Parallel Mapper algorithm for efficient topological data analysis.
problem Efficient parallel processing of Mapper for topological data analysis.
method Provable correct parallel algorithm for Mapper execution on multiple processors.
result Demonstrates the efficiency of parallel Mapper compared to sequential implementations.
Compact automorphism group of a topological parallelism in real projective 3-space proved.
problem Proving compactness of automorphism group for topological parallelism.
method Direct proof without relying on earlier results.
result Automorphism group is compact.
Improved bound on groups preserving Clifford parallelism in 4 dimensions.
problem Characterizing Clifford parallelism by automorphisms.
method Improving the bound on the dimension of groups preserving Clifford parallelism.
result Improved bound to 4 dimensions for groups preserving Clifford parallelism.
The paper explores parallelisms and spreads in 3D projective space.
problem Characterizing parallelisms and spreads in PG(3,R).
method Introducing topological parallelisms, analyzing their properties and relationships with spreads.
result There are more oriented parallelisms than ordinary parallelisms, and the automorphism group is SO(3).
Compactness proven for a group of transformations in 3D space.
problem Proving compactness of a group of transformations in 3D space.
method Conjecture and proof of identity component compactness.
result Identity component of the automorphism group is compact.
New parallelisms of 3D projective space found and classified.
problem Classifying parallelisms of 3D projective space with specific automorphism groups.
method Generalized existing constructions to include oriented parallelisms and showed that only Clifford parallelism remains for higher-dimensional automorphism groups.
result Only Clifford parallelism remains for topological parallelisms with automorphism groups of dimension 3 or larger.
The study classifies parallel mean curvature spheres in a sphere-hyperbolic product space.
problem Understanding surfaces with parallel mean curvature in a specific Riemannian product space.
method Analyzing the holomorphic quadratic differential and topological constraints.
result Classification of all parallel mean curvature spheres with vanishing differential.
The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports, and transports along paths satisfying some additional conditions, are constructed. In particular, the equivalence between the concepts "conn…
Decomposes Busemann spaces into simpler structures.
problem Understanding the structure of Busemann spaces.
method Topological decomposition and isometry analysis.
result Busemann spaces can be decomposed into products of simpler spaces.
New algorithm for collaborative deep learning over fixed networks.
problem Data parallelization and decentralized computation in deep learning.
method Consensus-based distributed SGD (CDSGD) and CDMSGD algorithms for fixed topology networks.
result Demonstrated improved performance over centralized and federated learning algorithms.
NEST optimizes deep learning training by placing devices efficiently across networks and memory.
problem Inefficient device placement in distributed deep learning leads to high communication and memory overhead.
method NEST uses network-, compute-, and memory-aware dynamic programming to optimize device placement.
result NEST achieves up to 2.43 times higher throughput and better memory efficiency.
Parallel algorithm speeds up Jones polynomial computation.
problem Efficient computation of knot complexity measures.
method First parallel algorithm for exact Jones polynomial computation.
result Reduces computational time by an exponential factor.
MATCHA speeds up decentralized SGD by parallelizing communication.
problem Error-runtime trade-off in decentralized SGD.
method MATCHA decomposes network topology into matchings for efficient communication.
result MATCHA reduces communication time by up to 5x compared to vanilla decentralized SGD.
Compact ECS manifolds have a simple topological structure.
problem Understanding the structure of compact ECS manifolds.
method Review of basic facts, construction of examples, and proof of a topological structure result.
result Compact rank-one ECS manifolds are bundles over S^1 with fibers being leaves of D⊥.
We classify locally homogeneous quasi-Sasakian manifolds in dimension five that admit a parallel spinor ψ of algebraic type F⋅ψ=0 with respect to the unique connection ∇ preserving the quasi-Sasakian structure and with totally skew-symmetric torsion. We introduce a certain conformal transformation of …
In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian m…
Program connects quantum computing and topological field theories.
problem Connecting quantum computing and topological field theories.
method Formalizes the connection using cobordisms and parallel transport.
result Realizes quantum circuits as cobordisms in a double category.
Develops a braid-theoretic framework to analyze chirality in molecular knots.
problem Analyzing chirality in molecular knots constructed using circuit topology.
method Translated circuit topology approach to knot engineering into braid-theoretic framework, calculating Jones polynomial for binary combinations.
result Jones polynomial provides a powerful tool for analyzing chirality of molecular knots.
