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We compute the compactly supported Euler characteristic of the space of degree $d$ irreducible polynomials in $n$ variables with real coefficients and show that the values are given by the digits in the so-called balanced binary expansion of the number of variables $n$.
The Euler characteristic of a semialgebraic set can be considered as a generalization of the cardinality of a finite set. An advantage of semialgebraic sets is that we can define negative sets to be the sets with negative Euler characteristics. Apply
Baez asks whether the Euler characteristic (defined for spaces with finite homology) can be reconciled with the homotopy cardinality (defined for spaces with finite homotopy). We consider the smallest infinity category $text{Top}^text{rx}$ containing
A well-known question by Gromov asks whether the vanishing of the simplicial volume of oriented closed connected aspherical manifolds implies the vanishing of the Euler characteristic. We study vario
We investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the matter representation of the associated Lie algebra. The main tool is the analysis and the
We resolve two long-standing and closely related problems concerning stably free $mathbb{Z} G$-modules and the homotopy type of finite 2-complexes. In particular, for all $k ge 1$, we show that there exists a group $G$ and a non-free stably free $mat