7 problems

Let $V$ have dimension $n$, let $\rho$ be an irreducible $GL_n({\cal O})$-component of $C^\infty(Gr_{n-i,n}(\mathbb F))$, and let ${\cal D}_i$, $P_i$, and…

Let $V$ have dimension $n$, let $Val_i^\infty(V)$ denote the degree-$i$ smooth translation-invariant valuations, and identify valuations represented by functions on the relevant Gr…

Mixed hard Lefschetz and Hodge–Riemann conjecture. $Val^\infty(V)$ satisfies the mixed hard Lefschetz theorem for the subset ${\cal K}$, and it satisfies the mixed Hodge–Riemann re…

Let $k$ be a field of arbitrary characteristic and let $\Delta$ be a connected oriented simplicial pseudomanifold over $k$ of dimension $d-1$ with vertex set $V$. Let $\ell$ be the…

Let $k$ be a field of arbitrary characteristic, and let $\Delta$ be a connected oriented simplicial pseudomanifold over $k$ of dimension $d-1$ with vertex set $V$. For…

Let $k$ be a field and let $\Delta$ be a connected oriented simplicial pseudomanifold over $k$. The constructions and results above, including the odd-multiplicity Hodge–Riemann th…

Almkvist–Lefschetz strengthening. (i) For each $m$, the algebra $A(m,n)$ satisfies the SLP for all $n\geq 11$. (ii) Moreover, if $m$ is even, then $A(m,n)$ satisfies the SLP for al…