5 problems

Convexity conjecture. For $1/2<s<1$, the function $f_s$ is convex on the set of positive semidefinite $n\times n$ matrices. Epstein's theorem gives concavity for $s\geq1$; the conj…

Cusp-form realization conjecture. Every cusp form on $SL(2,\mathbb Z)$, possibly with character, can be realized as a trace function $Z(v,q)$ for a primary field $v$ in $V$.

Arithmetic exponent pair hypothesis. For every $\epsilon>0$, provided that $|I|<q\delta$, one has

Let $v,w\in\mathfrak{B}_n$ avoid the patterns $3412$ and $4231$. Write $\Gamma(v)$ and $\Gamma(w)$ for their type-$\mathsf{BC}$ indifference graphs, and let…

Carlen–Lieb convexity conjecture. The functional $\Phi_p$ is convex for $1\le p\le 2$. Carlen and Lieb formulated this conjecture in 1999; its validity would imply a Minkowski ineq…