141 problems

Airy edge conjecture. The edge limit of the additions should still be the Airy $beta$-process under suitable rescaling, even when some $kappa_l$ are negative.

Let $M_n$ be drawn from the fixed-trace Gaussian Unitary Ensemble, and let $\lambda_1^n<\cdots<\lambda_n^n$ be the eigenvalues of $M_n/\sqrt n$. Define the centered empirical measu…

Let $(U_t)_{t\geq 0}$ and $(G_t)_{t\geq 0}$ denote the unitary and general linear multiplicative Brownian motions, respectively. The process $(U_t)_{t\geq 0}$ converges in $*$-dist…

For $r\in(1,\infty]$, let $L(\mathbb F_r)$ be the interpolated free group factor, and let the first-order fundamental group of $L(\mathbb F_r)$ be the group of positive scalars $t$…

For each permutation $\pi$, let $K_{\pi}$ be the universal Kerov character polynomial defined by … For a cycle of length $k$, write $K_k$ for the corresponding polynomial. Kerov's…

Let $\mu^{\mathcal{R}(\theta)}_{a_1,a_2}$ denote a solution of the product-formula measure problem for parameters $a_1,a_2\in\mathbb{R}^N$, with $\mathcal{R}$ a finite root system…

Free infinite divisibility conjecture. The free infinite divisibility of $\nu_{\alpha,q}$ is equivalent to the existence of its radial Bargmann measure.

Functional-relations conjecture. For every $2g-2+n\geq 0$, these differentials satisfy the functional relations of Theorem, after those relations are expressed as relations between…

Let $\chi_\mathrm{orb}$ denote orbital free entropy and let $i^*$ denote free mutual information for tuples of subalgebras in a tracial $W^*$-probability space. Orbital free entrop…

Let $D$ be the tensor rank, let ${\bm{\sigma}}=(\sigma_1,\ldots,\sigma_D)\in S_n^D$ be a trace-invariant, and let $K_\mathrm{p}({\bm{\sigma}},\eta)$ denote the number of pure conne…

Let $X$ be a free manifold in the sense of chapter 13, and consider an algebra of type $L^\infty(X)$. Free-manifold factor conjecture. The algebra $L^\infty(X)$ is a ${\rm II}_1$ f…

Let $p$ and $q$ be degree-$d$ polynomials, let $p\boxplus_d q$ denote their finite free convolution, and let $\operatorname{Dis}(p)$ denote the discriminant of $p$. Minkowski inequ…

Let $X$ be a tuple of non-commutative random variables, let $S$ be a free semicircular family free from $X$, and let $\chi$ and $\chi^*$ denote Voiculescu's microstates and non-mic…

Let $T$ be the DT-operator obtained as the limit in $*$-moments of the strictly upper triangular random matrices described above, with respect to the trace $\tau$. Dykema–Haagerup…

Let bi-non-crossing partitions be the combinatorial objects introduced by Mastnak and Nica, and let the associated $(l,r)$-cumulants be the cumulants indexed by these partitions. M…

Let $V$ be the double-well potential … where $g$ is a negative constant sufficiently close to zero, and let $\mu_t$ and $\mu_V$ denote the empirical measure process and the equilib…

Rotationally symmetric non-holomorphic N-transform conjecture. The two $N$-transforms are related by

Orbital free entropy mutual information conjecture. It would be further conjectured that

Projection mutual free information conjecture. The quantity $-\chi_\mathrm{proj}(p,q)$ should coincide with

q-Meixner conjecture. The Bernoulli distribution should be $q$-Meixner for every $q$ with

Let $r,s$ be positive integers, let $\Phi(x,j)$ be the generating-function expression used in the preceding propositions, and write $[x^{r+1}]$ and $[y^{s+1}]$ for coefficient extr…

Scaling conjecture. One might conjecture that

Let $\delta^*$ denote the non-microstates free entropy dimension and let $\delta_0$ denote the microstates free entropy dimension. Free entropy dimension equality conjecture. It is…

Let $d$ be the chaos degree, and let the upper estimate of Theorem C have a constant depending on $d$. Polynomial growth conjecture. The growth of this constant should be polynomia…

Let $X$ and $Y$ be colored graphs. Write $X*Y$ for the colored graph obtained by putting a copy of $X$ at each vertex of $Y$, let $\mu(Z)$ denote the spectral measure of the charac…