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Aperiodic crystals are the intermediates between strictly periodic crystalline matter and amorphous solids. The lack of translational symmetry combined with intrinsic long-range order endows aperiodic crystals with unique physical characteristics, while at the same time dramatically enriching the spectrum and localization properties. Here, we demonstrate exciton-polariton condensation in a two-dimensional Penrose tiling with $C_{10}$ rotational symmetry - the first signature of quasicrystalline order in a quantum fluid of light. We identify a regime, wherein near-perfect delocalization and synchronization of a quantum fluid of light occurs at mesoscopic length-scales extending beyond 100x the healing length and the size of each individual condensate. Realizing long-range order in fully reconfigurable aperiodic crystals of nonlinear, and open-dissipative quantum fluids, lays the foundations for testing a broad range of universality classes of continuous phase transitions beyond the limits of mathematically verifiable models in regular lattices.

The "Poor Man's Majorana" [Phys. Rev. B 86, 134528 (2012)] devoid of topological protection has been theoretically predicted to rely on the minimal Kitaev chain. Afterward, a pair of superconducting and spinless quantum dots turned the proposal practicable and differential conductance pinpointed consistent fingerprints with such a scenario [Nature 614, 445 (2023) and Nature 630, 329 (2024)]. In this work, we propose a model wherein the "Poor Man's Majorana" presents protection when one of the dots is exchange coupled to a quantum spin. If this quantum dot is perturbed by tuning the exchange coupling, the well-known spill over-like behavior of this Majorana surprisingly remains unchanged, and solely half of the fine structure is unexpectedly viewed. As a matter of fact, the "Poor Man's Majorana" zero mode consists in squeezing of the other half at zero frequency, which imposes its pinning there and prevents the mixing of the mode with the explicit fine structure. We claim that if the supposed unavoidable split of the zero mode by the fine structure is unexpectedly absent, then the "Poor Man's Majorana" can be considered robust against the quantum spin. In this way, it becomes protected and the lack of topological protection paradigm of the "Poor Man's Majorana" has been revisited, pushing this seemingly well-established issue into a new direction.

Exploring the non-Hermitian properties of semiconductor materials for optical applications is at the forefront of photonic research. However, the selection of appropriate systems to implement such photonic devices remains a topic of debate. In this work, we demonstrate that a perovskite crystal, characterized by its easy and low-cost manufacturing, when placed between two distributed Bragg reflectors with an air gap, can form a natural double microcavity. This construction shows promising properties for the realisation of novel, tunable non-Hermitian photonic devices through strong light-matter coupling. We reveal that such a system exhibits double-coupled polariton modes with dispersion including multiple inflection points. Owing to its non-Hermiticity, our system exhibits nonreciprocal properties and allows for the observation of exceptional points. Our experimental studies are in agreement with the theoretical analysis based on coupled mode theory and calculations based on transfer matrix method., Comment: Accepted manuscript (AM)

Bound states in the continuum have recently been utilized in photonic crystal gratings to achieve strong coupling and ultralow power-driven condensation of bosonic exciton-polariton quasiparticles with atypical Dirac-like features in their dispersion relation. Here, we develop the single- and many-body theory of these new effective relativistic exciton-polaritons modes and describe their mean field condensation dynamics facilitated by the interplay between protection from the radiative continuum and negative-mass pump induced optical trapping. Our theory accounts for many tunable grating parameters giving full control over the diffractive coupling properties between guided polaritons and the radiative continuum previously unexplored in the context of driven condensation. In particular, we discover stable cyclical condensate solutions mimicking a driven-dissipative analog of the zitterbewegung effect characterized by coherent superposition of both ballistic (rapid phase front) and trapped (slow phase front) polariton waves. Finally, important distinctions are drawn between the concepts of near field and far field in the photonic grating, clarifying recent experimental observations on the emission characteristics of these long lived nonlinear Dirac polaritons.

