Determinants of theta curves and symmetric graphs are studied.
problem Understanding the determinants of theta curves and symmetric graphs.
method Combinatorial approach using Kirchhoff's Matrix Tree Theorem and spanning tree enumeration.
result The determinant of a simple theta curve is the product of the determinants of its constituent knots.
Neural model parses non-projective dependency trees efficiently.
problem Parsing non-projective dependency trees.
method Probabilistic parsing model using neural representations and Kirchhoff's Matrix-Tree Theorem.
result State-of-the-art parsing performance on nine datasets.
Paper derives a formula for the determinant of Dirichlet-to-Neumann operator on Riemann surfaces.
problem Bounding asymptotics of a conformal invariant under degeneration of Riemann surfaces.
method Meyer-Vietoris formula, gluing, height function on moduli space, properness of height function, Steklov isospectral metrics, Laplacian with Dirichlet/Neumann boundary conditions.
result Properness of height function on moduli space of genus zero hyperbolic surfaces implies compactness theorem for Steklov isospectral metrics.
This talk is a report on joint work with A. Vaintrob [arXiv:math.CO/0109104 and math.GT/0111102]. It is organised as follows. We begin by recalling how the classical Matrix-Tree Theorem relates two different expressions for the lowest degree coefficient of the Alexander-Conway polynomial of a link. We then state our fo…
The classical Matrix-Tree Theorem allows one to list the spanning trees of a graph by monomials in the expansion of the determinant of a certain matrix. We prove that in the case of three-graphs (that is, hypergraphs whose edges have exactly three vertices) the spanning trees are generated by the Pfaffian of a suitably…
Extends classical results to virtual links, proving new properties of alternating and semi-alternating virtual links.
problem Classical results for virtual links, focusing on alternating and semi-alternating links.
method Inequality relating link determinant and crossing number, matrix-tree theorem, Tait conjectures for virtual and welded links.
result Alexander polynomial of almost classical alternating virtual links is alternating.
The spectral geometry of mesh matrices of graphs is explored, leading to new formulas and eigenvalue estimates.
problem Understanding the spectral properties of mesh matrices of graphs.
method Definition and study of mesh matrices, introduction of mesh Laplacian, derivation of characteristic polynomial formulas.
result Mesh Laplacian eigenvalues are all real and greater than or equal to 1, with a smallest positive eigenvalue estimated.
Bicycle paths form geodesics in 3D subspaces, related to Kirchhoff rods.
problem Optimizing bicycle paths between two points.
method Variational equations and geometric analysis of bicycle paths.
result Bicycle geodesics are contained in 3D subspaces and relate to Kirchhoff rods.
The abstract discusses the existence of complex structures on spheres and their implications.
problem The existence of complex structures on the six sphere and their implications.
method Analyzing the parallelism and H-space multiplication on the seven sphere associated with almost complex structures on the six sphere.
result The integrability condition of the almost complex structure on the six sphere does not imply the homotopy associativity of the multiplication on the seven sphere.
Introduces Lax-Kirchhoff moduli spaces for quivers and Lie groups.
problem Constructing moduli spaces for quivers and Lie groups.
method Introduces Lax equations and Kirchhoff conditions, constructs slices, and uses Marsden-Weinstein reduction.
result Proves M(Γ) is a finite-dimensional smooth symplectic manifold with a Hamiltonian action of G∂Γ. New energy model reveals knotted rod configurations.
problem Modeling elastic rods with stretch or inflation.
method Introduced a generalized functional on framed curves, used correspondence to infinite-dimensional Grassmann manifold.
result Explicit parameterizations of all periodic critical framed curves.
SHAKE-GNN scales GNNs for large graphs with multi-scale representations.
problem Scaling Graph Neural Networks (GNNs) to large graphs.
method SHAKE-GNN uses a hierarchy of Kirchhoff Forests for stochastic multi-resolution graph decompositions.
result SHAKE-GNN achieves competitive performance on large-scale graph classification benchmarks.
Study p-Willmore disks with boundary energies, finding equilibrium configurations.
problem Finding equilibrium configurations for p-Willmore disks with boundary energies.
method Model boundary as Kirchhoff elastic rod, interior term dependent on mean and Gaussian curvatures. Study among topological disks and p-Willmore examples.
result Equilibrium configurations for p-Willmore disks with boundary energies.
