One of my profs mentioned that sometimes people formulate theories about some type of object, but then later realize that those objects do not exist. Can someone given me an example of such a theory? I know this is vague, but I hope it makes sense.
This popular article describes something that fits the bill:
Researchers developed what Ravenel calls an entire “cosmology” of conjectures based on the assumption that manifolds with Arf-Kervaire invariant equal to 1 exist in all dimensions of the form 2n – 2. Many called the notion that these manifolds might not exist the “Doomsday Hypothesis,” as it would wipe out a large body of research. Earlier this year, Victor Snaith of the University of Sheffield in England published a book about this research, warning in the preface, “… this might turn out to be a book about things which do not exist.”
Just weeks after Snaith’s book appeared, Hopkins announced on April 21 that Snaith’s worst fears were justified: that Hopkins, Hill and Ravenel had proved that no manifolds of Arf-Kervaire invariant equal to 1 exist in dimensions 254 and higher. Dimension 126, the only one not covered by their analysis, remains a mystery.