We study the magneto-photoluminescence of an optically trapped exciton-polariton condensate in a planar semiconductor microcavity with multiple In0.08Ga0.92As quantum wells. Extremely high condensate coherence time and continuous control over the polariton confinement are among the advantages provided by optical trapping. This allows us to resolve magnetically induced {\mu}eV fine-energy shifts in the condensate and identify unusual dynamical regions in its parameter space. We observe polariton Zeeman splitting and, in small traps with tight confinement, demonstrate its full parametric screening when the condensate density exceeds a critical value, reminiscent of the spin-Meissner effect. For larger optical traps, we observe a complete inversion in the Zeeman splitting as a function of power, underlining the importance of condensate confinement and interactions with its background reservoir excitons.

We generalize the Bernstein-Walsh-Siciak theorem on polynomial approximation in $\mathbb{C}^n$ to the case where the polynomial ring $\mathcal{P}(\mathbb{C}^n)$ is replaced by a subring $\mathcal{P}^S(\mathbb{C}^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of $\mathbb{R}_+^n$ with $0 \in S$ and $m = 0, 1, 2, 3, \dots$, and uniform estimates of error in the approximation are replaced by weighted uniform estimates with respect to an admissible weight function., Comment: 10 pages, 1 figure

We report on the realization of all-optical planar microlensing for exciton-polariton condensates in semiconductor microcavities. We utilize spatial light modulators to structure a nonresonant pumping beam into a planoconcave lens-shape focused onto the microcavity plane. When pumped above condensation threshold, the system effectively becomes a directional polariton antenna, generating an intense focused beam of coherent polaritons away from the pump region. The effects of pump intensity, which regulates the interplay between gain and blueshift of polaritons, as well as the geometry of lens-shaped pump are studied and a strategy to optimize the focusing of the condensate is proposed. Our work underpins the feasibility to guide nonlinear light in microcavities using nonresonant excitation schemes, offering perspectives on optically reprogrammable on-chip polariton circuitry.

We study the separatrices at the origin of the vector field $-\nabla \log |f|$, where $f$ is any plane curve singularity. Under some genericity conditions on the metric, we produce a natural partition of the set of separatrices, $S$, into a finite collection smooth strata. As a byproduct of this theory, we construct a smooth fibration which is equivalent to the Milnor fibration, and lives on a quotient of the Milnor fibration at radius $0$. The strict transform of $S$ in this space induces a spine for each fiber of this fibration. These fibers are endowed with a vector field in such a way that the spine consists of trajectories which do not escape through the boundary., Comment: 102 pages, example added, spelling errors corrected

The classical Siciak-Zakharyuta theorem states that the Siciak-Zakharyuta function $V_{E}$ of a subset $E$ of $\mathbb C^n$, also called a pluricomplex Green function or global exremal function of $E$, equals the logarithm of the Siciak function $\Phi_E$ if $E$ is compact. The Siciak-Zakharyuta function is defined as the upper envelope of functions in the Lelong class that are negative on $E$, and the Siciak function is the upper envelope of $m$-th roots of polynomials $p$ in $\mathcal{P}_m(\mathbb C^n)$ of degree $\leq m$ such that $|p|\leq 1$ on $E$. We generalize the Siciak-Zakharyuta theorem to the case where the polynomial space ${\mathcal P}_m(\mathbb C^n)$ is replaced by ${\mathcal P}_m^S(\mathbb C^n)$ consisting of all polynomials with exponents restricted to sets $mS$, where $S$ is a compact convex subset of $\mathbb R^n_+$ with $0\in S$. It states that if $q$ is an admissible weight on a closed set $E$ in $\mathbb C^n$ then $V^S_{E,q}=\log\Phi^S_{E,q}$ on $\mathbb C^{*n}$ if and only if the rational points in $S$ form a dense subset of $S$., Comment: 18 pages, 0 figures

The main result of this paper is that an entire function $f$ that is in $L^2(\mathbb C^n,\psi)$ with respect to the weight $\psi(z)=2mH_S(z)+\gamma\log(1+|z|^2)$ is a polynomial with exponents in $m\widehat S_\Gamma$. Here $H_S$ is the logarithmic supporting function of a compact convex set $S\subset \mathbb R^n_+$ with $0\in S$, $\gamma\geq 0$ is small enough in terms of $m$, and $\widehat S_\Gamma$ is the hull of $S$ with respect to a certain cone $\Gamma$ depending on $S$, $m$ and $\gamma$. An example showing that in general $\widehat S_\Gamma$ can not be replaced by $S$ is constructed., Comment: 7 pages, 2 figures

