Vladimir Voevodsky has died

Vladimir Voevodsky has died
ias.edu

I became familiar with his work through HoTT --

> More recently he became interested in type-theoretic formalizations of mathematics and automated proof verification. He was working on new foundations of mathematics based on homotopy-theoretic semantics of Martin-Löf type theories. His new "Univalence Axiom" has had a dramatic impact in both mathematics and computer science.

And some of his online lectures outlining his reasoning and motivations became greatly influential on my own interests.

I've never heard of him so I'm curious, being a Field medalist and all.

It's quite technical to describe what that is, but the general idea is that he imported homotopy theory, a fundamental concept in mathematics which allows you to describe topology with an algebra of paths in that geometry, to algebraic geometry. This allowed him to solve some major open problems in the field.

Topology is essentially the study of continuity and the properties that are invariant of objects under continuous deformation. The prototypical example of a topological space is the continuum although the concept of continuity has been generalized a great deal. Homotopy is the study of continuous paths which are defined as continuous functions from the unit interval into topological spaces, as well as higher dimensional "paths". It turns out that these paths have a rich algebraic structure which give you a strong idea of what the "shape" of the space is like.

Classically algebraic geometry is the study of solutions to systems of polynomial equations over systems of numbers related to the integers, rationals, reals, and complex numbers. More modernly, it is the study of models of theories that resemble the previously mentioned theory in fundamental ways. For example, the theory of elliptic curves which underlies some of our modern cryptography is part of algebraic geometry.