**Aleksandar Milivojevi&cacute,
Max Planck Institute for Mathematics in Bonn**

**Title:**
Approaching Hirzebruch's prize question via rational surgery

**Abstract:** In his 1990's book on manifolds and modular forms,
Hirzebruch asked whether there exists a 24-dimensional closed spin
manifold satisfying certain conditions on its Pontryagin classes,
motivated by the observation that one could compute the dimensions of
the irreducible representations of the Monster group via certain
characteristic numbers of such a manifold. Hopkins and Mahowald showed
in the early 2000's that such manifolds exist by understanding the
bordism theory of manifolds admitting string structures (a further
"lift" of the special orthogonal group beyond spin). I will present an
alternative, relatively elementary construction of a manifold as asked
for by Hirzebruch, which also offers a large amount of flexibility in
constructing other solutions, using an adaptation of rational homotopy
theoretic results of Sullivan from the 1970's to spin manifolds.