Researchers explore valuations on polyhedra and topological arrangements without imposing algebraic structures.
problem Understanding valuations on polyhedra and their connections to topological arrangements.
method Generalizes the setting of valuations on convex polyhedra to collections of defining hyperplanes without imposing algebraic structures.
result Uncovered a close relationship between scissors congruence problems and finite hyperplane arrangements.
Parallel transport in a fibre bundle with respect to smooth paths in the base space B have recently been extended to representations of the smooth singular simplicial set Sing_{smooth}(B). Inspired by these extensions,I revisit the development of a notion of `parallel' transport in the topological setting of fibrations…
We consider the more general question as to when a connection is a metric connection. There are two aspects to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric in the neighbourhood of a given point and second, the characterization of the t…
We study recursive-cube-of-rings (RCR), a class of scalable graphs that can potentially provide rich inter-connection network topology for the emerging distributed and parallel computing infrastructure. Through rigorous proof and validating examples, we have corrected previous misunderstandings on the topological prope…
Enhances parallelism in decentralized learning for larger networks.
problem Scalability limitations in decentralized learning with increasing number of machines.
method Proposes Decentralized Anytime SGD, a novel algorithm that extends parallelism threshold.
result Establishes a theoretical upper bound on parallelism surpassing current state-of-the-art.
Proof of genus formula for 3-manifolds using arithmetic topology.
problem Proving a genus formula for finite abelian branched covers over integral homology 3-spheres.
method Utilizing analogies of arithmetic topology and Hasse norm principle.
result Presentation of an Iyanaga--Tamagawa type genus formula.
This paper explores parallels between minimal surfaces and Einstein manifolds.
problem Understanding Einstein manifolds, which are less studied.
method Synthesizes parallels between minimal surfaces and Einstein four-manifolds.
result Certain Einstein four-manifolds admit a minimal immersion into a higher-dimensional sphere.
We generalize the notion of parallel transport along paths for abelian bundles to parallel transport along surfaces for abelian gerbes using an embedded Topological Quantum Field Theory (TQFT) approach. We show both for bundles and gerbes with connection that there is a one-to-one correspondence between their local des…
Formality of foliation minimal model proved for co-Kähler manifolds.
problem Formality of foliation minimal model for co-Kähler manifolds.
method Demonstrated existence of parallel forms leading to simpler computation of cohomology.
result Formality of foliation minimal model proved for co-Kähler manifolds.
Improves parallel deep model performance by restructuring and pruning.
problem Latency in parallel deep model execution due to interdependency among sub-models.
method Layer-wise model restructuring and pruning, using ℓ0 optimization and Munkres assignment algorithm. result Significantly improves efficiency of distributed inference in terms of communication and computational complexity.
The study explores properties of homologically locally connected spaces and their connections to other topological concepts.
problem Characterizing and understanding homologically locally connected spaces.
method Investigates properties and characterizations of homologically UVn-maps and lcGn-spaces, comparing them to existing concepts. result Identifies similarities and parallels between homologically locally connected spaces and other topological concepts.
Compact pseudo-Riemannian manifolds that have parallel Weyl tensor without being conformally flat or locally symmetric are known to exist in infinitely many dimensions greater than 4. We prove some general topological properties of such manifolds, namely, vanishing of the Euler characteristic and real Pontryagin classe…
Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.
problem Characterizing and classifying Lorentzian four-manifolds with parallel spinors.
method Formulated parallel spinor flow equations and used parabolic pairs theory.
result Characterized all parallel Cauchy pairs on simply connected Cauchy surfaces and classified compact three-manifolds.
The paper explores the topology and curvature of isoparametric families in spheres.
problem Investigating the topology and curvature of isoparametric families in spheres.
method The paper investigates the topology and curvature of isoparametric families in spheres using homotopy, homeomorphism, diffeomorphism types, parallelizability, and Lusternik-Schnirelmann category.
result The paper determines conditions for non-negative sectional curvatures and positive Ricci curvatures in isoparametric families.
