Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. Proves existence of flat connection on theta functions for G-bundles.
problem Existence of flat connections on nonabelian theta functions for G-bundles.
method Proves existence of a flat projective connection on nonabelian theta functions on moduli space of parabolic G-bundles.
result Existence of a flat projective connection on nonabelian theta functions for parabolic G-bundles.
Connections between nodes of fully connected neural networks are usually represented by weight matrices. In this article, functional transfer matrices are introduced as alternatives to the weight matrices: Instead of using real weights, a functional transfer matrix uses real functions with trainable parameters to repre…
The paper uses distance correlation for brain connectivity and a novel multi-task learning model for age prediction.
problem Estimating age-related gender differences in brain functional connectivity.
method Estimates functional connectivity using distance correlation and proposes a non-convex multi-task learning model.
result The proposed non-convex multi-task learning model outperforms other models in age prediction and gender-specific connectivity.
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
We discuss generic smooth maps from smooth manifolds to smooth surfaces, which we call "Morse 2-functions", and homotopies between such maps. The two central issues are to keep the fibers connected, in which case the Morse 2-function is "fiber-connected", and to avoid local extrema over 1-dimensional submanifolds of th…
In this paper, we show that given a nontrivial concircular vector field u on a Riemannian manifold (M,g) with potential function f, there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇f, which we call the connecting function o…
Geometrically connects theta functions and WZNW blocks.
problem Identifying connections between theta functions and WZNW blocks.
method Constructing flat connections on bundles of theta functions and WZNW blocks.
result The Tsuchiya-Ueno-Yamada and Knizhnik-Zamolodchikov connections are identified.
The study characterizes 3D manifolds using specific Morse-Bott functions.
problem Characterizing 3D manifolds represented as connected sums of Lens spaces, S2imesS1, and torus bundles. method Using Morse-Bott functions to classify the manifolds.
result Explicit characterization of the manifolds via certain Morse-Bott functions.
Paper extends Simons theorem to F-Yang-Mills connections for instability.
problem Tackles instability of F-Yang-Mills connections. method Extends Simons theorem to F-Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method. result Derives a sufficient condition for instability of non-flat F-Yang-Mills connections. We state and prove a simple Theorem that allows one to generate invariant quantities in Metric-Affine Geometry, under a given transformation of the affine connection. We start by a general functional of the metric and the connection and consider transformations of the affine connection possessing a certain symmetry. We…
R-PLS improves analysis of brain functional connectivity matrices.
problem Improving analysis of functional connectivity matrices in brain imaging.
method Introducing R-PLS, a generalization of PLS for symmetric positive definite matrices.
result R-PLS identifies key functional connections in brain imaging datasets.
Derives continuum model from discrete ε-graphs with connectivity functional.
problem Modeling diffusion in networks with varying connectivity.
method Energy-based continuum limit derivation, neural-network reconstruction of connectivity.
result Error between discrete and continuum energies is O(ε), valid even with fluctuations. We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.
A classic theorem in the theory of connections on principal fiber bundles states that the evaluation of all holonomy functions gives enough information to characterize the bundle structure (among those sharing the same structure group and base manifold) and the connection up to a bundle equivalence map. This result and…
This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…
Defines a volume functional for Hermitian connections on manifolds, proving its properties and connections.
problem Defining and analyzing volume functionals for Hermitian connections on manifolds.
method Introduces the line bundle mean curvature flow and relates it to deformed connections and special submanifolds.
result Proves the mirror equality for mSpin(7)-dDT connections and deduces their properties. Paper introduces a new method to identify brain hubs using both structural and functional connectivity.
problem Hub node identification in brain networks using only functional connectivity.
method Graph signal processing framework that models functional activity as graph signals on structural connectivity.
result The proposed GraFHub framework identifies hub nodes more accurately than conventional methods.
ST-GCN improves rs-fMRI prediction accuracy by modeling spatio-temporal graph connectivity.
problem Existing rs-fMRI methods neglect functional connectivity or temporal dynamics.
method Spatio-temporal graph convolutional network (ST-GCN) trained on BOLD time series.
result ST-GCN predicts gender and age more accurately than common methods.
