Stability of Morse index for harmonic maps on degenerating surfaces analyzed.
problem Analyzing stability of Morse index for harmonic maps on degenerating Riemann surfaces.
method Analysis of second variation of energy, identification of conditions for upper semicontinuity, explicit contribution of geodesics.
result Sharper control of spectrum of Jacobi operator, explicit contribution of geodesic segments to Morse index.
We prove that an open manifold M of dimension at least 5 which admits a complete CAT(0) polyhedral metric is pseudo-collarable, its fundamental group at infinity is strongly perfectly semistable and has vanishing Chapman-Siebenmann obstruction τ∞(M). Moreover, this implies that M is topologically collap…
New examples show manifolds with noncompact boundaries that can't be pseudo-collarable.
problem Identifying manifolds with noncompact boundaries that cannot be pseudo-collarable.
method Provided examples of Z-compactifiable manifolds with noncompact boundaries that fail to be pseudo-collarable. result Found manifolds with noncompact boundaries that cannot be pseudo-collarable.
Positive representations on surfaces have positive cross-ratios and satisfy a collar lemma.
problem Characterizing representations of surface groups with positive properties.
method Proving a collar lemma and showing positivity of cross-ratios for Θ-positive representations. result Closed subsets of representation varieties are characterized by Θ-positive representations. The notion of an open collar is generalized to that of a pseudo-collar. Important properties and examples are discussed. The main result gives conditions which guarantee the existence of a pseudo-collar structure on the end of an open n-manifold (n > 6). This paper may be viewed as a generalization of Siebenmann's famo…
In this article, we prove an analog of the classical collar lemma in the setting of Hitchin representations.
We study the blow-up analysis and qualitative behavior for a sequence of harmonic maps with free boundary from degenerating bordered Riemann surfaces with uniformly bounded energy. With the help of Pohozaev type constants associated to harmonic maps defined on degenerating collars, including vertical boundary collars a…
Collar lemma proven for certain surface group representations.
problem Proving a collar lemma for specific surface group representations.
method Using partial hyperconvexity properties and Anosov representations.
result 'Positivity properties' hold for partially hyperconvex representations.
In this note we define the notion of collarable slices of Lagrangian submanifolds. Those are slices of Lagrangian submanifolds which can be isotoped through Lagrangian submanifolds to a cylinder over a Legendrian embedding near a contact hypersurface. Such a notion arises naturally when studying intersections of Lagran…
In this paper, we develop a geometric procedure for producing a reverse to Quillen's plus construction, a construction called a 1-sided h-cobordism or semi-h-cobordism. We then use this reverse to the plus construction to produce uncountably many distinct ends of manifolds called pseudo-collars, which are stackings of …
Derives generalizations of the long neck principle and spectral width inequality.
problem Understanding the spectral width of geodesic collar neighborhoods.
method Spinorial Callias operator approach and relative Gromov-Lawson pair.
result Generalizations of the long neck principle and spectral width inequality.
The paper extends the collar theorem to non-compact surfaces using new comparison theorems.
problem Proving the collar theorem for non-compact surfaces.
method Developed new Toponogov-type triangle comparison theorems.
result Eliminated the compactness hypothesis for the collar theorem.
Study parabolicity of Riemann surfaces via Fenchel-Nielsen parameters.
problem Determine conditions for a Riemann surface to be of parabolic type.
method Use Fenchel-Nielsen parameters and non-standard half-collars to study parabolicity.
result Obtain sufficient conditions for parabolicity in terms of Fenchel-Nielsen parameters.
Suppose δ is a generic immersed closed curve in the boundary of a 3-manifold M and δ is null-homotopic in M. Then δ can be displaced by a height function in a collar of the boundary so that the resulting simple closed curve in the collar bounds a disk in M.
This paper concerns a formula which relates the Lefschetz number L(f) for a map f:M --> M' to the fixed point index I(f) summed with the fixed point index of a derived map on part of the boundary of M. Here M is a compact manifold and M' is M with a collar attached.