This paper is a survey of plurisubharmonic theory where the usual polynomial ring is replaced by a polynomial ring $\mathcal P^S(\mathbb C^n)$ where the $m$-th degree polynomials have exponents restricted to $mS$, where $S\subseteq \mathbb R^n_+$ is compact, convex and $0\in S$. We assume no other conditions on $S$ as these are necessary for $\mathcal P^S(\mathbb C^n)$ to be a graded polynomial ring. We study the relationship between $\mathcal P^S(\mathbb C^n)$ and the class $\mathcal L^S(\mathbb C^n)$ of global plurisubharmonic functions where the growth is determined by the logarithmic supporting function of $S$. We present properties of their respective weighted extremal functions $\Phi_{K, q}^S$ and $V_{K, q}^S$ in connection with properties of $S$., Comment: 33 pages, 1 figure. Version 3: The article has been reorganized and various improvements made, most notably Theorem 3.6, Theorem 5.8 and results on regularity in Sections 4 and 5. Article arXiv:2305.06847, on characterization of polynomials by L^2-estimates, has merged into the article as Section 7

External driving of spinor degrees of freedom by magnetic or optical fields in quantum systems underpin many applications ranging from nuclear magnetic resonance to coherent state control in quantum computing. Although spinor polariton condensates are offering a flexible platform for spinoptronic applications, strong inter-particle interactions limit their spin coherence. Here, we introduce an all-optically driven spin precession in microcavity polariton condensates that eliminates depolarisation, through a radio frequency modulation of a spatially rotating, asymmetric exciton reservoir that both confines, and actively replenishes the polariton condensate. We realise several GHz driven spin precession with a macroscopic spin coherence time that is limited only by the extraneous to the condensate, frequency drift of the composite pumping sources. Our observations are supported by mean field modelling and evidence a driven-dissipative quantum fluidic analogue of the nuclear magnetic resonance effect.

We propose a vortex Chern insulator, motivated by recent experimental demonstrations on programmable arrangements of cavity polariton vortices by [Alyatkin et al., ArXiv:2207.01850 (2022)] and [Wang et al., National Sci. Rev. 10, Nwac096 (2022)]. In the absence of any external fields, time-reversal symmetry is spontaneously broken through polariton condensation into structured arrangements of localized co-rotating vortices. We characterize the response of the rotating condensate lattice by calculating the spectrum of Bogoliubov elementary excitations and observe the crossing of edge-states, of opposite vorticity, connecting bands with opposite Chern numbers. The emergent topologically nontrivial energy gap stems from inherent vortex anisotropic polariton-polariton interactions and does not require any spin-orbit coupling, external magnetic fields, or elliptically polarized pump fields.

The insensitivity of photons towards external magnetic fields forms one of the hardest barriers against efficient magneto-optical control, aiming at modulating the polarization state of light. However, there is even scarcer evidence of magneto-optical effects that can spatially modulate light. Here, we demonstrate the latter by exploiting strongly coupled states of semimagnetic matter and light in planar semiconductor microcavities. We nonresonantly excite two spatially adjacent exciton-polariton condensates which, through inherent ballistic near field coupling mechanism, spontaneously synchronise into a dissipative quantum fluidic supermode of definite parity. Applying a magnetic field along the optical axis, we continuously adjust the light-matter composition of the condensate exciton-polaritons, inducing a supermode switch into a higher order mode of opposite parity. Our findings set the ground towards magnetic spatial modulation of nonlinear light., Comment: Author's Accepted Manuscript: 9 pages, 6 figures

Vortices are topologically distinctive objects appearing as phase twists in coherent fields of optical beams and Bose-Einstein condensates. Structured networks and artificial lattices of coupled vortices could offer a powerful platform to study and simulate interaction mechanisms between constituents of condensed matter systems, such as antiferromagnetic interactions, by replacement of spin angular momentum with orbital angular momentum. Here, we realize such a platform using a macroscopic quantum fluid of light based on exciton-polariton condensates. We imprint all-optical hexagonal lattice that results into a triangular vortex lattice, with each cell having a vortex of charge l = +-1. We reveal that pairs of coupled condensates spontaneously arrange their orbital angular momentum antiparallel, implying a form of artificial orbital "antiferromagnetism". We discover that correlation exists between the emergent vortex patterns in triangular condensate lattices and the low-energy solutions of the corresponding antiferromagnetic Ising system. Our study offers a path toward spontaneously ordered vortex arrays with nearly arbitrary configurations and controlled couplings.

