沉痛悼念

据说死因是过量饮酒导致的肝病。介绍他的工作值得写一篇长文，以后找机会写

消息错误，死因是aneurism，动脉瘤，上面的是最早的小道消息

你们起的名字都好霸气，deligne scholze 黎曼

几个俄系数学家写了我认为比较好的回忆文章，下面贴出全文
VLADIMIR VOEVODSKY - WORK AND DESTINY
MARK BICKFORD, FEDOR BOGOMOLOV, AND YURI TSCHINKEL

Vladimir Voevodsky, who died in Princeton on September 30, 2017,
at the age of 51, was one of the most remarkable and highly original
mathematicians of our time. His achievements have been recognized
by the highest honor of the profession, the Fields medal, which he
received in 2002. His work transformed several elds of mathematics
and theoretical computer science.
Vladimir started his mathematical education as a high school stu-
dent, attending the Shafarevich seminar at the Steklov Institute of the
Russian Academy of Sciences in Moscow. He bypassed the \usual"
mathematical olympiad training and moved directly into research. His
exceptional talent and focus were apparent already then, to all who
interacted with him. As an undergraduate student at Moscow State
University he fully immersed himself into the study of Grothendieck's
anabelian geometry, formulated 1984 in Esquisse d'un programme. His
early work, joint with G. Shabat, concerned Dessins d'enfants, the
study of Galois groups of curves over number elds via their represen-
tation by special graphs on Riemann surfaces. The inspiration came
partially from a result of Belyi, who proved that all such curves admit
special meromorphic functions, with only three ramication points;
moreover, the existence of such functions characterizes these curves
among all complex projective curves. At that time, it seemed that this
result might open the door to the solution of major open problems in
arithmetic geometry, such as Mordell's conjecture and Fermat's last
theorem, as well as another important conjecture, which is still open:
the Section Conjecture of Grothendieck. Vladimir's interest in this
area showed his determination, early on, to tackle the most dicult
and challenging conjectures in mathematics.
His next project was the proof of reconstruction of hyperbolic curves,
over a natural class of ground elds, from their etale fundamental
groups. This is the rst step towards the Section Conjecture. His
joint papers with M. Kapranov, on what seemed to be not very popu-
lar issues in category theory (n-categories, 1-groupoids, higher braid
groups), turned out to be crucial for his work in algebraic geometry
over the next two decades, as well as for the Univalent foundations.

Already at that time, he said, casually: \If the categorical framework
works out, the Bloch-Kato conjecture will follow trivially."
His research interfered with his undergraduate work, he did not show
up for classes or exams. After eventually quitting Moscow State Uni-
versity, he moved, in 1990, to Harvard University, where he became a
Ph.D. student. He graduated in 1992, and, after one year at the In-
stitute for Advanced Study, returned to Harvard as a Junior Fellow of
the Harvard Society of Fellows.
All these years, he was relentlessly working on foundational prob-
lems; his meager publication record between 1991 and 1995 is in stark
contrast with the intensity of his investigations. Then came an avalanche
of papers that radically changed algebraic geometry, settling major
open conjectures (construction of the derived category of motives, Mil-
nor conjecture, the more general Bloch-Kato conjecture) and introduc-
ing powerful new techniques. These conjectures postulated a deep and
highly nontrivial connection between geometry of algebraic varieties
and their Galois symmetries. As Voevodsky's proof showed, this bridge
required radically new concepts; no simplications of his original proof
have emerged despite intense eorts by geometers and algebraists.
Voevodsky's main achievement was the creation of an amalgam of
homotopy theory and algebraic geometry. Both theories deal with ob-
jects of geometric origin, but on a basis of completely dierent concep-
tions: while homotopy theory emphasizes
exibility, algebraic geome-
try is rather rigid: algebraic varieties resist small, local perturbations.
Mixing these essentially incompatible worlds, in a meaningful context,
required a leap of faith and an enormous, prolonged, eort.
After a short period of teaching at Northwestern University, Voevod-
sky moved to the Institute of Advanced Study. In his words, he began
to \lose motivation" for work in algebraic geometry around 2003, hav-
ing completed a vast research program. He started taking computer
science courses at Princeton University; private conversations with him
frequently revolved around the nature of correctness, truth, and proof,
in mathematics. This was triggered by attempts and failures of several
mathematicians working with big categorical structures, too formida-
ble for paper-and-pencil analysis. Voevodsky was led to a more gen-
eral question of whether mathematicians had the right tools to explore
dicult new areas, for example, the highly complex theories he was
interested in.
Voevodsky learned that, starting with N.G. de Bruijn, computer sci-
entists and logicians had created automated proof assistants such as,
e.g., Mizar, Coq, or Nuprl. However, to be applicable in practice, the
mathematical proofs had to be rst fully formalized in logical systems

that the proof assistants implemented, e.g., set theory, or other type
theories. Already this step presents a daunting obstacle, in any min-
imally nontrivial situation. The analysis of existing proof assistants
convinced Voevodsky that computers could, in principle, check math-
ematical proofs, but that none of the available systems were up to this
task, on a fundamental, rather than just technical, level.
With his usual vigor and tenacity, Voevodsky decided to create the
foundations for this area at the interface of mathematics and computer
science. His main insight was based on his previous experience in math-
ematics: the introduction of ideas of homotopy theory into the theory
of types. His Univalence Axiom postulates that homotopy-equivalent
objects share the same formal properties. Voevodsky was convinced
that a systematic use of his Univalent Foundations would lead to the
construction of practical proof assistants.
Again, there was a substantial gap in his publication record, followed
by a burst of activity, starting in 2014, with 15(!) papers posted on
arXiv, 3 of them in June of this year alone. The introduction of univa-
lence has already created great excitement in the community working
with type theories in mathematics, philosophy, and computer science.
Initial prototype \univalent" proof assistants were created, and Vo-
evodsky and his collaborators embarked on a project to build a com-
prehensive library of mathematics, based on Univalent Foundations
rather than on set theory, and rigorously checked by computer.
Voevodsky always said that this work was only a prototype, and
that the ultimate foundations for computer-checked mathematics had
yet to be perfected. His vision sparked many ongoing research eorts,
for example, to nd a constructive interpretation of univalence and to
explore the use of univalence in various areas of mathematics.
His sudden and untimely death came as a shock to his colleagues and
friends. He stands out as one of the giants of modern mathematics. The
full impact of his ideas is still to be understood and appreciated.