Develops control and observer methods for complex systems.
problem Controlling and observing infinite-dimensional systems with boundary actuation.
method Energy-Casimir method and port-Hamiltonian system representation.
result Control law and observer designed for Kirchhoff-Love plate example.
The paper explores variational problems on Riemannian manifolds with special foliations, proving existence results.
problem Variational problems on Riemannian manifolds with singular Riemannian foliations.
method Application of Palais' Principle of Symmetric Criticality and Rellich-Kondrachov-Hebey-Vaugon Embedding Theorem.
result Existence of countably infinite weak solutions to variational problems.
Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy pol…
A Bayesian treatment of latent directed graph structure for non-iid data is provided where each child datum is sampled with a directed conditional dependence on a single unknown parent datum. The latent graph structure is assumed to lie in the family of directed out-tree graphs which leads to efficient Bayesian inferen…
Theory of point vortices extended to closed surfaces.
problem Extending point vortex dynamics to closed surfaces.
method Unified theory of point vortex dynamics on the plane, sphere, and closed surfaces.
result Comprehensive guide to point vortex dynamics on closed surfaces with genus zero and vanishing total vorticity.
Paper proves conditions for estimating precision matrices with Laplacian constraints.
problem Estimating high-dimensional precision matrices with Laplacian constraints.
method Minimizing Stein's loss with conditions on graph connectivity and Laplacian constraints.
result High-dimensional consistency achieved with Laplacian constraints, independent of graph structure.
New approach to electric group for knots and links.
problem No previous publication of electric invariant for knots and links.
method Simple and general approach to electric group for oriented knots and links, using proper colouring of knot diagrams.
result Each homomorphism from the electric group to an arbitrary finite group can be described by a proper colouring of the diagram.
The paper analyzes defects on structured surfaces and calculates stress and shape.
problem Analyzing defects on structured surfaces and their effects on stress and shape.
method Classified and quantified defects, derived strain incompatibility relations, and applied to shells.
result Determined internal stress field and deformed shape for shells with defects.
We consider the inference of the structure of an undirected graphical model in an exact Bayesian framework. More specifically we aim at achieving the inference with close-form posteriors, avoiding any sampling step. This task would be intractable without any restriction on the considered graphs, so we limit our explora…
Physics-guided neural network improves power flow analysis.
problem Infeasibility of traditional numerical approaches due to outdated or unavailable PF equations.
method Proposes a physics-guided neural network to learn PF mappings from historical data while constraining by physical laws.
result Physics-guided neural network achieves better performance and generalizability than unconstrained data-driven approaches.
Bayesian learning for forests and trees improves graph detection and structure learning.
problem Learning graph structures in non-decomposable graphs.
method Adapted MCMC and SSS algorithms for forests and trees, using the Chow-Liu algorithm and Matrix Tree Theorem.
result SSS with trees or forests outperforms SSS with decomposable graphs in certain cases.
Paper smooths classical tests for implicit models.
problem Learning rich implicit models from which densities are hard to evaluate.
method Smooth classical tests using graphical models.
result Smoothing increases test power, improving implicit model learning.
New sigma models compute graviton scattering amplitudes from quaternionic geometry.
problem Computing graviton scattering amplitudes from quaternionic geometry.
method Introducing new twistor sigma models that encode finite non-linear perturbations of flat structures.
result Provides a first-principles derivation of Hodges' formula for MHV graviton amplitudes.
We discuss several issues regarding material homogeneity and strain compatibility for materially uniform thin elastic shells from the viewpoint of a 3-dimensional theory, with small thickness, as well as a 2-dimensional Cosserat theory. A relationship between inhomogeneity and incompatibility measures under the two des…
We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand,…
Knot Theory is currently a very broad field. Even a long survey can only cover a narrow area. Here we concentrate on the path from Goeritz matrices to quasi-alternating links. On the way, we often stray from the main road and tell related stories, especially if they allow as to place the main topic in a historical cont…
The paper defines surface area for graphs and derives spectral estimates.
problem Understanding connectivity measures and spectral properties of graphs.
method Introducing surface area concepts related to inverse degree and deriving spectral bounds.
result An upper bound on the second eigenvalue for planar graphs.
This paper explores vortices and harmonic flows on compact surfaces, using Hodge decomposition.
problem Understanding the interplay between vortices and harmonic flows on compact surfaces.
method Hodge decomposition of Euler's equations, focusing on point vortices on compact Riemann surfaces.
result The harmonic part of the flow is constant on flat tori but not on non-flat tori.