Classifies Morse functions on 3-manifolds made from simple building blocks.
problem Classifying Morse functions on 3-dimensional manifolds.
method Examined Morse functions on 3-manifolds represented as connected sums of Heegaard genus one manifolds.
result Found conditions for the existence of Morse functions with specific properties.
This paper shows that every sublevel set of the loss function of a class of deep over-parameterized neural nets with piecewise linear activation functions is connected and unbounded. This implies that the loss has no bad local valleys and all of its global minima are connected within a unique and potentially very large…
We propose a novel denoising framework for task functional Magnetic Resonance Imaging (tfMRI) data to delineate the high-resolution spatial pattern of the brain functional connectivity via dictionary learning and sparse coding (DLSC). In order to address the limitations of the unsupervised DLSC-based fMRI studies, we u…
Quantizes functions on Kähler manifolds without formal deformation.
problem Deforming smooth functions on Kähler manifolds to non-formal quantization.
method Using Fedosov connections and prequantum line bundles.
result Quantizable functions form a sheaf of twisted differential operators.
A result (Corollary 4.3) in an article by Uhlenbeck (1985) asserts that the W1,p-distance between the gauge-equivalence class of a connection A and the moduli subspace of flat connections M(P) on a principal G-bundle P over a closed Riemannian manifold X of dimension d≥2 is bounded by a constant ti…
Study shows how SL2 Hitchin connection at level four behaves.
problem Understanding the behavior of SL2 Hitchin connection at level four. method Using Mumford-Welters connections and equivariant conformal embeddings, the connection's monodromy is shown to be finite.
result The monodromy of the SL2 Hitchin connection at level four is finite. Let Phi : M --> g^* be a proper moment map associated to an action of a compact connected Lie group, G, on a connected symplectic manifold, (M,ω). A collective function is a pullback via Φof a smooth function on g^*. In this paper we present four new results about the relationship between the collective functions and t…
This paper improves deep neural network approximation for fully connected networks, achieving optimal convergence rates.
problem Improving approximation of fully connected deep neural networks for optimal convergence rates.
method Deriving approximation bounds specifically for a narrower fully connected deep neural network.
result Achieves an optimal rate (up to a logarithmic factor) for fully connected deep neural networks.
A new deep metric learning method for defect classification in threaded pipe connections.
problem Defect classification in threaded pipe connections with limited and imbalanced multichannel functional data.
method COMPILED approach based on deep metric learning for imbalanced, multichannel, and partially observed functional data.
result Superior accuracy compared to existing benchmarks in a real-world case study.
The paper analyzes spectral properties of connection Laplacian on tori, proving convergence to real torus.
problem Spectral analysis of connection Laplacian on tori.
method Employing parallel orthonormal basis in pullback bundle, examining eigenvalues of connection Laplacian on real and discrete tori.
result Eigenvalues of connection Laplacian on discrete tori converge to those on real torus, with unique twist in torsion matrix.
Bi-forms extend contrast functions to handle torsion in information geometry.
problem Insufficient contrast-based approaches for geometric structures with torsion.
method Introducing contrast bi-forms, a generalization of contrast functions.
result Bi-forms provide a unified framework for statistical potentials.
Morse theory connects low energy submanifolds in 3-sphere.
problem Understanding low energy submanifolds in the 3-sphere.
method Morse-theoretic techniques and negative gradient flow.
result Constructs connections between low energy critical submanifolds.
If X is a full, finitely generated, projective module over a non-commutative torus, the Yang-Mills functional attains its minimum exactly on the flat connections on X. We classify the flat connections on modules admitting integrable connections.
The action of origin-preserving diffeomorphisms on a space of jets of symmetric connections is considered. Dimensions of moduli spaces of generic connections are calculated. Poincaré series of the geometric structure of symmetric connection is constructed, and shown to be a rational function.
