Abstract: The classical Luroth problem asks whether every subfield of a purely transcendental extension of a field k is
itself purely transcendental.  Much work was done in the 1970s to produce counterexamples to the Luroth problem.  I will
examine one approach (that of Artin and Mumford, as generalized by Colliot-Thelene and Ojanguren), which uses cohomological
invariants to detect counterexamples.  More precisely, I will discuss how techniques from the Morel-Voevodsky homotopy
theory of algebraic varieties allow one to use ``higher" cohomological invariants to detect new counterexamples to the
Luroth problem over the complex numbers.