M-theory and generalized cohomology

Anomalies and partition functions in physics have played a major role in uncovering the mathematics behind physical theories, such as quantum field theory. We will consider the case of string theory and M-theory and show how their anomalies and partition functions point to generalized cohomology theories such as Morava K-theory and variants of elliptic cohomology. Viewed as obstructions, anomalies also give rise to structures such as string structures and even lead to generalizations. We will also show that in cases of interest such structures, as well as the corresponding generalized cohomology theories, are in fact twisted.