Mathematisches Kolloquium

Abstract: A revolution is underway in mathematics that will see *set* replaced as the fundamental undefined term by *anima* or animated set. It is anima that give rise to infinity-categories and breath life into everything derived. While Lurie has given a definition within set theory of the infinity-category of anima, this does little to reveal their true nature. But whatever it is that animates anima, this something equips the homotopy groups of a commutative algebra in spectra with the structure of a commutative algebra in the symmetric monoidal category of *graded* abelian groups. Being commutative, these algebras form the affine building blocks of a geometry, which we call Dirac geometry. Informally, Dirac geometry constitutes a square root of equivariant geometry for the multiplicative group, and more concretely, the grading exhibits the hallmarks of spin in that it distinguishes symmetric and anti-symmetric behavior and provides the coherent cohomology of Dirac schemes and Dirac stacks with half-integer Serre twists. This is joint work with Piotr Pstragowski.

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