New theorem shows embedding restrictions for manifold skeletons.
problem Embedding restrictions for triangulated manifolds.
method Proves van Kampen-Flores theorem for manifolds with specific Stiefel-Whitney classes.
result Triangulated manifolds with non-trivial Stiefel-Whitney classes cannot embed into R2d. Computes a new metric quantity Y(M) for Riemannian 2d-manifolds.
problem No simple metric quantity exists for Riemannian manifolds.
method Defines Y(M) and Y_disc(M) involving sectional curvatures and computes them for specific manifolds.
result Y(M) and Y_disc(M) differ from the Euler characteristic and can be positive or negative.
We describe rules for building 2d theories labeled by 4-manifolds. Using the proposed dictionary between building blocks of 4-manifolds and 2d N=(0,2) theories, we obtain a number of results, which include new 3d N=2 theories T[M_3] associated with rational homology spheres and new results for Vafa-Witten partition fun…
Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
problem Derive bihamiltonian structure for rational reduction of 2D-Toda hierarchy
method Direct computations
result Derive local bihamiltonian structure
Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.
problem Local bihamiltonian structures and Frobenius manifolds for asymmetric rational reductions of 2D-Toda hierarchy.
method Construct three-dimensional generalized Frobenius manifold, relate to other hierarchies via transformations.
result Explicit relation between RR2T and bi-graded Toda and constrained KP hierarchies.
Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.
problem Determining the domain of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds with tangency points.
method Using tools from Lie groupoids, natural domains of perturbations are found.
result Method allows treatment of geometries with tangency points.
We build a connection between topology of smooth 4-manifolds and the theory of topological modular forms by considering topologically twisted compactification of 6d (1,0) theories on 4-manifolds with flavor symmetry backgrounds. The effective 2d theory has (0,1) supersymmetry and, possibly, a residual flavor symmetry. …
The Grove-Searle theorem on 2d manifolds with 8 or less symmetry groups has positive Euler characteristic.
problem Proving positive Euler characteristic for 2d manifolds with specific symmetry groups.
method Direct proof and analysis of fixed point components N with geodesic properties.
result Fixed point components N have amazing geodesic properties and can be S^2, RP^2, CP^d, HP^d, etc.
We propose that a simple, Lagrangian 2d N=(0,2) duality interface between the 3d N=2 XYZ model and 3d N=2 SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to flesh out a dictionary between more general triangulated 4-manifolds with boundar…
The study constructs AdS manifolds from Gromov-Thurston manifolds.
problem Creating hyperbolic and anti-de Sitter structures from Gromov-Thurston manifolds.
method Explicit correspondence between quasifuchsian AdS manifolds and compact quotients of Ø(2d,2)/U(d,1).
result Existence of quasifuchsian AdS manifolds and hyperbolic ends with specified boundary.
Index expectation curvature K(x) = E[i_f(x)] on a compact Riemannian 2d-manifold M is the expectation of Poincare-Hopf indices i_f(x) and so satisfies the Gauss-Bonnet relation that the interval of K over M is Euler characteristic X(M). Unlike the Gauss-Bonnet-Chern integrand, such curvatures are in general non-local. …
For integers d≥2 and ε=0 or 1, let S1,d−1(ε) denote the sphere product S1×Sd−1 if ε=0 and the twisted Sd−1 bundle over S1 if ε=1. The main results of this paper are: (a) if d≡ε (mod 2) then S1,d−1(ε) has a unique minimal triangulation using 2d+3 …
Following the previous authors works (joint with I.A.Dynnikov) we develop a theory of the discrete analogs of the differential-geometrical (DG) connections in the triangulated manifolds. We study a nonstandard discretization based on the interpretation of DG Connection as linear first order (''triangle'') difference eq…
Study on a specific type of Riemannian manifolds constructed from 2D space-forms.
problem Characterizing and understanding new types of Riemannian manifolds.
method Constructed as a product of a real line and a 2-dimensional Riemannian space-form, with metrics derived from cone and hyperbolic extensions.
result Characterized and studied in terms of their curvature properties.
Proposes TS-NMF for 2D clustering, preserving spatial info.
problem Loss of spatial information in 2D data.
method Semi-Nonnegative Matrix Factorization with manifold learning.
result Improves clustering performance compared to state-of-the-art.
