Conformal Wasserstein distances: comparing surfaces in polynomial time

We present a constructive approach to surface comparison realizable by a polynomial-time algorithm. We determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation problem differs from the standard case in that we require the solution to be invariant under global M\"{o}bius transformations. We present in detail the case where the surfaces to compare are disk-like; we also sketch how the approach can be generalized to other types of surfaces.Comment: 23 pages, 3 figure

Algebraic Geometry methods associated to the one-dimensional Hubbard model

In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective $\mathrm{R}$-matrix weights lie on an Abelian surface being birational to the product of two elliptic curves with distinct $\mathrm{J}$-invariants. One of the curves is isomorphic to that of the Lax operator but the other is solely fourfold isogenous. These results clarify the reason the $\mathrm{R}$-matrix can not be written using only difference of spectral parameters of the Lax operator.Comment: 24 page

Green functions for killed random walks in the Weyl chamber of Sp(4)

We consider a family of random walks killed at the boundary of the Weyl chamber of the dual of $\rm{Sp}(4)$, which in addition satisfies the following property: for any $n\geq 3$, there is in this family a walk associated with a reflection group of order $2n$. Moreover, the case $n=4$ corresponds to a process which appears naturally by studying quantum random walks on the dual of $\rm{Sp}(4)$. For all the processes belonging to this family, we find the exact asymptotic of the Green functions along all infinite paths of states as well as that of the absorption probabilities along the boundaries.Comment: 20 page

Efficient CSL Model Checking Using Stratification

For continuous-time Markov chains, the model-checking problem with respect to continuous-time stochastic logic (CSL) has been introduced and shown to be decidable by Aziz, Sanwal, Singhal and Brayton in 1996. Their proof can be turned into an approximation algorithm with worse than exponential complexity. In 2000, Baier, Haverkort, Hermanns and Katoen presented an efficient polynomial-time approximation algorithm for the sublogic in which only binary until is allowed. In this paper, we propose such an efficient polynomial-time approximation algorithm for full CSL. The key to our method is the notion of stratified CTMCs with respect to the CSL property to be checked. On a stratified CTMC, the probability to satisfy a CSL path formula can be approximated by a transient analysis in polynomial time (using uniformization). We present a measure-preserving, linear-time and -space transformation of any CTMC into an equivalent, stratified one. This makes the present work the centerpiece of a broadly applicable full CSL model checker. Recently, the decision algorithm by Aziz et al. was shown to work only for stratified CTMCs. As an additional contribution, our measure-preserving transformation can be used to ensure the decidability for general CTMCs.Comment: 18 pages, preprint for LMCS. An extended abstract appeared in ICALP 201

Stable modification of relative curves

We generalize theorems of Deligne-Mumford and de Jong on semi-stable modifications of families of proper curves. The main result states that after a generically \'etale alteration of the base any (not necessarily proper) family of multipointed curves with semi-stable generic fiber admits a minimal semi-stable modification. The latter can also be characterized by the property that its geometric fibers have no certain exceptional components. The main step of our proof is uniformization of one-dimensional extensions of valued fields. Riemann-Zariski spaces are then used to obtain the result over any integral base.Comment: 60 pages, third version, the paper was revised due to referee's report, section 2 was divided into sections 2 and 6, to appear in JA