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By Haynes Miller, explicitly credited for the supplementary-note collection on the MIT OpenCourseWare readings page. The collection accompanies Differential Equations, MIT 18.03, Spring 2010. This course context does not establish a note publication date. MIT OpenCourseWare hosts the collection.

Collection attribution: ocw.mit.edu/courses/18-03-differential-equations-spring-2010/pages/readings/supp_notes/

Chapter 15 in 18.03 Supplementary Notes by Haynes Miller. Authorship inherits the explicitly verified collection credit to Haynes Miller; it is not inferred from course instructor credits. No separate chapter publication date is verified.

Online text: ocw.mit.edu/courses/18-03-differential-equations-spring-2010/b06d6e91e362964ec92c9027dbbc28a9_MIT18_03S10_chapter_15.pdf

Cited passages: printed pp. 72–73, common homogeneous equation (1), initial-data system (3), determinant (4) and zero-or-nowhere-zero theorem. The final PDF page identifies the Spring 2010 course.

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

A standard form of the Sullivan conjecture, proved by Haynes Miller, states that for a finite group $G$ and a connected finite-dimensional CW complex $X$, the pointed mapping space from the classifying space of a discrete group $BG$ to $X$ is a weakly contractible space:Thus every basepoint-preserving continuous map $BG\to X$ is null-homotopic, and higher homotopy families of such maps are trivial as well. Miller's Theorem A allows the more general assumption that every finitely generated subgroup of $G$ is finite.^[ref]

The finite-dimensional target hypothesis is substantial. For $G=\mathbb{Z}/2\mathbb{Z}$, maps from $BG$ to a finite-dimensional sphere satisfy the theorem, while the identity map of $BG$ is not null-homotopic: it induces the identity on its nontrivial fundamental group. That target $BG$ is infinite-dimensional. The theorem concerns these unstable mapping spaces; it cannot simply be transferred to every stable mapping construction.