In a previous paper, we developed general techniques for constructing a variety of pseudo-collars, as defined by Guilbault and Tinsley, with roots in earlier work by Chapman and Siebenmann. As an application of our techniques, we exhibited an uncountable collection of pseudo-collars, all with the same boundary and simi…
In this paper, we recall Quillen's plus construction for high-dimensional smooth manifolds and the solution to the group extension problem. We then develop a geometric procedure due for producing a "reverse" to the plus construction, a one-sided s-cobordism called a semi-s-cobordism, when the total group of the group e…
In this paper, we obtain analogues of Jorgensen's inequality for non-elementary groups of isometries of quaternionic hyperbolic n-space generated by two elements, one of which is loxodromic. Our result gives some improvement over earlier results of Kim [10] and Markham [15]}. These results also apply to complex hyper…
We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at infinity. In this paper a primary goal is finally achieved; namely, a complete character…
For compact Riemann surfaces, the collar theorem and Bers' partition theorem are major tools for working with simple closed geodesics. The main goal of this paper is to prove similar theorems for hyperbolic cone-surfaces. Hyperbolic two-dimensional orbifolds are a particular case of such surfaces. We consider all cone …
Considering a solid 3-dimensional Klein bottle and a collaring of its boundary, can we extend a generic C∞ non-singular function defined on the collaring to the full solid Klein bottle without critical points? We give a condition on the Reeb graph of the given function that is necessary and sufficient for the e…
In this paper we prove a new version of the Schoenflies extension theorem for collared domains in Euclidean n-space: for 1 < p < n, locally bi-Lipschitz homeomorphisms between collared domains with locally p-integrable, second-order weak derivatives admit homeomorphic extensions of the same regularity. Moreover, the th…
We use Furuta's result, usually referred to as ``10/8-conjecture'', to show that for any compact 3-manifold M the open manifold $M\times\r$ has infinitely many different smooth structures. Another consequence of Furuta's result is existence of infinitely many smooth structures on open topological 4-manifolds with a t…
We consider cohomogeneity one homogeneous disk bundles and adress the question when these admit a nonnegatively curved invariant metric with normal collar, i.e., such that near the boundary the metric is the product of an interval and a normal homogeneous space. If such a bundle is not (the quotient of) a trivial bundl…
New examples of non-smoothable homeomorphisms of 4-manifolds with boundary found.
problem Finding non-smoothable homeomorphisms of 4-manifolds with boundary.
method Constructing specific examples of homeomorphisms that fix the boundary and act trivially on homology.
result First examples of non-smoothable self-homeomorphisms of smooth 4-manifolds with boundary.
Positive representations of surface groups in PO(p,q) form connected components of character varieties.
problem Characterizing representations of surface groups in special orthogonal groups PO(p,q).
method Using Anosov representations and root versus weight collar lemmas.
result Connected components of character varieties are formed by Θ-positive Anosov representations. The paper sets geometric lower bounds for low Steklov eigenvalues on manifolds.
problem Finding geometric lower bounds for low Steklov eigenvalues on manifolds.
method Using trace inequalities relating Steklov eigenvalues to Neumann eigenvalues of subdomains containing boundary collars.
result Geometric lower bounds for low Steklov eigenvalues, complementing earlier results.
In this paper we obtain a complete characterization of pseudo-collarable n-manifolds for n≥6. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as an extension of Siebenmann's dissertation that can be applied to mani…
Women belonging to the socially disadvantaged caste-groups in India have historically been engaged in labour-intensive, blue-collar work. We study whether there has been any change in the ability to predict a woman's work-status and work-type based on her caste by interpreting machine learning models using feature attr…
Arithmetic spaces simplified to simplicial complexes.
problem Understanding the complexity of arithmetic locally symmetric spaces.
method Homotopy equivalence to a simplicial complex with linearly bounded simplices, using a strengthened Margulis collar lemma.
result Arithmetic locally symmetric spaces are homotopy equivalent to simplicial complexes with linearly bounded simplices.
Proves a theorem for normal distributions on manifolds with boundary.
problem Normal distributions on manifolds with boundary require a new approach to integration.
method Introduces neat integral manifolds with boundary and conditions for integrability.
result Conditions for integrability expressed in terms of adapted collars and integrability on interior and boundary.
Deep learning classifies animal behavior from wearable accelerometers.
problem Classifying animal behavior from accelerometer data.
method End-to-end deep neural network with IIR and FIR filters.
result Outperforms state-of-the-art algorithms in real-time classification.
