arXiv research

Study shows spectral gaps limit points on surfaces.

Study finds limit points of bass notes on hyperbolic surfaces.

A limit point p of a discrete group of Mobius transformations acting on S^n is called a concentration point if for any sufficiently small connected open neighborhood U of p, the set of translates of U contains a local basis for the topology of S^n at p. For the case of Fuchsian groups (n = 1), every concentration point…

Study smooth convergence of metric flows from $\mathbb{F}$-limits.

2-regular points found in spaces with lower Ricci curvature bound.

Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.

Study shows how certain knots and tori are detected by ideal points in character varieties.

Suppose that f and g are Markov surjections, each defined on a wedge of circles, each fixing the branch point and having the branch point as the only critical value. We show that if the points in the inverse limit spaces associated with f and g corresponding to the branch point are distinguished then these inverse limi…

Given a hyperbolic subgroup $H$ of a hyperbolic group $G$ for which a Cannon-Thurston map $\hat i:\partial H \ra \partial G$ exists, we study the limit set $Λ_H$ of $H$ with respect to its action on $\partial G$. We prove that the set of conical limit points is exactly the subset of $Λ_H$ consisting of the points to wh…

The pentagram map's limit point is related to infinitesimal perturbations of polygons.

In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi diagrams and compactifications of configuration spaces. We study moving and colliding p…

New MGCPP model for order flow in financial markets.

The paper connects geodesic flows and limit sets on visibility manifolds.

We introduce a Markovian single point process model, with random intensity regulated through a buffer mechanism and a self-exciting effect controlling the arrival stream to the buffer. The model applies the principle of the Hawkes process in which point process jumps generate a shot-noise intensity field. Unlike the Ha…

Study the limits of discrete DPPs to continuous DPPs as set size grows.

Motivated by applications in Optimization, Game Theory, and the training of Generative Adversarial Networks, the convergence properties of first order methods in min-max problems have received extensive study. It has been recognized that they may cycle, and there is no good understanding of their limit points when they…

Study reveals three limiting regimes for neural network functionals.

We study blow-ups around fixed points at Type I singularities of the Ricci flow on closed manifolds using Perelman's W-functional. First, we give an alternative proof of the result obtained by Naber and Enders-Müller-Topping that blow-up limits are non-flat gradient shrinking Ricci solitons. Our second and main result …

A framework models order book dynamics using point processes and mass transport.

Study on the geometry of limit spaces of manifolds with boundary.

The problem of prescribing conformally the scalar curvature of a closed Riemannian manifold as a given Morse function reduces to solving an elliptic partial differential equation with critical Sobolev exponent. Two ways of attacking this problem consist in subcritical approximations or negative pseudo gradient flows. W…

We present a homogenization theorem for isotropically-distributed point defects, by considering a sequence of manifolds with increasingly dense point defects. The loci of the defects are chosen randomly according to a weighted Poisson point process, making it a continuous version of the first passage percolation model.…

The paper characterizes subgroup stability via limit sets on the Morse boundary.

We discuss the Euclidean limit of hyperbolic SU(2)-monopoles, framed at infinity, from the point of view of pluricomplex geometry. More generally, we discuss the geometry of hypercomplex manifolds arising as limits of pluricomplex manifolds.

The paper analyzes phase retrieval under limited samples, ensuring a benign local landscape for convergence.

The limit of energies of a sequence of harmonic maps as their annular domains approach the boundary of moduli space depends upon the boundary point approached. The infinite energy case is associated with limits of images containing ruled surfaces. The finite energy case yields a limit of images, under a suitable topolo…

New examples of manifolds in tangent cones of non-collapsed Ricci limit spaces.

In high dimensions, find paths connecting points with intermediate steps in a dense set.

We consider the Reidemeister torsion associated with SL(2, C)-representations of a knot group. A bifurcation point in the SL(2, C)-character variety of a knot group is a character which is given by both an abelian SL(2, C)-representation and a non-abelian one. We show that there exist limits of the non-acyclic Reidemei…

Study of convergence of point-object configurations to a charged dust continuum.

The paper explores the correspondence between gradient flow lines of a function and its Lagrange multiplier functional.

Study on critical faces convergence in a Poisson point process.

Critical points of approximations of the Dirichlet energy à la Sacks-Uhlenbeck are known to converge to harmonic maps in a suitable sense. However, we show that not every harmonic map can be approximated by critical points of such perturbed energies. Indeed, we prove that constant maps and the rotations of $S^2$ are th…

The adaptive optimizer for training neural networks has continually evolved to overcome the limitations of the previously proposed adaptive methods. Recent studies have found the rare counterexamples that Adam cannot converge to the optimal point. Those counterexamples reveal the distortion of Adam due to a small secon…

The study connects Hilbert entropy to non-differentiability points of limit sets in flag spaces.

We introduce a multivariate Hawkes process with constraints on its conditional density. It is a multivariate point process with conditional intensity similar to that of a multivariate Hawkes process but certain events are forbidden with respect to boundary conditions on a multidimensional constraint variable, whose evo…

The paper studies Lagrangian structures in Higgs bundle moduli spaces and their conformal limits.

We consider point clouds obtained as random samples of a measure on a Euclidean domain. A graph representing the point cloud is obtained by assigning weights to edges based on the distance between the points they connect. Our goal is to develop mathematical tools needed to study the consistency, as the number of availa…

Study on critical points in random neural networks, revealing three regimes based on activation function.

Study geometric structure of Ricci shrinker ends without global curvature assumptions.

We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) th…

We give two numerical methods for computing the first bifurcation point for Delaunay nodoids. With regard to methods for constructing constant mean curvature surfaces, we conclude that the bifurcation point in the analytic method of Mazzeo-Pacard is the same as a limiting point encountered in the integrable systems met…

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a regular domain G of R^3. We prove that the second fundamental form of the boundary surface of G is nonnegatively defined at every point of the limit set of such immersions.

The study examines the asymptotic behavior of extremal length in Teichmüller space.

Study on Teichmüller rays' asymptotic behavior and distances.

In the paper arXiv:1411.4887 [math.AP] it is shown that the set of Riemannian metrics which do not admit global limiting Carleman weights is open and dense, by studying the conformally invariant Weyl and Cotton tensors. In the paper arXiv:1011.2507 [math.DG] it is shown that the set of Riemannian metrics which do not a…

Rare Teichmüller disks converge to small limit sets.

Study geometric properties of surfaces with specific formulae.