arXiv subjects

Shachar Carmeli

Publications and source records attributed to Shachar Carmeli.

27 records · Page 2Linked to original sources

arXiv2021

$(A_\infty,2)$-categories and relative 2-operads

We define the notion of a 2-operad relative to an operad, and prove that the 2-associahedra form a 2-operad relative to the associahedra. Using this structure, we define the notions of an $(A_\infty,2)$-category and $(A_\infty,2)$-algebra in spaces and in chain complexes over a ring. Finally, we show that for any continuous map $A \to X$, we can associate an $(A_\infty,2)$-algebra $θ(A \to X)$ in $\textsf{Top}$, which specializes to $θ(\text{pt} \to X) = Ω^2 X$ and $θ(A \to \text{pt}) = ΩA \times ΩA$.

math.CT↗

arXiv2021

Relative de Rham Theory on Nash Manifolds

For a Nash submersion $ϕ\colon X\to Y$, we study the complex $\mathcal{SDR}(ϕ)$ of Schwartz sections of the relative de Rham complex of $ϕ$. We define the notion of Schwartz sections of constructible sheaves on Nash manifolds and prove that $\mathcal{SDR}(ϕ)$ is homotopy equivalent to the Schwartz sections of the proper push-forward $ϕ_!\mathbb{R}_X$ of the constant sheaf $\mathbb{R}_X$. Using this equivalence, we show that $\mathcal{SDR}(ϕ)$ depends (up to homotopy equivalence) only on the homology type of the map $ϕ$. We also deduce that $\mathcal{SDR}(ϕ)$ has Hausdorff homology spaces.

math.AG↗

arXiv2021

Tenfold Topology of Crystals: Unified classification of crystalline topological insulators and superconductors

The celebrated tenfold-way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of non-spatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly-interacting electrons' non-crystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct groundstates as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold-way of Altland-Zirnbauer non-spatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold-way.

cond-mat.mes-hall↗

arXiv2021

Counting Surfaces Singular Along a Line in $P^3$

We enumerate the number of surfaces of degree $d$ in $P^3$ having a singular line of order $k$, passing through $δ$ generic points (where $δ$ is the dimension of moduli space of such surfaces).

math.AG↗

arXiv2020

Ambidexterity and Height

We introduce and study the notion of \emph{semiadditive height} for higher semiadditive $\infty$-categories, which generalizes the chromatic height. We show that the higher semiadditive structure trivializes above the height and prove a form of the redshift principle, in which categorification increases the height by one. In the stable setting, we show that a higher semiadditive $\infty$-category decomposes into a product according to height, and relate the notion of height to semisimplicity properties of local systems. We place the study of higher semiadditivity and stability in the general framework of smashing localizations of $Pr^{L}$, which we call \emph{modes}. Using this theory, we introduce and study the universal stable $\infty$-semiadditive $\infty$-category of semiadditive height $n$, and give sufficient conditions for a stable $1$-semiadditive $\infty$-category to be $\infty$-semiadditive.

math.AT↗

arXiv2020

Ambidexterity in Chromatic Homotopy Theory

We extend the theory of ambidexterity developed by M. J. Hopkins and J. Lurie and show that the $\infty$-categories of $T(n)$-local spectra are $\infty$-semiadditive for all $n$, where $T(n)$ is the telescope on a $v_{n}$-self map of a type $n$ spectrum. This extends and provides a new proof for the analogous result of Hopkins-Lurie on $K(n)$-local spectra. Moreover, we show that $K(n)$-local and $T(n)$-local spectra are respectively, the minimal and maximal $1$-semiadditive localizations of spectra with respect to a homotopy ring, and that all such localizations are in fact $\infty$-semiadditive. As a consequence, we deduce that several different notions of "bounded chromatic height" for homotopy rings are equivalent, and in particular, that $T(n)$-homology of $π$-finite spaces depends only on the $n$-th Postnikov truncation. A key ingredient in the proof of the main result is a construction of a certain power operation for commutative ring objects in stable $1$-semiadditive $\infty$-categories. This is closely related to some known constructions for Morava $E$-theory and is of independent interest. Using this power operation we also give a new proof, and a generalization, of a nilpotence conjecture of J.P. May, which was proved by A. Mathew, N. Naumann, and J. Noel.

math.AT↗

arXiv2019

On the Stability and Gelfand Property of Symmetric Pairs

A symmetric pair of reductive groups $(G,H,θ)$ is called stable, if every closed double coset of $H$ in $G$ is preserved by the anti-involution $g\mapsto θ(g^{-1})$. In this paper, we develop a method to verify the stability of symmetric pairs over local fields of characteristic 0 (Archimedean and $p$-adic), using non-abelian group cohomology. Combining our method with results of Aizenbud and Gourevitch, we classify the Gelfand pairs among the pairs \begin{align*} &(SL_n(F), (GL_k(F) \times GL_{n - k}(F)) \cap SL_n(F)), (U(B_1 \oplus B_2),U(B_1) \times U(B_2)),\\ &(GL_n(F),O(B)), (GL_n(F),U(B)), (GL_{2n}(F), GL_n(E)),(SL_{2n}(F), SL_n(E)), \end{align*} and the pair $(O(B_1 \oplus B_2),O(B_1) \times O(B_2))$ in the real case.

math.RT↗

arXiv2019

Étale Homotopy Obstructions of Arithmetic Spheres

Let $K$ be a field of characteristic $\ne 2$ and let $X$ be the affine variety over $K$ defined by the equation $$ X:\ a_0x_0^2 + \cdots + a_nx_n^2 = 1 $$ where $n\ge 0$ and $a_i\in K$. In this paper we compute the lowest mod 2 étale homological obstruction class to the existence of a $K$-rational point on $X$, and show that it is the cup product of the form $$ o_{n+1} = [a_0]\cup\cdots\cup[a_n]. $$ Our computation is an étale-homotopy analogue of the topological fact that Stiefel-Whitney classes are the homological obstructions to find a section to the unit sphere bundle of a real vector bundle.

math.AG↗

arXiv2013

Rational curves on the compactified moduli space of rational curves with marked points

We investigate the possible homological classes of rational curves on the moduli space $X_n=\bar{\mathcal{M}_{0,n}}$ of rational nodal curves with $n$ marked points. In the case of $X_5$ and $X_6$ the relevant homology classes belong to certain polyhedral cones which we describe explicitly. Not all homology classes in the cone correspond to rational curves, but all classes sufficiently far from the walls of the cone do. The proof for $X_5$ is explicit, the proof for $X_6$ was carried out using computers.

math.AG↗