Major problems 1 — The biggest problem, in my opinion, is to come up with a specific vision of where homotopy theory shou…

Definitions and mathematical statement

Mark Hovey's first major item is a request to formulate a specific, coherent and somewhat accessible programme for homotopy theory, analogous in organizing role to the Weil conjectures in algebraic geometry and the Ravenel conjectures in the late 1970s. It is not a stated mathematical proposition with objects, hypotheses and a conclusion to prove.

A response in the source's intended spirit would articulate concrete conjectures and explain how their relations supply a shared direction for the field. The source gives no required list, formal adequacy criterion or unique target theorem, so no quantified existence assertion or new conjectural equivalence is substituted. "Analogous" means a comparable research-organizing role, not logical equivalence to the Weil conjectures.

Hovey says this was his motivation for making the list: a forum in which conjectures could collectively form such a programme. He distinguishes the talents of formulating conjectures from proving them, illustrates the point with Deligne/Weil and Hopkins/Ravenel, and uses Fields-Medal language rhetorically. Those personal illustrations are motivation, not mathematical conditions or a prize criterion. He also explicitly cautions that the proposed conjectures should be somewhat accessible and that the subsequent listed problems might not be.

The research summary names later chromatic-homotopy directions, Lurie's higher/derived-algebraic-geometric programme with derived moduli of formal groups or elliptic curves, and Stolz--Teichner links between elliptic cohomology and two-dimensional quantum field theory as related examples. They are reported illustrations, not a verified uniquely accepted answer or a supplied set of conjectures resolving this subjective request.

Short English statement

Hovey asks the community to formulate an accessible, unifying programme of concrete homotopy-theory conjectures. This is a forum-style research aspiration, with no objective theorem or unique solution specified.

Sources

• Local UnsolvedMath registry, permanent ID $11100001$, exact source code catalogue entry.
• Mark Hovey, Hovey's Algebraic Topology Problem List, Major problems, item 1, surviving mirror: https://www-users.cse.umn.edu/~tlawson/hovey/. Registry access 2026-07-29; html-ordered-list extraction.
• Weil, Ravenel, Deligne and Mike Hopkins appear as the source's motivational examples. The summary's chromatic, Lurie and Stolz--Teichner directions have no supplied precise bibliographic citations.