the motion of the surface itself, the influence of its own and the environment's curvatures on the motion, and also its application to other problems related to curvature and energy, to which members of the team I work with, have made significant contributions, and the relationship of these fluxes to the first eigenvalue of elliptic operators, from which we (with M. C. Domingo Juan and J. Zhu) discovered a surprising result in article 49. Teaching and dedication to my students has also been a substantial part of my work. Communicating with them has always been an incentive to try to better understand mathematics and to be able to explain it. Management has been less to my liking, but understanding its necessity, I have continued to serve as secretary or head of the department (of Geometry and Topology) when required. Furthermore, I have always been a member of some committee (CAT for one or more degree programs, recruitment committees, course coordinator, faculty boards, committees for some curricula, etc.).

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Guillermo Peñafort Sanchis, graduate in Mathematics from the University of Valencia and doctor in Mathematics from the University of Valencia. Works on singularity theory.