arXiv subjects

Nathan Perlmutter

Publications and source records attributed to Nathan Perlmutter.

11 recordsLinked to original sources

arXiv2022

Improved Batching Strategy For Irregular Time-Series ODE

Irregular time series data are prevalent in the real world and are challenging to model with a simple recurrent neural network (RNN). Hence, a model that combines the use of ordinary differential equations (ODE) and RNN was proposed (ODE-RNN) to model irregular time series with higher accuracy, but it suffers from high computational costs. In this paper, we propose an improvement in the runtime on ODE-RNNs by using a different efficient batching strategy. Our experiments show that the new models reduce the runtime of ODE-RNN significantly ranging from 2 times up to 49 times depending on the irregularity of the data while maintaining comparable accuracy. Hence, our model can scale favorably for modeling larger irregular data sets.

cs.LG↗

arXiv2017

Parametrized Morse Theory and Positive Scalar Curvature

We use the cobordism category constructed in arXiv:1703.01047 to the study the homotopy type of the space of positive scalar curvature metrics on a spin manifold of dimension > 4. Our methods give an alternative proof and extension of a recent theorem of Botvinnik, Ebert, and Randal-Williams from arXiv:1411.7408.

math.AT↗

arXiv2017

Cobordism Categories and Parametrized Morse Theory

Fix a tangential structure $\theta: B \longrightarrow BO(d+1)$ and an integer $k < d/2$. In this paper we determine the homotopy type of a cobordism category $\mathbf{Cob}^{\text{mf}, k}_{\theta}$, where morphisms are given by $\theta$-cobordisms $W: P \rightsquigarrow Q$ equipped with a choice of proper Morse function $h_{W}: W \longrightarrow [0, 1]$, with the property that all critical points $c \in W$ of $h_{W}$ satisfy the condition: $k < \text{index}(c) < d-k+1$. In particular, we prove that there is a weak homotopy equivalence $B\mathbf{Cob}^{\text{mf}, k}_{\theta} \simeq\Omega^{\infty}\mathbf{hW}^{k}_{\theta}$, where $\mathbf{hW}^{k}_{\theta}$ is a Thom spectrum associated to the space of Morse jets on $\mathbb{R}^{d+1}$. In the special case that $k = -1$, the equivalence $B\mathbf{Cob}^{\text{mf}, -1}_{\theta} \simeq\Omega^{\infty}\mathbf{hW}^{-1}_{\theta}$ follows from the work of Madsen and Weiss used in their celebrated proof of the Mumford conjecture. Following the methods of Madsen and Weiss we use the weak equivalence $B\mathbf{Cob}^{\text{mf}, k}_{\theta} \simeq\Omega^{\infty}\mathbf{hW}^{k}_{\theta}$ to give an alternative proof the "high-dimensional Madsen-Weiss theorem" of Galatius and Randal-Williams which identifies the homology of the moduli spaces, $BDiff((S^{n}\times S^{n})^{\# g}, D^{2n})$, in the limit $g \to \infty$.

math.AT↗

arXiv2016

Cobordism categories and moduli spaces of odd dimensional manifolds

We prove that the stable moduli space of $(n-1)$-connected, $n$-parallelizable, $(2n+1)$-dimensional manifolds is homology equivalent to an infinite loopspace for $n \geq 4, n \neq 7$. The main novel ingredient is a version of the cobordism category incorporating surgery data in the form of Lagrangian subspaces.

math.AT↗

arXiv2015

Homological Stability for Diffeomorphism Groups of High Dimensional Handlebodies

In this paper we prove a homological stability theorem for the diffeomorphism groups of high dimensional manifolds with boundary, with respect to forming the boundary connected sum with the product $D^{p+1}\times S^{q}$ for $|q - p| < \min\{p, q\} - 2$. In a recent joint paper with Boris Botvinnik (see arXiv:1509.03359 ), we identify the homology of $colim_{g\to \infty}BDiff((D^{n+1}\times S^{n})^{\natural g}, \; D^{2n})$ with that of the infinite loopspace $Q_{0}BO(2n+1)\langle n\rangle_{+}$, in the case that $n \geq 4$. Combining this "stable homology" calculation with this paper's homological stability theorem enables one to compute the (co)homology groups of $BDiff((D^{n+1}\times S^{n})^{\natural g}, D^{2n})$ in degrees $k \leq \tfrac{1}{2}(g - 4)$. This leads to the determination of the characteristic classes in degrees $k \leq \tfrac{1}{2}(g - 4)$ for all smooth fibre-bundles with fibre diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$.

