arXiv subjects

Tomer M. Schlank

Publications and source records attributed to Tomer M. Schlank.

At least 37 records · Page 2Linked to original sources

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.

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arXiv2019

Monochromatic homotopy theory is asymptotically algebraic

In previous work, we used an $\infty$-categorical version of ultraproducts to show that, for a fixed height $n$, the symmetric monoidal $\infty$-categories of $E_{n,p}$-local spectra are asymptotically algebraic in the prime $p$. In this paper, we prove the analogous result for the symmetric monoidal $\infty$-categories of $K_{p}(n)$-local spectra, where $K_{p}(n)$ is Morava $K$-theory at height $n$ and the prime $p$. This requires $\infty$-categorical tools suitable for working with compactly generated symmetric monoidal $\infty$-categories with non-compact unit. The equivalences that we produce here are compatible with the equivalences for the $E_{n,p}$-local $\infty$-categories.

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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.

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arXiv2019

On d-Categories and d-Operads

We extend the theory of d-categories, by providing an explicit description of the right mapping spaces of the d-homotopy category of an $\infty$-category. Using this description, we deduce an invariant $\infty$-categorical characterization of the d-homotopy category. We then proceed to develop an analogous theory of d-operads, which model $\infty$-operads with (d -1)-truncated multi-mapping spaces, and prove analogous results for them.

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arXiv2018

Exact maximum-entropy estimation with Feynman diagrams

A classical longstanding open problem in statistics is finding an explicit expression for the probability measure which maximizes entropy with respect to given constraints. In this paper a solution to this problem is found, using perturbative Feynman calculus. The explicit expression is given as a sum over weighted trees.

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arXiv2018

On Bias and Rank

Given a hypersurface $X\subset \mathbb{P}^{N+1}_{\mathbb{C}}$ Dimca gave a proof showing that the cohomologies of X are the same as the projective space in a range determined by the dimension of the singular locus of X. We prove the analog of Dimca's result case when $\mathbb{C}$ is replaced with an algebraically closed field of finite characteristic and singular cohomology is replaced with $\ell$-adic étale cohomology. The Weil conjectures allow relating results about éatle cohomology to counting problems over a finite field. Thus by applying this result, we are able to get a relationship between the algebraic properties of certain polynomials and the size of their zero set.

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arXiv2017

The unramified inverse Galois problem and cohomology rings of totally imaginary number fields

We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the unramified inverse Galois problem. That is, we show that our methods can be used to determine that certain groups cannot be realized as the Galois groups of unramified extensions of certain number fields. To demonstrate the power of our methods, we give an infinite family of totally imaginary quadratic number fields such that $\text{Aut}(\text{PSL}(2,q^2))$ for $q$ an odd prime power, cannot be realized as an unramified Galois group over $K,$ but its maximal solvable quotient can. To prove this result, we determine the ring structure of the étale cohomology ring $H^*(\text{Spec }\mathcal{O}_K;\mathbb{Z}/ 2\mathbb{Z})$ where $\mathcal{O}_K$ is the ring of integers of an arbitrary totally imaginary number field $K.$

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arXiv2017

The Brauer-Manin obstruction to the local-global principle for the embedding problem

We study an analogue of the Brauer-Manin obstruction to the local-global principle for embedding problems over global fields. We will prove the analogues of several fundamental structural results. In particular we show that the (algebraic) Brauer-Manin obstruction is the only one to weak approximation when the embedding problem has abelian kernel. As a part of our investigations we also give a new, elegant description of the Tate duality pairing and prove a new theorem on the cup product.

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arXiv2017

Sylow theorems for $\infty$-groups

Viewing Kan complexes as $\infty$-groupoids implies that pointed and connected Kan complexes are to be viewed as $\infty$-groups. A fundamental question is then: to what extent can one "do group theory" with these objects? In this paper we develop a notion of a finite $\infty$-group: an $\infty$-group with finitely many non-trivial homotopy groups which are all finite. We prove a homotopical analog of the Sylow theorems for finite $\infty$-groups. We derive two corollaries: the first is a homotopical analog of the Burnside's fixed point lemma for $p$-groups and the second is a "group-theoretic" characterization of (finite) nilpotent spaces.

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arXiv2016

From weak cofibration categories to model categories

In [BaSc2] the authors introduced a much weaker homotopical structure than a model category, called a "weak cofibration category". We further showed that a small weak cofibration category induces in a natural way a model category structure on its ind-category, provided the ind-category satisfies a certain two out of three property. The purpose of this paper is to serve as a companion to the papers above, proving results which say that if a certain property or structure exists in the weak cofibration category, then the same property or structure also holds in the induced model structure on the ind-category. Namely, we consider the property of being left proper and the structures of a monoidal category and a category tensored over a monoidal category (in a way that is compatible with the weak cofibration structure). For the purpose of future reference, we consider the more general situation where we only have an "almost model structure" on the ind-category.

