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Let $B\subset \mathbb{P}^3$ be an slc quartic surface. The existence of an embedding $\mathbb{G}_m^3\hookrightarrow \mathbb{P}^3\setminus B$ implies that $B$ has coregularity zero. In this article, we initiate the classification of coregularity zero slc quartic surfaces $B\subset \mathbb{P}^3$ for which $\mathbb{P}^3\setminus B$ contains an algebraic torus $\mathbb{G}_m^3$. Equivalently, the classification of cluster type pairs $(\mathbb{P}^3,B)$. Along the way, we give criteria for a log Calabi--Yau pair $(X,B)$ over a toric variety $T$ to be of cluster type., Comment: 24 pages

Light field cameras and multi-camera arrays have emerged as promising solutions for accurately estimating depth by passively capturing light information. This is possible because the 3D information of a scene is embedded in the 4D light field geometry. Commonly, depth estimation methods extract this information relying on gradient information, heuristic-based optimisation models, or learning-based approaches. This paper focuses mainly on explicitly understanding and exploiting 4D geometrical cues for light field depth estimation. Thus, a novel method is proposed, based on a non-learning-based optimisation approach for depth estimation that explicitly considers surface normal accuracy and occlusion regions by utilising a fully explainable 4D geometric model of the light field. The 4D model performs depth/disparity estimation by determining the orientations and analysing the intersections of key 2D planes in 4D space, which are the images of 3D-space points in the 4D light field. Experimental results show that the proposed method outperforms both learning-based and non-learning-based state-of-the-art methods in terms of surface normal angle accuracy, achieving a Median Angle Error on planar surfaces, on average, 26.3\% lower than the state-of-the-art, and still being competitive with state-of-the-art methods in terms of Mean Squared Error $\vc{\times}$ 100 and Badpix 0.07.

In this work, we present new constructions for topological subsystem codes using semi-regular Euclidean and hyperbolic tessellations. They give us new families of codes, and we also provide a new family of codes obtained through an already existing construction, due to Sarvepalli and Brown. We also prove new results that allow us to obtain the parameters of these new codes.

This paper aims to study the birational geometry of log Calabi-Yau pairs$(\mathbb{P}^3, D)$ of coregularity 2, where in this case $D$ is an irreducible normal quartic surface with canonical singularities. We completely classify which toric weighted blowups of a point will initiate a volume preserving Sarkisov link starting with this pair. Depending on the type of singularity, our results point out that some of these weights do not work generically for a general member of the corresponding coarse moduli space of quartics., Comment: 40 pages, 4 figures. Derived from the PhD thesis of the author. Comments are very welcome!

This paper aims to study the decomposition group of a nonsingular plane cubic under the light of the log Calabi-Yau geometry. Using this approach we prove that an appropriate algorithm of the Sarkisov Program in dimension 2 applied to an element of this group is automatically volume preserving. From this, we deduce some properties of the (volume preserving) Sarkisov factorization of its elements. We also negatively answer a question posed by Blanc, Pan and Vust asking whether the canonical complex of a nonsingular plane cubic is split. Within a similar context em dimension 3, we exhibit in detail an interesting counterexample for a possible generalization of a theorem by Pan in which there exists a Sarkisov factorization obtained algorithmically that is not volume preserving., Comment: 30 pages, 17 figures. Derived from the PhD thesis of the author. Comments are very welcome!

We give a complete description of the Poisson boundary of wreath products $A\wr B= \bigoplus_{B} A\rtimes B$ of countable groups $A$ and $B$, for probability measures $\mu$ with finite entropy where lamp configurations stabilize almost surely. If, in addition, the projection of $\mu$ to $B$ is Liouville, we prove that the Poisson boundary of $(A\wr B,\mu)$ is equal to the space of limit lamp configurations, endowed with the corresponding hitting measure. In particular, this answers an open question asked by Kaimanovich, and Lyons-Peres, for $B=\mathbb{Z}^d$, $d\ge 3$, and measures $\mu$ with a finite first moment., Comment: 39 pages

In this paper, we utilize the framework of Markov processes to attain a more probabilistic perspective on the theory of transfer operators. In doing so, we establish a functional central limit theorem (FLCT) for an $O(N)$ model associated with Dyson potential on the one-dimensional lattice. We also proof a FLCT on a non-compact alphabet setting, for a model associated with a Dyson type potential on the one-dimensional lattice. A Breimann ergodic theorem for equilibrium measures arising from the Ruelle Operator Formalism is proved. Furthermore, we obtain a qualitative criterion (strong transitivity) to determine when the conformal measure has full support. As an application, we show how to connect the Ruelle operator framework with the perspective of the Hopf theory of Markov Processes., Comment: In this second version, we have corrected grammatical errors and made some revisions in the writing of section 5, making it more clear to read

We study random walks on the lampshuffler group $\mathrm{FSym}(H)\rtimes H$, where $H$ is a finitely generated group and $\mathrm{FSym}(H)$ is the group of finitary permutations of $H$. We show that for any step distribution $\mu$ with a finite first moment that induces a transient random walk on $H$, the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for $H=\mathbb{Z}$, the above convergence completely describes the Poisson boundary of the random walk $(\mathrm{FSym}(\mathbb{Z})\rtimes \mathbb{Z},\mu)$., Comment: 25 pages, no figures