The homotopy type of the polyhedral product for shifted complexes

We prove a conjecture of Bahri, Bendersky, Cohen and Gitler: if K$K$ is a shifted simplicial complex on n$n$ vertices, _X1,…,_Xn$X_1,\dots ,X_n$ are pointed connected CW$CW$-complexes and C_Xi$C{X}_i$ is the cone on  _Xi$X_i$, then the polyhedral product determined by K$K$ and the pairs (C_Xi,_Xi)$(C{X}_i,X_i)$ is homotopy equivalent to a wedge of suspensions of smashes of the _Xi$X_i$’s. Earlier work of the authors dealt with the special case where each _Xi$X_i$ is a loop space. New techniques are introduced to prove the general case. These have the advantage of simplifying the earlier results and of being sufficiently general to show that the conjecture holds for a substantially larger class of simplicial complexes. We discuss connections between polyhedral products and toric topology, combinatorics, and classical homotopy theory.     

Homotopy of area decreasing maps by mean curvature flow

Let f:M→N$f:M\to N$ be a smooth area decreasing map between two Riemannian manifolds (M,_gM)$(M,{\mathrm{g}}_M)$ and (N,_gN)$(N,{\mathrm{g}}_N)$. Under weak and natural assumptions on the curvatures of (M,_gM)$(M,{\mathrm{g}}_M)$ and (N,_gN)$(N,{\mathrm{g}}_N)$, we prove that the mean curvature flow provides a smooth homotopy of f to a constant map.     

Unilateral small deviations of processes related to the fractional Brownian motion

Let x(s)$x\left(s\right)$, s∈R^d$s\in R^d$ be a Gaussian self-similar random process of index H$H$. We consider the problem of log-asymptotics for the probability _pT$p_T$ that x(s)$x\left(s\right)$, x(0)=0$x\left(0\right)=0$ does not exceed a fixed level in a star-shaped expanding domain T⋅Δ$T\cdot \Delta$ as T→∞$T\to \infty$. We solve the problem of the existence of the limit, θ?lim(−log_pT)/(logT)^D$\theta ?lim(-log{p}_T)/{(logT)}^D$, T→∞$T\to \infty$, for the fractional Brownian sheet x(s)$x\left(s\right)$, s∈[0,T]²$s\in {[0,T]}^2$ when D=2$D=2$, and we estimate θ$\theta$ for the integrated fractional Brownian motion when D=1$D=1$.     

Symplectic Calabi–Yau 6-manifolds

In this article, we construct simply connected symplectic Calabi–Yau 6-manifolds by applying Gompf's symplectic fiber sum operation along T⁴${\mathbb{T}}^4$. Using our method, we also construct symplectic non-Kähler Calabi–Yau 6-manifolds with fundamental group Z$\mathbb{Z}$. This paper also produces the first examples of simply connected and non-simply connected symplectic Calabi–Yau 6-manifolds with fundamental groups _Zp×_Zq${\mathbb{Z}}_p\times {\mathbb{Z}}_q$, and Z×_Zq$\mathbb{Z}\times {\mathbb{Z}}_q$ for any p≥1$p\ge 1$ and q≥2$q\ge 2$via co-isotropic Luttinger surgery.     

Depths and Cohen–Macaulay properties of path ideals

Given a tree T on n vertices, there is an associated ideal I   of R[_x1,…,_xn]$R[x_1,\dots ,x_n]$ generated by all paths of a fixed length ? of T  . We classify all trees for which R/I$R/I$ is Cohen–Macaulay, and we show that an ideal I whose generators correspond to any collection of subtrees of T satisfies the König property. Since the edge ideal of a simplicial tree has this form, this generalizes a result of Faridi. Moreover, every square-free monomial ideal can be represented (non-uniquely) as a subtree ideal of a graph, so this construction provides a new combinatorial tool for studying square-free monomial ideals.     

Asymptotics of a Brownian ratchet for protein translocation

Protein translocation in cells has been modelled by Brownian ratchets  . In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion _(Xt)t≥0${\left(X_t\right)}_{t\ge 0}$ with a changing reflection point _(Rt)t≥0${\left(R_t\right)}_{t\ge 0}$. The rate of change of _Rt$R_t$ is γ(_Xt−_Rt)$\gamma (X_t-R_t)$ and the new reflection boundary is distributed uniformly between _Rt−$R_{t-}$ and _Xt$X_t$. The asymptotic speed of the ratchet scales with γ^1/3${\gamma }^{1/3}$ and the asymptotic variance is independent of γ$\gamma$.     