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arXiv2016

Sieves and the Minimal Ramification Problem

The minimal ramification problem may be considered as a quantitative version of the inverse Galois problem. For a nontrivial finite group $G$, let $m(G)$ be the minimal integer $m$ for which there exists a Galois extension $N/\mathbb{Q}$ that is ramified at exactly $m$ primes (including the infinite one). So, the problem is to compute or to bound $m(G)$. In this paper, we bound the ramification of extensions $N/\mathbb{Q}$ obtained as a specialization of a branched covering $ϕ\colon C\to \mathbb{P}^1_{\mathbb{Q}}$. This leads to novel upper bounds on $m(G)$, for finite groups $G$ that are realizable as the Galois group of a branched covering. Some instances of our general results are: $$ 1\leq m(S_m)\leq 4 \quad \mbox{and} \quad n\leq m(S_m^n) \leq n+4, $$ for all $n,m>0$. Here $S_m$ denotes the symmetric group on $m$ letters, and $S_m^n$ is the direct product of $n$ copies of $S_m$. We also get the correct asymptotic of $m(G^n)$, as $n \to \infty$ for a certain class of groups $G$. Our methods are based on sieve theory results, in particular on the Green-Tao-Ziegler theorem on prime values of linear forms in two variables, on the theory of specialization in arithmetic geometry, and on finite group theory.

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arXiv2015

A Projective Model Structure on Pro Simplicial Sheaves, and the Relative Étale Homotopy Type

In this work we shall introduce a new model structure on the category of pro-simplicial sheaves, which is very convenient for the study of étale homotopy. Using this model structure we define a pro-space associated to a topos, as a result of applying a derived functor. We show that our construction lifts Artin and Mazur's étale homotopy type [AM] in the relevant special case. Our definition extends naturally to a relative notion, namely, a pro-object associated to a map of topoi. This relative notion lifts the relative étale homotopy type that was used in [HaSc] for the study of obstructions to the existence of rational points. This relative notion enables to generalize these homotopical obstructions from fields to general base schemas and general maps of topoi. Our model structure is constructed using a general theorem that we prove. Namely, we introduce a much weaker structure than a model category, which we call a "weak fibration category". Our theorem says that a weak fibration category can be "completed" into a full model category structure on its pro-category, provided it satisfies some additional technical requirements. Our model structure is obtained by applying this result to the weak fibration category of simplicial sheaves over a Grothendieck site, where the weak equivalences and the fibrations are local in the sense of Jardine [Jar].

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arXiv2015

Model Structures on Ind Categories and the Accessibility Rank of Weak Equivalences

In a recent paper we introduced a much weaker and easy to verify structure than a model category, which we called a "weak fibration category". We further showed that a small weak fibration category can be "completed" into a full model category structure on its pro-category, provided the pro-category satisfies a certain two out of three property. In the present paper we give sufficient intrinsic conditions on a weak fibration category for this two out of three property to hold. We apply these results to prove theorems giving sufficient conditions for the finite accessibility of the category of weak equivalences in combinatorial model categories. We apply these theorems to the standard model structure on the category of simplicial sets, and deduce that its class of weak equivalences is finitely accessible. The same result on simplicial sets was recently proved also by Raptis and Rosický, using different methods.

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arXiv2014

A new model for pro-categories

In this paper we present a new way to construct the pro-category of a category. This new model is very convenient to work with in certain situations. We present a few applications of this new model, the most important of which solves an open problem of Isaksen [Isa] concerning the existence of functorial factorizations in what is known as the strict model structure on a pro-category. Additionally we explain and correct an error in one of the standard references on pro-categories.

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arXiv2014

A transchromatic proof of Strickland's theorem

In "Morava E-theory of symmetric groups", Strickland proved that the Morava E-theory of the symmetric group has an algebro-geometric interpretation after taking the quotient by a certain transfer ideal. This result has influenced most of the work on power operations in Morava E-theory and provides an important calculational tool. In this paper we give a new proof of this result as well as a generalization by using transchromatic character theory. The character maps are used to reduce Strickland's result to representation theory.

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arXiv2013

Functorial Factorizations in Pro Categories

In this paper we prove a few propositions concerning factorizations of morphisms in pro categories, the most important of which solves an open problem of Isaksen concerning the existence of certain types of functorial factorizations. On our way we explain and correct an error in one of the standard references on pro categories.

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arXiv2011

On the Brauer-Manin Obstruction Applied to Ramified Covers

The Brauer-Manin obstruction is used to explain the failure of the local-global principle for algebraic varieties. In 1999 Skorobogatov gave the first example of a variety that does not satisfy the local-global principle which is not explained by the Brauer-Manin obstruction. He did so by applying the Brauer-Manin obstruction to étale covers of the variety, and thus defining a finer obstruction. In 2008 Poonen gave the first example of failure of the local-global principle which cannot be explained for by Skorobogatov's étale-Brauer obstruction. However, Poonen's construction was not accompanied by a definition of a new finer obstruction. In this paper I shall present a possible definition for such an obstruction by allowing to apply the Brauer-Manin obstruction to some ramified covers as well, and show that this new obstruction can explain Poonen counterexample in the case of a totally imaginary number field.

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arXiv2011

The Étale Homotopy Type and Obstructions to the Local-Global Principle

In 1969 Artin and Mazur defined the étale homotopy type of an algebraic variety \cite{AMa69}. In this paper we define various obstructions to the local-global principle on a variety $X$ over a global field using the étale homotopy type of $X$ and the concept of homotopy fixed points. We investigate relations between those "homotopy obstructions" and connect them to various known obstructions such as the Brauer -Manin obstruction, the étale-Brauer obstruction and finite descent obstructions. This gives a reinterpretation of known arithmetic obstructions in terms of homotopy theory.

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