We study the stability of symmetric trajectories of a particle on the Lie group SO(3) whose motion is governed by an SO(3)×SO(2) invariant metric and an SO(2)×SO(2) invariant potential. Our method is to reduce the number of degrees of freedom at {\em singular} values of the SO(2)×SO(2) momentu…
The paper compares PINN methods for solving drift-diffusion equations on metric graphs.
problem Solving drift-diffusion equations on metric graphs using machine learning.
method Comparison of physics-informed neural networks (PINNs) for solving drift-diffusion equations on metric graphs.
result PINNs offer a flexible and versatile tool for solving parameter identification or optimization problems on metric graphs.
The aim of this paper is to give a formulation of the dynamics of nonlinear RLC circuits as a geometric Birkhoffian system and to discuss in this context the concepts of regularity, conservativeness, dissipativeness. An RLC circuit, with no assumptions placed on its topology, will be described by a family of Birkhoffia…
A new framework approximates covariance matrices using tree decompositions.
problem Approximating covariance matrices for Gaussian distributions.
method Cascade of tree decompositions with Cholesky factorization.
result The proposed framework guarantees convergence and outperforms KL divergence.
New approach links 2D fluid dynamics to matrix theory.
problem Understanding swirling patterns in 2D fluids.
method Matrix hydrodynamics linking 2D fluid dynamics to matrix theory.
result Established connections between 2D hydrodynamics and matrix Lie theory.
Theorem analogues proven using Artin's approximation theorem.
problem Proving analogues of Moser's Theorem.
method Using Artin's approximation theorem.
result Few analogues of Moser's Theorem proven.
AI generates theorems and proofs for training theorem provers.
problem Limited human-written theorems and proofs for supervised learning.
method Proposes a neural generator to automatically synthesize theorems and proofs.
result Synthetic data improves automated theorem proving in Metamath.
Global inverse function theorem proved easily using Riemannian geometry.
problem Global inverse function theorem in Riemannian geometry.
method Hopf--Rinow theorem in Riemannian geometry.
result Hadamard's global inverse function theorem is proven easily.
A new comparison theorem for geometric spaces.
problem Geometric space comparison theorems.
method Relative form of Toponogov comparison theorem.
result New geometric space comparison theorem established.
The paper proves a new theorem in Riemannian geometry and offers a new proof for Toponogov's theorem in Alexandrov geometry.
problem Proving new theorems in Riemannian and Alexandrov geometries.
method Inspired by the proof of the Schur-Toponogov theorem, a new proof of Toponogov's theorem is provided.
result A new theorem in Riemannian geometry and a new proof of Toponogov's theorem in Alexandrov geometry.
Revises a theorem by Thurston, finding a counter-example and a weaker version.
problem The bounded image theorem in Haken manifolds.
method Providing a counter-example and a weaker version of the second statement of Thurston's theorem.
result A counter-example and a weaker version of the second statement of Thurston's theorem are presented.
The paper proves three circles theorems and Liouville type theorems for subharmonic and holomorphic functions.
problem Establishing theorems for subharmonic and holomorphic functions on specific geometric structures.
method Using subharmonic and holomorphic functions on Riemannian manifolds and gradient shrinking Ricci solitons.
result Proves Liouville type theorems as applications of the established theorems.
Paper develops formulas and theorems in Hermitian geometry.
problem None explicitly stated in the abstract.
method Develops second variational formulas and index forms in Hermitian geometry.
result Establishes results analogous to classical theorems in Riemannian geometry.
Fixed-point theorems for set-valued maps using homological methods.
problem Finding fixed points for specific types of set-valued maps.
method Homological selection theorems applied to finite-dimensional spaces.
result Established fixed-point theorems for usco homologically UV^n set-valued maps.
Proves Markov theorem for trivalent braids using L-move approach.
problem Proving Markov theorem for trivalent braids.
method Follows L-move approach to prove Markov theorem.
result Proves one-move Markov-type theorem and algebraic Markov-type theorem for trivalent braids.
Proofs for Moon's theorem and its generalization.
problem Proving Moon's theorem and its generalization.
method Proofs based on key lemmas.
result Generalization of the four-vertex theorem.
New measure proves Poncelet-type theorems.
problem Proving Poncelet-type theorems.
method Introducing a new invariant measure on the circle.
result Simple proof of Emch closing theorem.