New classes of semimetals predict torsion Fermi arcs.
problem Understanding the subtle interplay between local and global charges in topological semimetals.
method Using geometric exact sequences and cohomological semimetal invariants.
result Prediction of torsion Fermi arcs in new classes of semimetals.
We study the boundary conditions in the topologically twisted Chern-Simons matter theories with the Lie 3-algebraic structure. We find that the supersymmetric boundary conditions and the gauge invariant boundary conditions can be unified as the complexified gauge invariant boundary conditions which lead to the supergro…
New proof shows affine manifolds with parallel volume are Riemannian-flat.
problem Characterize compact affine manifolds with parallel volume.
method Construct a representative metric with Levi-Civita connection, using Hessian of volume-normalized distance functions.
result Affine manifolds with parallel volume are Riemannian-flat.
Following recent work by Ghomi, Solomon and Tabachnikov, we study geometry and topology of skew branes. A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any pair of distinct points are not parallel. We prove that if an oriented closed manifold has a non-zero Euler characterist…
New metric tensor field on symmetric matrices simplifies eigenvector computation.
problem Complex eigenvector computation for 2x2 symmetric matrices.
method Introducing a metric tensor field on the space of symmetric matrices, resulting in a curved manifold.
result Parallel transport simplifies eigenvector computation for one-parameter families of matrices.
New theory couples Chern-Simons to matter, topological.
problem Developing a new topological theory in 3D.
method Coupling Chern-Simons to matter, using transverse holomorphic foliation.
result The theory is equivalent to an N=2 supersymmetric Chern-Simons matter theory.
We study the differential geometry of principal G-bundles whose base space is the space of free paths (loops) on a manifold M. In particular we consider connections defined in terms of pairs (A,B), where A is a connection for a fixed principal bundle P(M,G) and B is a 2-form on M. The relevant curvatures, parallel tran…
PhyloVAE learns tree topologies without supervision.
problem Learning accurate tree representations from evolutionary data.
method Unsupervised learning via variational autoencoders with efficient tree generation.
result PhyloVAE generates high-resolution tree topologies efficiently.
CodedReduce combines tree topology and gradient coding for efficient and resilient gradient aggregation.
problem Efficient and robust gradient aggregation in distributed learning.
method CodedReduce combines tree topology and gradient coding to overcome bandwidth bottlenecks and straggler delays.
result CodedReduce achieves up to 27.2x speedup over benchmarks GC and RAR.
Develops topological concepts for Morrey-Sobolev bundles in high dimensions.
problem Lack of continuity in transition maps for Morrey-Sobolev bundles.
method Introduces topological isomorphism classes and uses connection-oriented approach.
result Derives approximability results for bundles and connections in Morrey-Sobolev setting.
New spectral sequence connects to topological Hochschild homology.
problem Connecting spectral sequences to topological Hochschild homology.
method Developed a spectral sequence and applied Tate diagonal techniques.
result Spectral sequence converges to localized topological Hochschild homology.
PipeDream-2BW accelerates large model training by 20x with minimal memory usage.
problem Training large models requires memory beyond single accelerator capacity.
method Pipeline parallelism, weight gradient coalescing, double buffering.
result Accelerates large model training by up to 20x.
Very few results are known about the topology of the strata of the moduli space of quadratic differentials. In this paper, we prove that any connected component of such strata has only one topological end. A typical flat surface in a neighborhood of the boundary is naturally split by a collection of parallel short sadd…
The paper explores almost paracomplex structures on 4-manifolds and their properties.
problem Existence and properties of isometric and anti-isometric almost paracomplex structures on pseudo-Riemannian 4-manifolds.
method Analyzes the conditions under which these structures are parallel and their implications for the scalar curvature and Einstein metrics.
result If an isometric or anti-isometric almost paracomplex structure on a conformally flat manifold is parallel, the scalar curvature of the metric must be zero.
We consider compact connected minimal surfaces, with a pair of boundary curves (not necessarily convex) in distinct planes, that have least-area amongst all orientable surfaces with the same boundary. When the planes containing these two boundary curves are either parallel or sufficiently close to parallel, and when th…
TKFT models computation via smooth vector fields, simulating functions in a single dynamical step.
problem Modeling computation in a single step.
method Established Topological Kleene Field Theory (TKFT) as a new model of computation.
result Any computable function can be simulated in a single go of a dynamical system.