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Let me describe the simplest non-trivial case of what I have in mind. Let $V$ be a 2-dimensional $\mathbb{R}$-vector space and fix an isomorphism $V \cong \mathbb{R}^2$, where $\mathbb{R}^2$ is ...

Is the set of real $2n \times 2n$ skew-symmetric matrices having positive Pfaffians path connected? By definition, the Pfaffian is a polynomial in the entries $a_{ij}$ ($i<j$) such that $Pf(A)^2=\...