We introduce and develop the theory of Newton nondegenerate local Weil divisors $(X,0)$ in toric affine varieties. We characterize in terms of the toric combinatorics of the Newton diagram different properties of such singular germs: normality, Gorenstein property, or being an Cartier divisor in the ambient space. We discuss certain properties of their (canonical) resolution $\tilde{X}\to X $ and the corresponding canonical divisor. We provide combinatorial formulae for the delta--invariant $\delta(X,0)$ and for the cohomology groups $H^i(\tilde X,\mathcal{O}_{\tilde X})$ for $i>0$. In the case $\dim(X,0)=2$, we provide the (canonical) resolution graph from the Newton diagram and we also prove that if such a Weil divisor is normal and Gorenstein, and the link is a rational homology sphere, then the geometric genus is given by the minimal path cohomology, a topological invariant of the link., Comment: 37 pages, 13 figures

In this paper, we describe a new national language technology programme for Icelandic. The programme, which spans a period of five years, aims at making Icelandic usable in communication and interactions in the digital world, by developing accessible, open-source language resources and software. The research and development work within the programme is carried out by a consortium of universities, institutions, and private companies, with a strong emphasis on cooperation between academia and industries. Five core projects will be the main content of the programme: language resources, speech recognition, speech synthesis, machine translation, and spell and grammar checking. We also describe other national language technology programmes and give an overview over the history of language technology in Iceland., Comment: Accepted at LREC 2020

We compare three naturally occurring multi-indexed filtrations of ideals on the local ring of a Newton nondegenerate hypersurface surface singularity with rational homology sphere, which in many cases are all distinct. These are the divisorial, the order, and the image filtrations. These filtrations are indexed by the lattice associated with a toric partial resolution of the singularity, or equivalently, the free abelian group generated by the compact facets of the Newton polyhedron. We prove that there exists a top dimensional cone contained in the Lipman cone having the property that the three ideals indexed by order vectors from this cone coincide. As a corollary, if a periodic constant can be associated with the Hilbert series associated with these filtrations on the Lipman cone, then they coincide. In some cases, the Poincar\'e series associated with one of these filtrations has been shown to coincide with a zeta function associated with the topological type of the singularity. In the end of the article, we show that this is the case for all three filtrations considered in the case of a Newton nondegenerate suspension singularity. As a corollary, in this case, the zeta function provides a direct method of determining the Newton diagram from the link., Comment: 19 pages, 2 figures

T\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed t\^ete-\`a-t\^ete graphs provide a generalization which define mixed t\^ete-\`a-t\^ete twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed t\^ete-\`a-t\^ete twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of t\^ete-\`a-t\^ete twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities., Comment: 21 pages, 14 figures. Minor corrections. Version as accepted in journal. arXiv admin note: text overlap with arXiv:1706.05580

Let $f,g\in\mathbb{C}\{x,y\}$ be germs of functions defining plane curve singularities without common components in $(\mathbb{C}^2,0)$ and let $\Phi(x,y,z) = f(x,y) + zg(x,y)$. We give an explicit algorithm producing a plumbing graph for the boundary of the Milnor fiber of $\Phi$ in terms of a common resolution for $f$ and $g$., Comment: 22 pages, added references to introduction

