arXiv research

Simplified presentations for non-orientable surface mapping class groups.

The paper provides infinite presentations for surface groups.

This paper provides an infinite presentation for a subgroup of mapping class groups of non-orientable surfaces.

Study infinite group presentations and their Dehn functions.

We give an infinite presentation for the mapping class group of a non-orientable surface with boundary components. The presentation is a generalization of the presentation given by the second author [15].

We construct a finitely presented group with infinitely many non-homeomorphic asymptotic cones. We also show that the existence of cut points in asymptotic cones of finitely presented groups does, in general, depend on the choice of scaling constants and ultrafilters.

Accessible groups with infinitely many ends have infinitely many twisted conjugacy classes.

We give an infinite presentation for the mapping class group of a non-orientable surface. The generating set consists of all Dehn twists and all crosscap pushing maps along simple loops.

Develops trace class operators and inverse Laplacian theory for infinite dimensions.

In earlier work, we provided a general description of the forces of attraction and repulsion, encountered by two parallel vertical plates of infinite extent and of possibly differing materials, when partially immersed in an infinite liquid bath and subject to surface tension forces. In the present study, we examine som…

For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fréchet-Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if \cX is an orbifold, presented by a proper étale Lie groupoid, then Map(M…

In this article we present infinitely many 3-manifolds admitting infinitely many universally tight contact structures each with trivial Ozsvath-Szabo contact invariants. By known properties of these invariants the contact structures constructed here are non weakly symplectically fillable.

Introduces infinite-dimensional differential geometry using Bastiani calculus.

Study on infinite energy maps from surfaces to CAT(0) spaces.

We define a limiting $\mathfrak{sl}_N$ Khovanov-Rozansky homology for semi-infinite positive multi-colored braids, and we show that this limiting homology categorifies a highest-weight projector for a large class of such braids. This effectively completes the extension of Cautis' similar result for infinite twist braid…

We revisit the issue of the existence of infinitely many distinct prime knots with the same Alexander invariant. We present infinitely many distinct families, each family made up of infinitely many distinct knots. Within each family, the Alexander invariant is the same. Unlike other examples in the literature, ours are…

Study infinite Euclidean distance discriminants of algebraic varieties.

This paper presents a natural extension of stagewise ranking to the the case of infinitely many items. We introduce the infinite generalized Mallows model (IGM), describe its properties and give procedures to estimate it from data. For estimation of multimodal distributions we introduce the Exponential-Blurring-Mean-Sh…

Cyclohedra are a well-known infinite familiy of finite-dimensional polytopes that can be constructed from centrally symmetric triangulations of even-sided polygons. In this article we introduce an infinite-dimensional analogue and prove that the group of symmetries of our construction is a semidirect product of a degre…

In this paper we present in a topological way the construction of the orientable surface with only one end and infinite genus, called \emph{The Infinite Loch Ness Monster}. In fact, we introduce a flat and hyperbolic construction of this surface. We discuss how the name of this surface has evolved and how it has been h…

Study finds infinite non-fibered twisted torus knots.

New invariant detects infinite order cabled knots.

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense sub…

The paper studies the combinatorial p-th Calabi flow for finite and infinite circle patterns.

Study of Torelli group action on complex of cycles yields infinite generation result.

Study of infinitely deep but narrow neural networks using NTK theory.

Lectures on deep learning properties in infinite and large-width networks.

Survey of spectral theory and dynamics for infinite volume hyperbolic manifolds.

We present the Infinite Latent Events Model, a nonparametric hierarchical Bayesian distribution over infinite dimensional Dynamic Bayesian Networks with binary state representations and noisy-OR-like transitions. The distribution can be used to learn structure in discrete timeseries data by simultaneously inferring a s…

The abstract discusses transversality for infinite dimensional manifolds.

We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing t…

Develops control and observer methods for complex systems.

Study confirms infinitely many non-characterizing slopes for various knots.

Infinite Hidden Markov Models (iHMM's) are an attractive, nonparametric generalization of the classical Hidden Markov Model which can automatically infer the number of hidden states in the system. However, due to the infinite-dimensional nature of transition dynamics performing inference in the iHMM is difficult. In th…

The paper finds pseudo-Anosov-like maps on an infinite ladder surface.

In the present note, we will show that there are infinitely many composite twisted torus knots.

This article presents some methods to control the bottom of the spectrum of the Laplacian $λ_0$ on hyperbolic surfaces with infinite volume. Our first result bounds the $λ_0$ of a geometrically finite surface in terms of the geometry of its convex core. We then focus on infinite type periodic hyperbolic surfaces built …

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid gr…

We present examples of foliations with infinite dimensional basic symplectic and com- plex cohomologies, along with a general sufficient condition for such phenomena. This puts re- strictions on possible generalizations of several finiteness results from Riemannian foliations to any broader class. The examples are also…

Develops theory for conditional optimal transport in infinite-dimensional spaces.

In the present paper, the notion of $\mathbb{R}$-complex Finsler space with Infinite Series ($α, β$)- metric $\dfrac{β^2}{β- α}$ is defined. The Fundamental metric fields $g_{ij}$, $g_{i\bar{j}}$, their determinants and the inverse of these tensor fields are obtained. Also some properties of these spaces are studied.

We consider an infinite horizon portfolio problem with borrowing constraints, in which an agent receives labor income which adjusts to financial market shocks in a path dependent way. This path-dependency is the novelty of the model, and leads to an infinite dimensional stochastic optimal control problem. We solve the …

The report analyzes infinite-dimensional output space regression.

In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all "separating twists", all "bounding pair maps", and all "commutators of simply intersecting pairs" and whose relations all come from a short list of topological …

New method for quandle presentations of surface knots in 4-manifolds.

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams o…

This research develops approximation theory for OOMs of infinite-dimensional processes.

The paper explores infinite-dimensional nonholonomic and vakonomic systems.