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Haynes Miller works in algebraic topology, homotopy theory and homological algebra. His early research helped develop chromatic homotopy theory, and his solution of the Sullivan conjecture connected questions about maps of spaces with the structure of Lie groups. His lecture notes and expository writing develop the algebraic topology used to turn geometric questions into calculations with algebraic invariants.^[ref]

Identity source: math.mit.edu/directory/profile.html?pid=186

Higher category theory studies structures with morphisms between objects, morphisms between those morphisms, and further layers of comparison. In a two-dimensional example, categories are objects, functors are one-morphisms and natural transformations are two-morphisms. A two-morphism can be composed along a shared functor or alongside another natural transformation; the compatibility of these compositions is part of the structure.^[ref]

Homotopy theory gives the motivating picture: points are connected by paths, paths are compared by homotopies, and homotopies themselves can be compared by further homotopies. Replacing every path by only its homotopy class produces the fundamental groupoid and forgets these higher comparisons. Weak higher categories retain coherent comparison data when associativity holds only up to equivalence. One must specify a model and whether higher morphisms are required to be invertible; “higher category” does not impose one universal convention.

Jacob Lurie is a mathematician whose work develops higher category theory and its applications to algebraic geometry and homotopy theory. His 2009 paper sketches a proof of the cobordism hypothesis, a classification of extended topological quantum field theories.^[ref]

The organizing idea is that the value on a point, with the appropriate duality properties, can determine the field theory on higher-dimensional manifolds and their cobordisms. Cutting and gluing become composition laws in a higher category. This explains why passing from whole manifolds to pieces of every dimension changes the algebraic framework. His chromatic lecture notes give another bridge between algebra and topology, constructing Morava K-theory from structured topological spectra.^[ref]

Niko Naumann works in arithmetic geometry and homotopy theory. With Annette Huber and Guido Kings, he refined the Lazard isomorphism, extending a comparison between continuous group cohomology and Lie algebra cohomology to integral coefficients under suitable hypotheses.^[ref]

The distinction between integral and rational coefficients matters: tensoring with ${\mathbb{Q}}_p$ can discard torsion elements, so a rational comparison does not automatically recover integral information. The same work also treats analytic cochains over finite extensions of ${\mathbb{Q}}_p$. Naumann's author page lists arithmetic algebraic geometry, motivic homotopy theory, and joint work on nilpotence and descent in equivariant stable homotopy theory.^[ref]

A subset $S$ of a real vector space is a star-shaped set if some point $p\in S$ sees every point of $S$ along a segment lying in $S$: $(1-t)p+tx\in S$ for all $x\in S$ and $0\le t\le 1$. Every convex set is star-shaped about each of its points; a star-shaped set need only have one such point.

For example, the union of the horizontal and vertical line segments through the origin is star-shaped about the origin but is not convex. In a topological vector space, the map $H(x,t)=(1-t)x+tp$ contracts a star-shaped set to $p$. This makes it contractible, a useful bridge from elementary geometry to homotopy theory.