17 problems

Let $(athcal{M},\mathcal{W}^{a,b}_p)$ be the Wasserstein metric space of nonnegative measures on a finite Markov chain with state space $\mathcal{X}$, and let $t\mapsto\mu_t$ be a…

Density conjecture for compositions of optimal maps. The set of compositions of $k=4$ optimal maps is dense in the set of orientation-preserving diffeomorphisms, whereas the analog…

Let $(\mathcal{P}_2,\mathcal{W}_2)$ denote the Wasserstein space of probability measures with finite second moment, and let $\tilde g_{e,2J}$ be the cost metric considered in the p…

Mesa limit and uniform projection conjecture. The flows should converge to a Hele–Shaw or mesa-type free-boundary evolution whose terminal state is the Wasserstein projection of…

Subdifferential conjecture. The regularity property (A3) might be weakened or removed altogether by employing the Wasserstein subdifferential.

Let the density manifold be equipped with the Otto metric, and let the exponential connection be defined as the connection dual to the mixture connection with respect to this metri…

Non-equivalence conjecture. In general, spatial medians have no relation to the maximizers of $\mathrm{MSD}(Q;{\bf P})$.

Wasserstein-space rate conjecture. The sub-optimal rate obtained when the Wasserstein proximal coordinate gradient method with random updates is specialized to the Euclidean case i…

Pedrotti's conjecture. Positive Ollivier curvature, together with non-negative $\ell_\infty$ Wasserstein curvature, implies a modified log-Sobolev inequality.

The paper considers posterior contraction rates in statistical models whose parameter spaces may be equipped with Wasserstein structure, and distinguishes parameter spaces with a l…

Let $P$ be a distribution and let the polymean in the $k$-skeleton be viewed in Wasserstein space. For i.i.d. samples with distribution $P$, the conditions governing the convergenc…

Let the $q$-skeleton be considered as a subspace of $p$-Wasserstein space, and let its “medial axis” denote the singular set relevant to the distance from that skeleton. A medial-a…

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$CD(0,m)$ characterization conjecture. Then the $CD(0,m)$-condition holds, that is,

Subdifferential condition. For almost every $t\in(0,\infty)$,

Let $M$ be the compact Riemannian manifold underlying the Wasserstein space, let $(\mathcal P(M),d^{W})$ denote the space of Borel probability measures on $M$ equipped with the Was…

Let $M$ be a compact Riemannian manifold, let $\beta>0$, and let $\mathbb{P}^{\beta}\in\mathcal P(\mathcal P(M))$ be the higher-dimensional entropic measure constructed as a backgr…