arXiv research

We give an alternate proof of Wise's Malnormal Special Quotient Theorem (MSQT), avoiding cubical small cancellation theory. We also show how to deduce Wise's Quasiconvex Hierarchy Theorem from the MSQT and theorems of Hsu--Wise and Haglund--Wise.

Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.

Proves better rigidity theorems for special solitons.

The paper proves drilled bundles over graphs are virtually special cubulable.

Proves rigidity of ancient solutions in mean curvature flow.

In this paper we extend Witten-Helffer-Sjöstrand theory from selfadjoint Laplacians based on fiber wise Hermitian structures, to non-selfadjoint Laplacians based on fiber wise non-degenerate symmetric bilinear forms. As an application we verify, up to sign, the conjecture about the comparison of the Milnor-Turaev torsi…

In this on-going work, I explore certain theoretical and empirical implications of data transformations under the PCA. In particular, I state and prove three theorems about PCA, which I paraphrase as follows: 1). PCA without discarding eigenvector rows is injective, but looses this injectivity when eigenvector rows are…

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $w\in F$ be a non-primitive element. The primitivity rank of $w$, $π(w)$, is the smallest rank of a subgroup of $F$ containing $w$ as an imprimitive element. Then any subgroup of the one-relator group $G=F/\langle…

New theorem for generalized group sparsity improves consistency and convergence rates.

The paper proves Sard's theorem for polynomial maps in infinite dimensions.

This paper studies how key tensor properties are inherited in subtensors of tensor train decompositions.

By using certain idea developed in minimal submanifold theory we study rigidity problem for self-shrinkers in the present paper. We prove rigidity results for squared norm of the second fundamental form of self-shrinkers, either under point-wise conditions or under integral conditions.

This work focuses on the estimation of multiple change-points in a time-varying Ising model that evolves piece-wise constantly. The aim is to identify both the moments at which significant changes occur in the Ising model, as well as the underlying graph structures. For this purpose, we propose to estimate the neighbor…

We extend several results of Helfer, Wise, Louder and Wilton related to coherence in one-relator groups to the more general setting of one-relator products of locally indicable groups. The methods developed to do so also give rise to a new proof of a theorem of Brodsky.

This work analyzes how different layers in deep neural networks contribute to generalization error.

Study derives new equation for reserves in non-monotone information scenarios.

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise and deduce that such manifolds are virtually fibered. We generalise a theorem…

We derive point-wise and integral rigidity/gap results for a closed manifold with harmonic Weyl curvature in any dimension. In particular, there is a generalization of Tachibana's theorem for non-negative curvature operator. The key ingredients are new Bochner-Weitzenböck-Lichnerowicz type formulas for the Weyl tensor,…

We study some basic problems of translating solitons: the volume growth, generalized maximum principle, Gauss maps and certain functions related to the Gauss maps, finally we carry out point-wise estimates and integral estimates for the squared norm of the second fundamental form. Those estimates give rigidity theorems…

Paper compares total quotient curvature and proves bounds for Einstein metric.

MPE framework proves universal approximation for quantum data distribution.

The paper finds non-isotopic exact Lagrangians in symplectic manifolds with $\mathbb{C}^*$-actions.

Considering Wirtinger's inequality for piece-wise equipartite functions we find a discrete version of this classical inequality. The main tool we use is the theorem of classification of isometries. Our approach provides a new elementary proof of Wirtinger's inequality that also allows to study the case of equality. Mor…

This paper proves that every finite volume hyperbolic 3-manifold M contains a ubiquitous collection of closed, immersed, quasi-Fuchsian surfaces. These surfaces are ubiquitous in the sense that their preimages in the universal cover separate any pair of disjoint, non-asymptotic geodesic planes. The proof relies in a cr…

Paper presents a new insurance model equation for diverse structures.

The paper studies how more data affects prediction risk in high-dimensional models.

Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory to obtain a simple version of the three geodesics theorem for the 2-sphere [5]. In general these three geodesics may be unstable, corresponding, for example, to the three equators of an ellipsoid. Using a…

FLANs process each feature separately for better interpretability.

Many real datasets contain values missing not at random (MNAR). In this scenario, investigators often perform list-wise deletion, or delete samples with any missing values, before applying causal discovery algorithms. List-wise deletion is a sound and general strategy when paired with algorithms such as FCI and RFCI, b…

Lattices in PSL(2,C) are omnipotent, acting on geodesics and homology.

Ricci flow controls curvature on manifolds with bounds.

xDeepInt learns both vector-wise and bit-wise feature interactions.

The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of this result. The first one is by a characterization of rank-$1$ isometries by Ham…

In this paper we show that given any 3-manifold N and any non-fibered class in H^1(N;Z) there exists a representation such that the corresponding twisted Alexander polynomial is zero. This is obtained by extending earlier work of the authors, together with results of Agol and Wise on separability of 3-manifold groups. …

Proposes SROF for row-wise fusion in federated learning for multivariate responses.

Let $1 \to H \to G \to Q \to 1$ be an exact sequence where $H= π_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide sufficient conditions to prove that $G$ is cubulable and construct examples satis…

Defines fiber-wise linear differential operators on vector bundles.

New bounds show limitations of sample-wise information-theoretic generalization.

Develops theory of relatively geometric actions on CAT(0) cube complexes.

Layer-wise preconditioning methods improve neural network optimization and feature learning.

Proposes QEP to mitigate quantization error propagation in layer-wise post-training quantization.

A rather complete investigation of anisotropic Bessel potential, Besov, and Hölder spaces on cylinders over (possibly) noncompact Riemannian manifolds with boundary is carried out. The geometry of the underlying manifold near its 'ends' is determined by a singularity function which leads naturally to the study of weigh…

We address the question of whether the property of being virtually special (in the sense of Haglund and Wise) is algorithmically decidable for finite, non-positively curved cube complexes. Our main theorem shows that it cannot be decided locally, i.e. by examining one hyperplane at a time. Specifically, we prove that t…

Let $G$ be a right-angled Artin group with defining graph $Γ$ and let $H$ be a finitely generated group quasi-isometric to $G(Γ)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $Γ$ does not have induced 4-cycle; (3) $Γ$ is star-rigid; then $H$ is commensurable to $G$. We show condition (2) is…

Study improves understanding of why agentic theorem provers succeed.

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic functi…

A new conformal prediction framework for two-stage models identifies stage-wise uncertainty.