The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.
problem Analyzing heat flow and constants on graphs.
method Introducing concepts, recalling graph theory, and proposing new discrete Morse flows.
result Weak discrete Morse flows for heat flow on finite graphs under suitable assumptions.
We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.
problem Classifying steady Euler flows with Morse-Bott Bernoulli functions.
method Constructing non-vanishing steady solutions using integrable systems and topology.
result Steady Euler flows with Morse-Bott Bernoulli functions exist only on graph three-manifolds.
Study on planar graphs in Poincare model of hyperbolic geometry.
problem Investigating Morse flows on a 2-disk using planar graphs.
method Using planar graphs and spherical graphs to describe topological structures.
result Listed all planar graphs with at least 3 edges and described those with 4 edges.
For a closed oriented 3-manifold Y we define n(Y) to be the minimal non-negative number such that in each homotopy class of non-singular vector fields of Y there is a Morse-Smale vector field with less or equal to n(Y) periodic orbits. We combine the construction process of Morse-Smale flows given in [2] with h…
In this paper, we first develope the concept of Lyapunov graph to weighted Lyapunov graph (abbreviated as WLG) for nonsingular Morse-Smale flows (abbreviated as NMS flows) on S3. WLG is quite sensitive to NMS flows on S3. For instance, WLG detect the indexed links of NMS flows. Then we use WLG and some other tool…
Combines techniques to remove tameness condition in Morse-Smale flows.
problem Tameness condition in Morse-Smale flows.
method Combines Shilnikov's ODE techniques with Latschev's ideas.
result Removes tameness hypothesis from homotopy formula.
Kleinberg introduced three natural clustering properties, or axioms, and showed they cannot be simultaneously satisfied by any clustering algorithm. We present a new clustering property, Monotonic Consistency, which avoids the well-known problematic behaviour of Kleinberg's Consistency axiom, and the impossibility resu…
The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow Φ:R×X→X on pfaffian set X is tame if the graph of Φ is a pfaffian subset of R×X×X. Any compact tame set admits plenty tame flows. We prove …
A new approach to Morse theory using folded ribbon trees.
problem Applying Morse theory on symmetric products of surfaces.
method Introducing an A-infinity category with objects as κ-tuples of Morse functions, and showing conditions for the endomorphism to be a Hecke algebra.
result The endomorphism of a specific type of κ-tuple of Morse functions on T*R^2 is the Hecke algebra associated to the symmetric group.
A graph clustering method that moves nodes to highest-degree neighbors.
problem Graph clustering for data with Morse regularity.
method Max-degree hill-climbing on graph nodes.
result Asymptotically consistent for random geometric graphs.
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, an…
In this paper we define and study the moduli space of metric-graph-flows in a manifold M. This is a space of smooth maps from a finite graph to M, which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on M. Using the model of Gromov-Witten theory, with this moduli spa…
The paper studies Morse flows on 3-manifold boundaries with fixed points.
problem Classifying Morse flows on 3-manifold boundaries.
method Constructing a Pr-diagram as a topological invariant.
result A complete topological invariant of Morse flows on 3-manifold boundaries.
A Morse complex for Axiom A flows on smooth manifolds.
problem Constructing a finite-dimensional cohomological complex for Axiom A flows.
method Defining anisotropic Sobolev spaces and spectral projectors.
result The cohomology of the constructed complex is isomorphic to De Rham cohomology.
Study Morse functions on projective plane using Reeb graphs.
problem Investigate topological structure of Morse functions on projective plane.
method Use Reeb graphs to describe and prove properties of simple Morse functions on RP2. result Prove that Reeb graphs are a complete topological invariant for simple Morse functions on RP2. Study describes Morse flows on a torus with up to six singular points.
problem Understanding the structure of Morse flows on a torus with a hole.
method Used separatrix diagrams to describe topological structures and saddle-node bifurcations.
result Identified all possible topological structures of Morse flows with at most six singular points.
Study on singularity behavior of mean curvature flow with bounded curvature and index.
problem Understanding singularity formation in mean curvature flow with constraints.
method Analyzing flow with bounded mean curvature and Morse index.
result Either mean curvature or Morse index blows up at first singular time.
Introduces a Morse complex on symplectic manifolds using gradient flows and proves its cohomology is independent of metrics and Morse functions.
problem Cohomology of symplectic manifolds under different metrics and Morse functions.
method Symplectic Morse complex with gradient flows and Witten deformation.
result Cohomology of the complex is isomorphic to Tsai, Tseng, and Yau's cohomology and independent of metrics and Morse functions.
Paper connects knot invariants and Morse flow loops.
problem Connecting quantum group invariants and Morse flow loops for knot study.
method Defining a two-variable series invariant by counting Morse flow loops in knot complements and proving it agrees with quantum group BPS series.
result Correspondence proven for all braid-homogeneous knots.
Graph products inherit Morse local-to-global property from their components.
problem Generalizing local-to-global property to graph products of infinite groups.
method Generalizing maximization procedure for relatively hierarchically hyperbolic groups and showing stable embeddings.
result Graph products of infinite Morse local-to-global groups have the Morse local-to-global property.
Classifies Morse flows on 3-sphere with specific saddle connections.
problem Classifying Morse-Smale flows on a 3-sphere with specific saddle connections.
method Used generalized Heegaard diagrams (Pr-diagrams) to classify flows.
result Found all possible, up to homeomorphism, ways to embed two circles in a 2-sphere with no more than 10 points of transversal intersection.
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.
Defines and calculates foliation homology from flows.
problem Homology of foliations defined by flows.
method Definition and calculation of foliation homology.
result Homology naturally associated with Seifert fibration.
