**
Esbriel Avilés Cruz and I, ca. 1965.**

My late uncle Esbriel (from my mother's side) was the light of my childhood. He was born in 1935 in Oficina Santa Laura, Chile (see http://www.albumdesierto.cl/ingles/2stalau.htm) and worked for the nitrate companies until 1956, at which time he and the rest of the family (including my mother) moved to Arica, Chile. My dear uncle died on January 15th, 1985, after suffering from a terrible disease. His memory lives on in me and in others who loved him. One of my papers (number 16 below) is dedicated to his memory.

Here is a (much more recent) picture of me taken in July of 2009, in Salamanca, Spain.

I was born on the 25th of June, 1961, in Arica, Chile. I studied Mathematics at Universidad de Chile (in Santiago) from 1980 to 1986, where I was trained by Oscar Barriga and Ricardo Baeza, among others. In early 1987 I began my doctoral studies at The Ohio State University in Columbus, Ohio, graduating from that institution in early 1994. From 1994 to the present I have worked at various institutions in Chile. I currently work at Universidad de La Serena, which is located about 450 km north of Santiago. La Serena is the best place to live in Chile. Many locals call it "paradise", and I'm beginning to agree with them (except during the summer months, when the city becomes overcrowded with tourists. But then the wheather is very nice...).

(In reverse chronological order)

17) "Quasi-abelian crossed modules and nonabelian cohomology" Quasi-abelian crossed

16) "On Néron class groups of abelian varieties".
Neron_Class_Groups

15) "On Néron-Raynaud class groups of tori and the Capitulation Problem". Class gps tori

14) "Arithmetic duality theorems for 1-motives over function
fields". J. reine angew. Math. **632** (2009), 203-231. 1-motives

13) "Finiteness theorems for algebraic cycles of small codimension on
quadric fibrations over curves". Cent. Eur. J. Math.** 7**, no. 4 (2009),
606-616. CEJM article

12) "The generalized Cassels-Tate dual exact sequence for 1-motives"
(with K.-S.Tan). Math. Res. Lett. **16**, no. 5 (2009), 829-841. GCTdes

11) "Chevalley's ambiguous class number for an arbitrary torus".
Math. Res. Lett. **15** (2008), no.6, 1149-1165. Chevalley

10) "On K2 of varieties over number fields". J. K-Theory **1 **(2008),
no.1, 175-183. K2 varieties

9) "Algebraic cycles on Severi-Brauer schemes of prime degree over a curve".
Math. Res. Lett. **15** (2008), no.1, 51-56. Cycles-SBS

8) "Capitulation, ambiguous classes and the cohomology of the units"
J. reine angew. Math. **613** (2007) , 75-97. Camb

7) "A generalization of the Cassels-Tate dual exact sequence" (with
K.S. Tan). Math. Res. Lett. **14** (2007), 295-302. CTdes

6) "On the Hasse principle for zero-cycles on Severi-Brauer fibrations".
Int. Math. Res. Not. **2005**, no.48, 2969-2982. Hasse-SBF

5) "Finite modules over non-semisimple group rings". Israel J.Math.
**144** (2004), 61-92. G-mod

4) "Brauer groups and Tate-Shafarevich groups". J. Math. Sci., Univ.
Tokyo, **10 **(2) (2003), 391-419. BTS

3) "On Tate-Shafarevich groups of abelian varieties". Proc. Amer.
Math. Soc. **128**, No. 4 (2000), 953-961. Sha

2) "On the conjecture of Birch and Swinnerton-Dyer". Trans. Amer.
Math. Soc. **349** (1997), 4181-4200. BSD

1) "Class numbers of quadratic function fields and continued fractions".
J. Number Theory **40**, No.1 (1992), 38-59.

1) Handwritten preparatory notes for my master's degree thesis (in Spanish). resumen.pdf

These notes were probably written in early 1986. They were used in the writing of my master's degree thesis (see below). Although they are full of scratches and are an ugly sight to behold, I post them here in case they are of any use to anyone (they contain some material which was left out of my thesis).

2) My master's degree thesis (in Spanish), Universidad de Chile, 1986. tesis.pdf

3) My dissertation, The Ohio State University, 1994.

4) "A generalization of the Picard-Brauer exact sequence" NewPicBr.pdf

(this is the latest version of the 2003 original)

This preprint **has **been checked by someone. Here is the referee
report. Unfortunately, I have been unable to find the time (so far) to implement
the suggestions made by the referee (the technical points mentioned in the report
are really not a problem, but developing the applications alluded to in the
report requires quite a bit of time. Eventually, I hope to be able to generalize
this preprint using Rost's* cycle complexes,* thereby increasing the chances
of finding interesting applications of the resulting generalized exact sequence).

5) "Integral 3 x 3 matrices with prescribed integral eigenvalues" Integral matrices.pdf

I wrote this note in late 2004 for my own use in the teaching of linear algebra. I did not check the literature before embarking on the project (which is never a smart thing to do), and all calculations were done by hand. But after the pain was over, I never had to look anywhere else to find interesting diagonalization exercises for my linear algebra students.

* *The main obstacle to achieving happiness is our own ego. Oriental
philosophers have known this for centuries. Many in science strive to attain
recognition, prestige, awards, etc., in order to satisfy their own demanding
egos. But letting your ego control your personality (rather than the other way
around) is not very smart, since an out-of-control ego is insatiable. No matter
how many awards, honors, recognition, etc., you may get, your neighbor's lawn
will always seem to you to be greener than your own. You might even feel jealous
at the success of others (which is a most repellent human characteristic), and
be bitter and unhappy. So, humbly, I give you the following advice: free yourself
from the tiranny of your own ego. If you do, your chances of leading a truly
happy life will increase dramatically.

Oscar was one of my teachers between 1980 and 1986, while I was an undergraduate student at Universidad de Chile in Santiago. He showed me the pristine beauty of mathematics early on, thereby convincing me that I should become a mathematician. Sadly, he passed away much too soon, in December of 2001. One of my papers (number 8 above) is dedicated to his memory.

1) "If you are going through hell, keep going." Winston Churchill

2) "There is nothing as corrosive as praise. It sweetens the palate, but corrupts the soul." Simón Bolívar.

3) "Talent without probity is a scourge." Simón Bolívar.

4) "Mathematics, rightly viewed, possesses not only truth, but supreme beauty. A beauty cold and austere, like that of sculpture." Bertrand Russell.

5) "Only two things are infinite. The universe and human stupidity. And I'm not sure about the former". Albert Einstein.