Isa Fer, from the UGR to the ICM

The university is one of the best stages in the life of a student, at least under my point of view. In this epoch of the academic life one penetrates into a completely new world, in which multitude of histories lives and in the one that meets many people.

At least this was my case. I was lucky to meet very good persons in my university stage in Granada, persons who helped me very much in those moments and with whom I shared unforgettable experiences. Unfortunately they always remain the people to whom you do not go so far as to be related so much, although there is no reason for it. The people who shares every day with you but with whom you do not have so much contact.

Although it has been a few years since it already finished this period I keep on remembering many of my partners, so much to the most nearby (clearly) as those who it were not so much. Isa belongs, unfortunately, to the latter group. And I say unfortunately because she always looked like to me a magnificent person, always with a smile in the mouth, always ready to throw a hand. And, penetrating already into the academic part, because she was always a brilliant student. And when I say brilliant I mean tremendously brilliant. Lola, one of his friends in that epoch (I do not know if before already beginning the career you knew each other), can confirm that Isa was always over all that we share class with her. For this motive it does not surprise me that it has come up to where it has come. And for being as it is an atrocity I am glad.

Who is Elizabeth Fernández?

Elizabeth Fernández

Elizabeth Fernández born in Linares on August 16, 1979, began his Licentiate in Mathematical Sciences in the course 1997-98 in the University of Granada. According to its own words, from the beginning there attracted attention of him the geometry (to none of that we know your sublime Geometry course III with Paco Martín misses this to us). Even such a point came the thing that he enjoyed in his last two years of career of two fellowships in the department of Geometry and Topology in the above mentioned university.

After happening for Murcia and Badajoz, at present she is an Employed Teacher She confers a doctorate of the department of Applied Mathematics on I of the University of Seville.

In the direction of the ICM

Very well: and what is what Isa has obtained? Since a little so important as to be the first Spanish woman who receives an invitation to help like speaker in the ICM. Almost not at all.

Pablo Mira

The above mentioned invitation came to him for his works on surfaces of constant average curvature and so much she like his partner Pablo Mira is going to be who communicate the results that they have obtained in this most important congress.

The news on this invitation came to me to the ICM across Wiggle me (the linkage is at the end of this article). Nothing more to see it I started looking for a way of contacting with Isa to congratulate it and to mention to him that he wanted to write something on her in Gaussianos, blog that, certainly, already knew across Lola (thank you). Later a few post office crossed, Isa mentioned to me that he would try to write something telling of what there consists the work that Pablo and she they have realized and that has served to them to go to the ICM. Who better than she to explain it?

The works of Isa and Pablo

Very well, it is already being time for we to know why they have invited the ICM to Isa and Pablo. The following text is what Isa has written for me and for all of you explaining his works to us.

Surfaces of constant average curvature (CMC)

A very important concept at the time of studying surfaces is that of the average curvature, which gives us a measurement of how the surface bows in the space. The idea is the following one: for every point P of the surface we consider all the normal sections of the surface that happen for this point, which there are the curves that are obtained on having cut the surface with all the planes perpendicular to the same one in the point P. Of all these curves we remain with those who have minor and major curvature (the called principal directions), these directions mark the maximum thing that we can bow towards a side or towards other in the surface.

If we call and to the curvatures of two principal directions, the average curvature in this point is, precisely, the arithmetical average between two o'clock:

The surfaces that have the same average curvature in all his points name half constant (CMC) surfaces of curvartura, and have geometric properties that make them very interesting.

For example, the surfaces of equal CMC to zero are named a minimal surfaces, name that comes from the fact of which these surfaces are those who have minor area of between all the surfaces with the same outline (locally, that is to say, considering sufficiently small pieces of the surface). This property is precisely the one that it characterizes to the movies of soap (it is the first one of the famous laws of Plateu, which govern the behavior of the movies of soap). This allows us to characterize the minimal surfaces as those in that, if we cut away a small piece of the surface and put the rest of the surface in water with soap, the movie that forms in the hollow left by the clipping has exactly the same form as the original piece.

Very well, everything previous was referring to surfaces through that they live inside the usual space (the space euclídeo three-dimensional), but the CMC surfaces in general, and the minimal surfaces in particular, exist in any type of spaces, and a branch of the surface Theory of big relevancy at present is the study of the surfaces of CMC in homogeneous spaces. And what is a homogeneous space? Since said to big features, it is a space that is equal in all his points, that is to say, it does not have special points (although yes there can be special directions). Obviously the space euclídeo is a homogeneous space, but there is more. Let's think for example about a cylinder on a sphere, that is to say, the space product. In this cylinder all the points are equal, but it turns out to be clear that the vertical direction (that of the factor) is special. The three-dimensional homogeneous spaces are very studied, and his classification keeps many relation with the famous geometries of Thurston, related to the conjecture of Poincaré.

And this is the field in which Pablo Mira and it was me who have been employed at last years (from 2005) and on that they have invited us to give a conference in the ICM (that probably will take for title Thusrton 3 - dimensional geometries).

What why we? Since basically thanks to two articles that we published on the topic and in which we solved one of the problems opened on minimal surfaces in homogeneous spaces of more actuality in this moment. I will try to tell briefly of what it consists.

One of the basic problems on minimal surfaces is the called problem of Bernstein, which consists of classifying the minimal surfaces that are grafos especially a plane. In the space euclídeo this problem was solved in 1915 by the proper Bernstein, which demonstrated that the only ones grafos minimal entire are the planes.

One of the most studied homogeneous spaces is the Heisenberg space. This space is (topológicamente) like the usual space, but with a different metrics. Namely the way of measuring distances (and therefore everything related to the curvature) is different. In this space the vertical direction is special, and it has properties different from the horizontal directions. In the Heisenberg space he has therefore sense there appears the problem of Bernstein on that we comment previously, that is to say, to classify the minimal entire grafos under the Heisenberg space.

To solve this problem has been our biggest contribution to this theory. In 2007, and thanks to the first work of 2005, Pablo and I classify the family of grafos minimal points of the space of Heisenberg that, contrary to what it happens in the space euclídeo, is a very big family, parametrizada in terms of quadratic differential holomorfas, which are obtained from a harmonic application on the surface, but this is already another topic …

Related linkage:

• He interviews Elizabeth Fernández in The Country.
• Isa and Pablo in Mathematics and his Borders.