On Mann iteration in Hilbert spaces

Let K$K$ be a compact convex subset of a real Hilbert space H$H$; T:K→K$T:K\to K$ a hemicontractive map. Let {_αn}$\left\{{\alpha }_n\right\}$ be a real sequence in [0,1] satisfying appropriate conditions; then for arbitrary _x0∈K$x_0\in K$, the sequence {_xn}$\left\{x_n\right\}$ defined iteratively by _xn=_αnxn−1+(1−_αn)T_xn$x_n={\alpha }_n{x}_{n-1}+(1-{\alpha }_n)T{x}_n$, n≥1$n\ge 1$ converges strongly to a fixed point of T$T$.     

Isoperimetric domains of large volume in homogeneous three-manifolds

Given a non-compact, simply connected homogeneous three-manifold X   and a sequence _{Ωn}n${\{{\Omega }_n\}}_n$ of isoperimetric domains in X   with volumes tending to infinity, we prove that, as n→∞$n\to \infty$:

(1)

The radii of the _Ωn${\Omega }_n$ tend to infinity.     

Zero separation results for solutions of second order linear differential equations

The oscillation of solutions of f^″+Af=0$f^{″}+Af=0$ is discussed by focusing on four separate situations. In the complex case A$A$ is assumed to be either analytic in the unit disc D$\mathbb{D}$ or entire, while in the real case A$A$ is continuous either on (−1,1)$(-1,1)$ or on (0,∞)$(0,\infty )$. In all situations A$A$ is expected to grow beyond bounds that ensure finite oscillation for all (non-trivial) solutions, and the separation between distinct zeros of solutions is considered.     

Voevodsky's mixed motives versus Kontsevich's noncommutative mixed motives

We prove that the quotient of Voevodsky's category of geometric mixed motives _DMgm${\mathrm{DM}}_{\mathrm{gm}}$ by the endofunctor −⊗Q(1)[2]$-\otimes \mathbb{Q}(1)[2]$ embeds fully-faithfully into Kontsevich's category of noncommutative mixed motives KMM. We show moreover that this embedding is compatible with the one between Chow motives and noncommutative Chow motives. Along the way, we relate KMM with Morel–Voevodsky's stable A¹$A^1$-homotopy category, recover the twisted algebraic K-theory of Kahn–Levine from KMM, and extend Elmendorf–Mandell's foundational work on multicategories to a broader setting.     

A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator _THar$T_{\mathrm{Har}}$ was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Ω$\Omega$ of some Euclidean space. In this present work the authors define an extensive class of _THar$T_{\mathrm{Har}}$-like self-adjoint operators on the Hilbert function space _L2(Ω);$L_2(\Omega );$ but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω$\Omega$ now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such _THar$T_{\mathrm{Har}}$-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in _L2(Ω)$L_2(\Omega )$ that does not lie within the usual Sobolev Hilbert function space W²(Ω)$W^2(\Omega )$. These _THar$T_{\mathrm{Har}}$-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω$\mathrm{\partial }\Omega$, and may have non-empty essential spectra.     

The asymptotic behaviour of symbolic generic initial systems of generic points

Consider the ideal I⊆K[x,y,z]$I\subseteq K[x,y,z]$ corresponding to points _p1,…,_pr$p_1,\dots ,p_r$ of P²${\mathbb{P}}^2$. We study the symbolic generic initial system  _{gin_(I(m))}m${\left\{\text{gin}\left(I^{\left(m\right)}\right)\right\}}_m$ of such an ideal and its behaviour as m$m$ gets large. In particular, we describe the limiting shape   of this system explicitly when _p1,…,_pr$p_1,\dots ,p_r$ are generic, assuming that the Uniform SHGH Conjecture holds for r≥9$r\ge 9$. The symbolic generic initial system and its limiting shape reflect information about the asymptotic behaviour of Hilbert functions of fat point ideals.