Paper connects AJ conjecture and colored Jones polynomial potential function.
problem Relationship between A-polynomial and colored Jones polynomial. method Connects AJ conjecture and colored Jones polynomial potential function.
result Establishes connection between A-polynomial and colored Jones polynomial potential function. For any compact Lie group G and closed, smooth Riemannian manifold (X,g) of dimension d≥2, we extend a result due to Uhlenbeck (1985) that gives existence of a flat connection on a principal G-bundle over X supporting a connection with Lp-small curvature, when p>d/2, to the case of a connection with …
We consider local invariants of general connections (with torsion). The group of origin-preserving diffeomorphisms acts on a space of jets of general connections. Dimensions of moduli spaces of generic connections are calculated. Poincaré series of the geometric structure of connection is constructed, and shown to be a…
Previous work has questioned the conditions under which the decision regions of a neural network are connected and further showed the implications of the corresponding theory to the problem of adversarial manipulation of classifiers. It has been proven that for a class of activation functions including leaky ReLU, neur…
Studies in recent years have demonstrated that neural organization and structure impact an individual's ability to perform a given task. Specifically, individuals with greater neural efficiency have been shown to outperform those with less organized functional structure. In this work, we compare the predictive ability …
Anti-self-dual (ASD) connections for a compact smooth four manifold arise as critical values for the Yang-Mills action functional. Nahm transform is a nice correspondence between a vector bundle with ASD connections and a vector bundle with ASD connections over Picard torus associated to X. In this talk we propose a no…
Extending Lévi-Civita's concept to non-quadratic spaces, this study finds extremal compatible linear connections.
problem Extending the Lévi-Civita connection to non-quadratic spaces.
method Hybrid conditional extremum problem, Lagrange multipliers, geometric approach.
result Existence and characterization of extremal compatible linear connections.
The paper studies stability of F-Yang-Mills connections on complex projective spaces.
problem Stability of F-Yang-Mills connections on complex projective spaces.
method Inspired by Lawson-Simons, the paper proves stability conditions and structures for F-Yang-Mills connections.
result Conditions for weakly stable F-Yang-Mills connections on complex projective spaces.
Background: Depression has become a major health burden worldwide, and effective detection depression is a great public-health challenge. This Electroencephalography (EEG)-based research is to explore the effective biomarkers for depression recognition. Methods: Resting state EEG data was collected from 24 major depres…
Functional dimension varies in ReLU networks, with implications for symmetry and connectivity.
problem Understanding the functional dimension of ReLU neural networks.
method Careful definition and analysis of functional dimension, study of quotient space and fibers.
result Functional dimension is inhomogeneous and can be non-constant, with implications for symmetry and connectivity.
Constructs flow lines connecting unstable to stable self-expanders.
problem Existence of monotone Morse flow lines for expander functionals.
method Constructs a singular Morse flow line connecting unstable to stable self-expanders.
result Constructs a monotone flow line with a small singular set.
Paper connects risk consistency to L_p consistency for broader loss functions.
problem Establishing risk consistency for a wider class of loss functions.
method Analyzes the connection between risk consistency and L_p-consistency for various loss functions.
result Shifted loss functions do not reduce assumptions as much as other results.
Spontaneous brain activity, as observed in functional neuroimaging, has been shown to display reproducible structure that expresses brain architecture and carries markers of brain pathologies. An important view of modern neuroscience is that such large-scale structure of coherent activity reflects modularity properties…
The paper explores connections between braids, links, and cobordisms using algebraic methods.
problem Investigating functions on manifolds and their connections to braids, links, and cobordisms.
method Algebraic methods including group theory, sheaves, and formal groups.
result Constructs Lazard's one-dimensional universal commutative formal group and applies it to cobordism theory.
We study the level sets of the distance function from a boundary point of a convex set in Euclidean space. We provide a lower bound for the range of connectivity of the level sets, in terms of the critical points of the distance function in the sense of Grove-Shiohama-Gromov-Cheeger.