Complexity from spinning primaries

We define circuits given by unitary representations of Lorentzian conformal field theory in 3 and 4 dimensions. Our circuits start from a spinning primary state, allowing us to generalize formulas for the circuit complexity obtained from circuits starting from scalar primary states. These results are nicely reproduced in terms of the geometry of coadjoint orbits of the conformal group. In contrast to the complexity geometry obtained from scalar primary states, the geometry is more complicated and the existence of conjugate points, signaling the saturation of complexity, remains open.

Integrated Preprocessing of Multitemporal Very-High-Resolution Satellite Images via Conjugate Points-Based Pseudo-Invariant Feature Extraction

Multitemporal very-high-resolution (VHR) satellite images are used as core data in the field of remote sensing because they express the topography and features of the region of interest in detail. However, geometric misalignment and radiometric dissimilarity occur when acquiring multitemporal VHR satellite images owing to external environmental factors, and these errors cause various inaccuracies, thereby hindering the effective use of multitemporal VHR satellite images. Such errors can be minimized by applying preprocessing methods such as image registration and relative radiometric normalization (RRN). However, as the data used in image registration and RRN differ, data consistency and computational efficiency are impaired, particularly when processing large amounts of data, such as a large volume of multitemporal VHR satellite images. To resolve these issues, we proposed an integrated preprocessing method by extracting pseudo-invariant features (PIFs), used for RRN, based on the conjugate points (CPs) extracted for image registration. To this end, the image registration was performed using CPs extracted using the speeded-up robust feature algorithm. Then, PIFs were extracted based on the CPs by removing vegetation areas followed by application of the region growing algorithm. Experiments were conducted on two sites constructed under different acquisition conditions to confirm the robustness of the proposed method. Various analyses based on visual and quantitative evaluation of the experimental results were performed from geometric and radiometric perspectives. The results evidence the successful integration of the image registration and RRN preprocessing steps by achieving a reasonable and stable performance.

Effective GUP-modified Raychaudhuri equation and black hole singularity: four models

The classical Raychaudhuri equation predicts the formation of conjugate points for a congruence of geodesics, in a finite proper time. This in conjunction with the Hawking-Penrose singularity theorems predicts the incompleteness of geodesics and thereby the singular nature of practically all spacetimes. We compute the generic corrections to the Raychaudhuri equation in the interior of a Schwarzschild black hole, arising from modifications to the algebra inspired by the generalized uncertainty principle (GUP) theories. Then we study four specific models of GUP, compute their effective dynamics as well as their expansion and its rate of change using the Raychaudhuri equation. We show that the modification from GUP in two of these models, where such modifications are dependent of the configuration variables, lead to finite Kretchmann scalar, expansion and its rate, hence implying the resolution of the singularity. However, the other two models for which the modifications depend on the momenta still retain their singularities even in the effective regime.

Closed geodesics on surfaces without conjugate points

We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points.

Complexity growth in integrable and chaotic models

We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O($$ \sqrt{N} $$ N ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.