Given a normal surface singularity (X,0), its link, M is a closed differentiable three dimensional manifold which carries much analytic information. It is an interesting question to ask whether, under suitable analytic and topological conditions, the geometric genus (or other analytic invariants) can be recovered from the link. The Casson invariant conjecture predicts that p_g can be identified using the Casson invariant in the case when (X,0) is a complete intersection and M has trivial first homology with integral coefficients. The Seiberg-Witten invariant conjecture predicts that the geometric genus of a Gorenstein singularity, whose link has trivial first homology with rational coefficients, can be calculated as a normalized Seiberg-Witten invariant of the link. The first conjecture is still open, but counterexamples have been found for the second one. We prove here the Seiberg-Witten invariant conjecture for hypersurface singularities given by a function with Newton nondegenerate principal part. We provide a theory of computation sequences and of the way they bound the geometric genus. Newton nondegenerate singularities can be resolved explicitly by Oka's algorithm, and we exploit the combinatorial interplay between the resolution graph and the Newton diagram to show that in each step of the computation sequence we construct, the given bound is sharp. Our method recovers the geometric genus of (X,0) explicitly from the link, assuming that (X,0) is indeed Newton nondegenerate with a rational homology sphere link. Assuming some additional information about the Newton diagram, we recover part of the spectrum, as well as the Poincar\'e series associated with the Newton filtration. Finally, we show that the normalized Seiberg-Witten invariant associated with the canonical spin^c structure on the link coincides with our identification of the geometric genus., Comment: 97 pages, 10 figures, PhD thesis at Central European University, abstract abridged

We give a description of the Milnor fiber and the monodromy of a singularity of the form f+zg = 0 where f and g define plane curves and have no common components. The description depends only on the topological type of the two plane curve germs defined by f and g. In particular, this gives a description of the boundary of the Milnor fiber. As a corollary, we give a simple formula for the monodromy zeta function and the Euler characteristic of the fiber., Comment: 13 pages

We describe the framework and strategy of the \^G infrared search for extraterrestrial civilizations with large energy supplies, which will use the wide-field infrared surveys of WISE and Spitzer to search for these civilizations' waste heat. We develop a formalism for translating mid-infrared photometry into quantitative upper limits on extraterrestrial energy supplies. We discuss the likely sources of false positives, how dust can and will contaminate our search, and prospects for distinguishing dust from alien waste heat. We argue that galaxy-spanning civilizations may be easier to distinguish from natural sources than circumstellar civilizations (i.e., Dyson spheres), although Gaia will significantly improve our capability to identify the latter. We present a "zeroth order" null result of our search based on the WISE all-sky catalog: we show, for the first time, that Kardashev Type III civilizations (as Kardashev originally defined them) are very rare in the local universe. More sophisticated searches can extend our methodology to smaller waste heat luminosities, and potentially entirely rule out (or detect) both Kardashev Type III civilizations and new physics that allows for unlimited "free" energy generation., Comment: 15 pages, 4 figures, accepted for publication in the Astrophysical Journal

We motivate the \^G infrared search for extraterrestrial civilizations with large energy supplies. We discuss some philosophical difficulties of SETI, and how communication SETI circumvents them. We review "Dysonian SETI", the search for artifacts of alien civilizations, and find that it is highly complementary to traditional communication SETI; the two together might succeed where either one, alone, has not. We discuss the argument of Hart (1975) that spacefaring life in the Milky Way should be either galaxy-spanning or non-existent, and examine a portion of his argument that we dub the "monocultural fallacy". We discuss some rebuttals to Hart that invoke sustainability and predict long Galaxy colonization timescales. We find that the maximum Galaxy colonization timescale is actually much shorter than previous work has found ($< 10^9$ yr), and that many "sustainability" counter-arguments to Hart's thesis suffer from the monocultural fallacy. We extend Hart's argument to alien energy supplies, and argue that detectably large energy supplies can plausibly be expected to exist because life has potential for exponential growth until checked by resource or other limitations, and intelligence implies the ability to overcome such limitations. As such, if Hart's thesis is correct then searches for large alien civilizations in other galaxies may be fruitful; if it is incorrect, then searches for civilizations within the Milky Way are more likely to succeed than Hart argued. We review some past Dysonian SETI efforts, and discuss the promise of new mid-infrared surveys, such as that of WISE., Comment: 18 pages, accepted for publication in the Astrophysical Journal

Using the path lattice cohomology we provide a conceptual topological characterization of the geometric genus for certain complex normal surface singularities with rational homology sphere links, which is uniformly valid for all superisolated and Newton non--degenerate hypersurface singularities.