arXiv research

Regularizers change the geometric properties of loss functions in neural networks.

In this paper, we prove a Morse index theorem for the index form of regular Lagrangian system with selfadjoint boundary condition.

It is proved that every discrete Morse function in the sense of Forman on a finite regular CW complex can be represented by a polyhedral Morse function in the sense of Banchoff on an appropriate embedding in Euclidean space of the barycentric subdivision of the CW complex; such a representation preserves critical point…

Exponential growth of stable subgroups in Morse geodesics.

Counterexample disproves Borde-Sorkin conjecture on causal continuity of Morse spacetimes.

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furt…

The paper classifies Morse functions on 3-manifolds with specific level sets.

In this paper, we revisit implicit regularization from the ground up using notions from dynamical systems and invariant subspaces of Morse functions. The key contributions are a new criterion for implicit regularization---a leading contender to explain the generalization power of deep models such as neural networks---a…

New Morse theory for shapes at distances.

Random walk constructs Morse functions on surfaces.

New condition for reconstructing Morse functions on 3D manifolds.

In this paper we study Morse homology and cohomology with local coefficients, i.e. "twisted" Morse homology and cohomology, on closed finite dimensional smooth manifolds. We prove a Morse theoretic version of Eilenberg's Theorem, and we prove isomorphisms between twisted Morse homology, Steenrod's CW-homology with loca…

Motivated by the use of degenerate Jacobi metrics for the study of brake orbits and homoclinics, we develop a Morse theory for geodesics in conformal metrics having conformal factors vanishing on a regular hypersurface of a Riemannian manifold.

We discuss a PL analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong…

The Morse-Novikov number MN(L) of a smooth link L in the three-dimensional sphere is by definition the minimal possible number of critical points of a regular circle-valued Morse function on the link complement (the term regular means that the Morse function must have nice behaviour in a tubular neighbourhood of L). No…

Proves unique symplectic Lefschetz fibration from Morse functions.

We prove that finite Morse index solutions to the Allen-Cahn equation in $\R^2$ have {\bf finitely many ends} and {\bf linear energy growth}. The main tool is a {\bf curvature decay estimate} on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second order regularit…

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 $g_0$ 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)…

Constructs Morse homology for complex algebraic varieties.

Stability of branched immersions with energy constraints.

We introduce a novel combinatorial method to study $Q^{**}$-transformations of group presentations or, equivalently, 3-deformations of CW-complexes of dimension 2. Our procedure is based on a refinement of discrete Morse theory that gives a Whitehead simple homotopy equivalence from a regular CW-complex to the simplifi…

Generic metrics on manifolds yield simple Steklov eigenvalues and Morse boundary functions.

Study of hyperbolic directions in convex projective geometry.

Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a…

Study of circle arrangements related to Morse-Bott functions.

In studies of smooth maps with good differential topological conditions such as immersions, embeddings, Morse functions and their higher dimensional versions including fold maps and application to geometry, especially algebraic and differential topology of manifolds, liftings or desingulizations of maps of appropriate …

The study identifies all possible vector field structures on specific 2D shapes.

Researchers simplify the computation of diffeomorphism groups for Morse-Bott foliations.

Study the space of simple polygons and their moduli.

Classical Morse theory proceeds by considering sublevel sets $f^{-1}(-\infty, a]$ of a Morse function $f: M \to R$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1}(a)$ and give conditions under which the topology of $f^{-1}(a)$ changes when passing a cri…

For any compact Lie group $G$ and closed, smooth Riemannian manifold $(X,g)$ of dimension $d\geq 2$, 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 $L^p$-small curvature, when $p>d/2$, to the case of a connection with …

The paper develops algorithms and topological invariants for distinguishing dynamic systems.

We prove the transversality result necessary for defining local Morse chain complexes with finite cyclic group symmetry. Our arguments use special regularized distance functions constructed using classical covering lemmas, and an inductive perturbation process indexed by the strata of the isotropy set. A global existen…

A graph clustering method that moves nodes to highest-degree neighbors.

Study on planar graphs in Poincare model of hyperbolic geometry.

New nonlocal minimal surfaces on manifolds, proving Yau's conjecture.

Given a closed manifold N and a self-indexing Morse function f: N --> R with up to four distinct Morse indices, we construct a symplectic Lefschetz fibration pi: E --> C which models the complexification of f on the disk cotangent bundle, f_C : D(T*N) --> C, when f is real analytic. By construction, pi: E --> C comes w…

In this paper, we study Vanishing Mean Oscillation vector fields on a compact manifold with boundary. Inspired by the work of Brezis and Niremberg, we construct a topological invariant - the index - for such fields, and establish the analogue of Morse's formula. As a consequence, we characterize the set of boundary dat…

The tame flows are ``nice'' flows on ``nice'' spaces. The nice (tame) sets are the pfaffian sets introduced by Khovanski, and a flow $Φ: \mathbb{R}\times X\to X$ on pfaffian set $X$ is tame if the graph of $Φ$ is a pfaffian subset of $\mathbb{R}\times X\times X$. Any compact tame set admits plenty tame flows. We prove …

The distance function to a generic submanifold behaves well under small perturbations.

In this paper, we study Lipschitz-Fredholm vector fields on Bounded-Fréchet-Finsler manifolds. In this context we generalize the Morse-Sard-Brown theorem, asserting that if $M$ is a connected smooth bounded-Fréchet-Finsler manifold endowed with a strengthened connection $\mathcal{K}$ and if $ξ$ is a smooth Lipschitz-Fr…

Tripod spiders' energy control analyzed for Hooke and Coulomb potentials.

We consider the configuration space of planar $n$-gons with fixed perimeter, which is diffeomorphic to the complex projective space $\mathbb{C}P^{n-2}$. The oriented area function has the minimal number of critical points on the configuration space. We describe its critical points (these are regular stars) and compute …

Paper constructs continuous families of topological Morse functions.

New formulae connect topological and geometric properties of singular spaces.

New proof for discrete Morse theory using combinatorial construction.

We introduce a notion of Morse shellings (and tilings) on finite simplicial complexes which extends the classical one and its relation to discrete Morse theory.Skeletons and barycentric subdivisions of Morse shellable (or tileable) simplicial complexes are Morse shellable (or tileable). Moreover, every triangulated clo…

Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.