We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…
FPP preserves sublinear Morse boundaries in geodesic graphs.
problem Preserving sublinear Morse boundaries in FPP.
method First passage percolation on geodesic graphs with i.i.d. passage times.
result Sublinear Morse boundaries are invariant under FPP.
Connected components of Morse boundaries are studied in graph of groups.
problem Understanding the structure of Morse boundaries in graph of groups.
method Analyzes connected components of Morse boundaries, considering edge and vertex groups properties.
result Connected components of Morse boundaries are derived from vertex groups under certain conditions.
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.
Constructs flows of tori in sphere perturbations for Morse homology.
problem Understanding tori in sphere perturbations.
method Constructs eternal mean curvature flows of tori.
result Constructs flows of tori in sphere perturbations.
In this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric g0 of constant scalar curvature, and and for Porous media equation on a bounded regular domain in the plane. We show that with a suitable assumption about $g(0)…
The paper solves graph realization problems for Reeb graphs of Morse functions.
problem Realizing graphs as Reeb graphs with specific preimage configurations.
method Constructing Morse functions with prescribed preimages.
result Solved realization problems for certain types of graphs.
The counting function on the natural numbers defines a discrete Morse-Smale complex with a cohomology for which topological quantities like Morse indices, Betti numbers or counting functions for critical points of Morse index are explicitly given in number theoretical terms. The Euler characteristic of the Morse filtra…
By studying spaces of flow graphs in a closed oriented manifold, we construct operations on its cohomology, parametrized by the homology of the moduli spaces of compact Riemann surfaces with boundary marked points. We show that the operations satisfy the gluing axiom of an open homological conformal field theory. This …
In~\cite{rotvandervorst} a homology theory --Morse-Conley-Floer homology-- for isolated invariant sets of arbitrary flows on finite dimensional manifolds is developed. In this paper we investigate functoriality and duality of this homology theory. As a preliminary we investigate functoriality in Morse homology. Functor…
In Garside groups, axes of Morse elements are strongly contracting.
problem Understanding the dynamics of Morse elements in Garside groups.
method Analyzing the Cayley graph of Garside groups modulo their center, using Garside generators.
result Morse elements act loxodromically on the additional length graph of Garside groups.
On a smooth, compact and oriented manifold without boundary, we give a complete description of the correlation function of a Morse-Smale gradient flow satisfying a certain nonresonance assumption. This is done by analyzing precisely the spectrum of the generator of such a flow acting on certain anisotropic spaces of cu…
The works of Donaldson and Mark make the structure of the Seiberg-Witten invariant of 3-manifolds clear. It corresponds to certain torsion type invariants counting flow lines and closed orbits of a gradient flow of a circle-valued Morse map on a 3-manifold. We study these invariants using the Morse-Novikov theory and H…
We investigate the probability of detecting combinatorial Morse flows on a simplicial complex via a random search. We prove that it is really small, in a quantifiable way.
Study geodesics on graphs with random lengths, proving bi-infinite paths exist.
problem Existence of bi-infinite geodesic paths on graphs with random edge lengths.
method Sublinear Morse geodesics and first passage percolation analysis.
result Proves the existence of bi-infinite geodesic paths in graphs with specific properties.
Study compares thimbles to Morse theory on Lie theory models.
problem Exploring thimbles in Landau-Ginzburg models using Morse theory.
method Constructing real Lagrangian thimbles and comparing to gradient flow manifolds.
result Explicit construction and comparison of thimbles to gradient flow manifolds.
Examples of Morse functions with integrable gradient flows on some classical Riemannian manifolds are considered. In particular, we show that a generic height function on the symmetric embeddings of classical Lie groups and certain symmetric spaces is a perfect Morse function, i.e. has as many critical points as the ho…
A map from 3-manifold skein to Lagrangian skein via holomorphic curve counting.
problem Counting holomorphic curves in cotangent bundles for 3-manifold skein.
method Skein-valued counting of holomorphic curves in branched covers.
result Wall-crossing formula for skein traces in branched covers.
The study of Morse functions on 3-manifolds and their Reeb graphs.
problem Existence of Morse functions with high Betti numbers on 3-manifolds.
method Analysis of Reeb graphs and Heegaard splittings.
result Calculation of a new invariant for 3-manifold groups.
Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.
problem Restricting Morse indices of saddles in gradient-like flows.
method Analyzing invariant manifolds and their intersections for gradient-like flows.
result Morse indices of saddles are either 1 or n-1, no other indices possible.
The paper proves index theorems for graph-based optimal control problems.
problem Optimal control problems on graphs with constraints.
method Proves Morse index theorems for a broad class of variational problems on graphs.
result Formulas compute the difference of Hessians related to different graphs or boundary conditions.
Study of circle arrangements related to Morse-Bott functions.
problem Understanding the geometry and singularity theory of Morse-Bott functions.
method Systematic construction of circle arrangements centered at existing circles, studying local changes in Reeb graphs.
result Reeb graphs of Morse-Bott functions are spaces of all components of preimages of single points.
Proves properties of Morse vector fields on compact manifolds.
problem Properties of gradient vector fields of Morse functions.
method Analyzes connectedness of critical points and shrinkage of flow.
result Shows connectedness of critical points through orbits and exponential shrinkage.
We study the L2 gradient flow of the Yang--Mills functional on the space of connection 1-forms on a principal G-bundle over the sphere S2 from the perspective of Morse theory. The resulting Morse homology is compared to the heat flow homology of the space ΩG of based loops in the compact Lie group G. An iso…
Heat flow on lens spaces settles into Morse functions with four critical points.
problem Understanding the behavior of heat flow on lens spaces.
method Analyzing the asymptotic spectral expansion of the heat flow.
result Generic heat evolutions on lens spaces \(L(p,q)\) with \(p\geq2\) and \(1\leq q\leq p/2\) tend to settle into Morse functions with exactly four critical points.