Let F be a real closed field. We define the notion of a maximal framing for a representation of the fundamental group of a surface with values in Sp(2n,F). We show that ultralimits of maximal representations in Sp(2n,R) admit such a framing, and that all maximal framed represen…
Given a closed Riemannian manifold (M,gM) of dimension n≥3, we prove the existence of a conformally compact Einstein metric g+ defined on a collar neighborhood M×(0,1] whose conformal infinity is [gM].
New coordinates for Teichmüller space compactification.
problem Compactification of Teichmüller space.
method Defined new coordinates.
result Thurston compactification is radial compactification of Euclidean space.
In this paper we continue an earlier study of ends non-compact manifolds. The over-arching goal is to investigate and obtain generalizations of Siebenmann's famous collaring theorem that may be applied to manifolds having non-stable fundamental group systems at infinity. In this paper we show that, for manifolds with c…
We give a lower bound for the widths of the collars of certain short partial pants decomposition of the surface. Then we apply this to obtain upper bounds of the renormalized volume of certain Schottky manifolds in terms of the hyperbolic length of compressible curves.
Researchers approximate spectral targets on manifolds with constant negative curvature.
problem Prescribing an arbitrary finite portion of the Laplace-Beltrami spectrum on manifolds of constant negative curvature.
method Constructing macroscopically heterogeneous hyperbolic covering manifolds in d≥3 and using discrete spectral limit theorems in d=2. result Any finite strictly increasing target list can be approximated to arbitrary precision by a closed manifold of constant negative curvature.
We show that if the totally geodesic boundary of a compact hyperbolic 3-manifold M has a large collar of depth d, then the diameter of the skinning map of M is no more than A exp(-d) for some A depending only on the genus and injectivity radius of the boundary of M.
This paper presents a curvature-free version of the Log(2k-1) Theorem of Anderson, Canary, Culler & Shalen [ACCS96]. It generalizes a result by Hou [Hou01] and its proof is rather straightforward once we know the work by Lim [Lim08] on volume entropy for graphs. As a byproduct we obtain a curvature-free version of the …
We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only depends on the length of the geodesic. As an application, we show that there is a lo…
We establish an energy quantization result for sequences of Willmore surfaces when the underlying sequence of Riemann surfaces is degenerating in the moduli space. we notably exhibit a new residue which quantifies the potential loss of energy in collar regions. Thanks to these residues, we also prove compactness of Wil…
Bayesian deep learning faces posterior collapse due to likelihood vs. prior competition.
problem Posterior collapse in Bayesian deep learning models.
method Identified competition between likelihood and prior regularization in a linear latent variable model.
result Posterior collapse is related to neural and dimensional collapse, suggesting a broader learning issue.
Guilbault and author prove sufficient conditions for Z-compactifiability of certain manifolds.
problem Establishing sufficient conditions for Z-compactifiability of manifolds. method Introducing additional conditions to prove Z-compactifiability of Mnimes[−2,2] implies Z-compactifiability of Mn. result There exist infinitely many non-pseudo-collarable 4-manifolds which are Z-compactifiable. We consider hyperbolic 3-manifolds with either non-empty compact geodesic boundary, or some toric cusps, or both. For any such M we analyze what portion of the volume of M can be recovered by inserting in M boundary collars and cusp neighbourhoods with disjoint embedded interiors. Our main result is that this portion c…
Proves weakly non-collapsed RCD spaces are strongly non-collapsed.
problem Proving the equivalence of weakly non-collapsed and strongly non-collapsed RCD spaces.
method Analyzes properties of RCD spaces and uses auxiliary results.
result Confirms conjecture about RCD spaces being strongly non-collapsed.
We show that in any triangulation of a solid torus, there is a pre-core curve that lies in the 2-skeleton and that intersects the interior of each face in at most 10 straight arcs. By definition, a pre-core curve is a simple closed curve that becomes a core curve when a collar is attached to the boundary of the solid t…
Study on Neural Collapse limits in deep learning.
problem Understanding the limits of Neural Collapse in deep learning.
method Investigated Neural Collapse in the context of generalization and feature learning, refining conjectures and conducting experiments.
result Neural Collapse primarily occurs on the train set and not on the test set, suggesting it is an optimization phenomenon with unclear connections to generalization.