math.AT↗

arXiv2015

Stable Moduli Spaces of High Dimensional Handlebodies

We study the moduli space of handlebodies diffeomorphic to $(D^{n+1}\times S^{n})^{\natural g}$, i.e. the classifying space $BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n})$ of the group of diffeomorphisms that restrict to the identity near a $2n$-dimensional disk embedded in the boundary, $\partial(D^{n+1}\times S^n)^{\natural g}$. We construct a map $colim_{g\to\infty}BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}) \longrightarrow Q_{0}BO(2n+1)\langle n \rangle_{+}$ and prove that it induces an isomorphism on integral homology in the case that $2n+1 \geq 9$. Above, $BO(2n+1)\langle n \rangle$ denotes the $n$-connective cover of $BO(2n+1)$. The (co)homology of the space $Q_{0}BO(2n+1)\langle n \rangle_{+}$ is well understood and so our results enable one to compute the homology groups $H_{k}(BDiff((D^{n+1}\times S^n)^{\natural g}, D^{2n}))$ in a range of degrees when $k << g$. Our main theorem can be viewed as an analogue of the Madsen-Weiss theorem for the moduli spaces of surfaces and the recent theorem of Galatius and Randal-Williams for the moduli spaces of manifolds of dimension $2n \geq 6$.

math.AT↗

arXiv2014

Linking forms and stabilization of diffeomorphism groups of manifolds of dimension 4n+1

Let $n \geq 2$. We prove a homological stability theorem for the diffeomorphism groups of $(4n+1)$-dimensional manifolds, with respect to forming the connected sum with $(2n-1)$-connected, $(4n+1)$-dimensional manifolds that are stably parallelizable. Our techniques involve the study of the action of the diffeomorphism group of a manifold $M$, on the linking form associated to the homology groups of $M$. In particular, we construct a geometric model for the linking form using the intersections of embedded and immersed $\mathbb{Z}/k$-manifolds. In addition to our main homological stability theorem, we prove several disjunction results for the embeddings and immersions of $\mathbb{Z}/k$-manifolds that could be of independent interest.

math.AT↗

arXiv2014

Homological Stability For The Moduli Spaces of Products of Spheres

We prove a homological stability theorem for moduli spaces of high-dimensional, highly connected manifolds, with respect to forming the connected sum with the product of spheres $S^{p}\times S^{q}$, for $p < q < 2p - 2$. This result is analogous to recent results of S. Galatius and O. Randal-Williams regarding the homological stability for the moduli spaces of manifolds of dimension $2n > 4$, with respect to forming connected sums with $S^{n}\times S^{n}$.

math.AT↗

arXiv2014

Homological Stability For Moduli Spaces of Odd Dimensional Manifolds

We prove a homological stability theorem for the moduli spaces of manifolds diffeomorphic to $\#^{g}(S^{n+1}\times S^{n})$, provided $n \geq 4$. This is an odd dimensional analogue of a recent homological stability result of S. Galatius and O. Randal Williams for the moduli space of manifolds diffeomorphic to $\#^{g}(S^{n}\times S^{n})$ for $n \geq 3$.

math.AT↗

arXiv2013

Cobordism categories of manifolds with Baas-Sullivan singularities, Part 2

For a given list of closed manifolds $\Sigma_k=(P_1,...,P_k)$, we construct a cobordism category $\mathbf{Cob}_{d}^{\Sigma_{k}}$ of embedded manifolds with Baas-Sullivan singularities of type $\Sigma_k$. Our main results identify the homotopy type of the classifying spaces $B\mathbf{Cob}_{d}^{\Sigma_{k}}$ of these cobordism categories with that of the infinite loop-space of a certain spectrum, $\mathsf{MT}_{\Sigma_{k}}(d)$. Our results generalize of the work of Galatius, Madsen, Tillmann, and Weiss from arXiv:math/0605249. It also generalizes the work of Genauer on the cobordism categories of manifolds with corners from arXiv:0810.0581.

math.AT↗

arXiv2012

Cobordism Category of Manifolds With Baas-Sullivan Singularities, Part I

For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectrum $\text{MT}(d)$ introduced in [arXiv:math/0605249]. We obtain an analogue of the Bockstein-Sullivan exact couple that arises between the classical bordism theories $MO$ and $MO_{P}$ on the level of cobordism categories.

math.AT↗