Researchers study Killing superalgebras in 2D manifolds.
problem Characterizing Killing superalgebras in 2D pseudo-Riemannian manifolds.
method Computing Spencer cohomology and filtered deformations of superalgebras.
result Killing superalgebras arise as solutions for geometric and skew-Killing spinors.
Existing techniques to compress point cloud attributes leverage either geometric or video-based compression tools. We explore a radically different approach inspired by recent advances in point cloud representation learning. Point clouds can be interpreted as 2D manifolds in 3D space. Specifically, we fold a 2D grid on…
The paper develops a theory linking Hamiltonian and quasi-Hamiltonian manifolds.
problem Understanding the deformation of Hamiltonian quasi-Poisson manifolds to Hamiltonian Poisson manifolds.
method Introduces a generalized Hamiltonian deformation theory and constructs a topological quantum field theory.
result Shows that the imploded cross section of the double $D(G)_\imp$ deforms to the implosion of the cotangent bundle $T^*G_\imp$.
We show how wall-crossing formulas in coupled 2d-4d systems, introduced by Gaiotto, Moore and Neitzke, can be interpreted geometrically in terms of the deformation theory of holomorphic pairs, given by a complex manifold together with a holomorphic vector bundle. The main part of the paper studies the relation between …
This paper is concerned with lower bounds for the connectivity of graphs (one-dimensional skeleta) of triangulations of compact manifolds. We introduce a structural invariant b_M for simplicial d-manifolds M taking values in the range 0 <= b_M <= d-1. The main result is that b_M influences connectivity in the following…
Generic singularities of line fields have been studied for lines of principal curvature of embedded surfaces. In this paper we propose an approach to classify generic singularities of general line fields on 2D manifolds. The idea is to identify line fields as bisectors of pairs of vector fields on the manifold, with re…
This paper proposes grid cells encode position via a conformal isometric embedding of 2D physical space.
problem Hexagonal grid firing patterns in grid cells.
method Learning a distance-preserving position embedding in neural space using a recurrent neural network.
result The conformal isometric embedding of 2D physical space into neural space explains hexagonal grid firing patterns.
We show that the smooth geometry of a hyperbolic 3-manifold emerges from a classical spin system defined on a 2d discrete lattice, and moreover show that the process of this "dimensional oxidation" is equivalent with the dimensional reduction of a supersymmetric gauge theory from 4d to 3d. More concretely, we propose a…
New geometric perspective for optimal learning on hexagonal structures.
problem Optimal learning process on hexagonal structures.
method Local trivial fibrations and Ceva's theorem.
result Learning can be defined on hexagonal structures.
Formula for sectional curvature on 2D Lorentzian manifolds derived.
problem Calculating sectional curvature on 2D Lorentzian manifolds.
method Obtained a formula for sectional curvature.
result Formula for sectional curvature on 2D Lorentzian manifolds.
In this work we reduce undersampling artefacts in two-dimensional (2D) golden-angle radial cine cardiac MRI by applying a modified version of the U-net. We train the network on 2D spatio-temporal slices which are previously extracted from the image sequences. We compare our approach to two 2D and a 3D Deep Lear…
2D CNNs approximate Korobov functions with near-optimal rates.
problem Approximating Korobov functions using 2D CNNs.
method Constructive approach for 2D CNNs with ReLU activations and fully connected layers.
result 2D CNNs achieve near-optimal approximation rates for Korobov functions.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
The study classifies discrete pseudomanifolds with up to 2d+7 vertices.
problem Understanding discrete pseudomanifolds with a small number of vertices.
method Proved existence of at least 2(d+1) vertices, classified up to 2d+6 vertices, established equivalence with edge graphs of flag normal pseudomanifolds.
result Every flag normal d-pseudomanifold with at most 2d+7 vertices is either a simplicial d-sphere or a flag triangulation of the (d-2)-fold suspension of RP^2.
Arboricity of manifolds is explored, with specific results for 2D surfaces.
problem Understanding the arboricity of different types of manifolds.
method Analyzing discrete 2-spheres, other 2D surfaces, and d-manifolds of higher dimensions.
result Arboricity of 2D surfaces is 3 or 4, and for higher dimensions, it can be arbitrarily large.
We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobeniu…
Proposes a model to generate 3D-aware images from 2D images.
problem Generating 3D-aware images from 2D images.
method Likelihood-based top-down model using Neural Radiance Fields and energy-based latent variables.
result Model can infer 3D object structures from 2D images and generate novel views.
Method counts connected 2D stratifolds with singular curves and components.
problem Counting 1-connected trivalent 2-stratifolds. method Describes a method for counting.
result Counts 1-connected trivalent 2-stratifolds. Enhances 2D face recognition with 3D features using active illumination.
problem Improving robustness of 2D face recognition to spoofing attacks and low-light conditions.
method Projecting a high spatial frequency pattern onto the face to recover 3D information and a 2D image simultaneously.
result Significantly boosts face recognition performance and dramatically improves robustness to spoofing attacks.
Quantization on even-dimensional compact manifolds using cell decomposition.
problem Quantization of compact even-dimensional manifolds.
method Cell decomposition and embedding in CP^d, inducing local Poisson structure and star product.
result Achieved Berezin-type quantization on compact even-dimensional manifolds.
In this paper we address several aspects of flat Bogomolnyi-Prasad-Sommerfeld (BPS) domain walls together with their Lorentz invariant vacua of 4d N=1 supergravity coupled to a chiral multiplet. The scalar field spans a one-parameter family of 2d Kähler manifolds satisfying a Kähler-Ricci flow equation. We find that BP…
For many automated driving functions, a highly accurate perception of the vehicle environment is a crucial prerequisite. Modern high-resolution radar sensors generate multiple radar targets per object, which makes these sensors particularly suitable for the 2D object detection task. This work presents an approach to de…
New method reconstructs 3D shapes from 2D images using Kendall's shape space.
problem Reconstruct 3D shapes from 2D images, especially for rare specimens.
method Kendall's shape space approach with prior information.
result More robust and plausible shapes compared to previous methods.
The paper establishes T-duality for 2D σ-models with H-flux.
problem T-duality for 2D σ-models with H-flux.
method Localization and graded T-duality map (graded Hori morphism).
result Establishes the most general version of T-duality for Type II String Theory.
Study natural invariants for third order nonlinear operators on 2D manifolds.
problem Equivalence problem of third order nonlinear differential operators.
method Description of rational natural differential invariants.
result Application of natural invariants to equivalence problem.
Researchers prove rigidity of 2D manifolds from boundary geodesic lengths.
problem Reconstructing a Riemann surface from boundary geodesic lengths.
method Re-casting lens data as generalized Riemannian circles and solving a system of equations.
result Essentially optimal results on boundary and lens rigidity for 2D manifolds.
Transformer-M learns molecular data in 2D or 3D formats.
problem Learning models for molecules are limited to specific data formats.
method Developed a Transformer-based model that can handle 2D and 3D molecular data.
result Transformer-M achieves strong performance on both 2D and 3D molecular tasks.
Study 2D viscoelastic equations using Lie group theory.
problem Symmetry classification and reduction of 2D viscoelastic equations.
method Investigation through Lie group theory, including algebra of symmetries and optimal subalgebras.
result Classification of reductions of similarities related to Lie subalgebras.
DISPR uses diffusion models to predict 3D cell shapes from 2D images.
problem Predicting 3D cell shapes from 2D microscopy images.
method Diffusion model trained to predict 3D shapes from 2D microscopy images as a prior.
result Adding DISPR predictions to minority cell classes improves classification accuracy.
In this paper, we present our general results about traversing flows on manifolds with boundary in the context of the flows on surfaces with boundary. We take advantage of the relative simplicity of 2D-worlds to explain and popularize our approach to the Morse theory on smooth manifolds with boundary, in which the bo…
iSTFTNet2 improves iSTFTNet's speed and lightness with 1D-2D CNN.
problem Efficiently synthesizing high-fidelity speech.
method Improved iSTFTNet using 1D-2D CNNs for temporal and spectrogram structures.
result iSTFTNet2 is faster and more lightweight with comparable speech quality.
Paper extends 2D ZSAD to 3D MRI without training, achieving robust anomaly detection.
problem Challenges in extending zero-shot anomaly detection to 3D medical images.
method Constructs localized volumetric tokens by aggregating 2D slices processed by 2D foundation models.
result Training-free, batch-based ZSAD effectively extends from 2D encoders to full 3D MRI volumes.
Optimizes master faces for 2D and 3D face verification using evolutionary algorithms and neural networks.
problem Impersonation attacks using master faces for face-based identity authentication.
method Evolutionary algorithm in latent space of StyleGAN, neural network to direct search, 2D and 3D face reconstruction.
result Obtains high impersonation rates with fewer master faces for 2D and 3D face verification.