Lior Yanovski

(math homepage)

I am a senior lecturer (equivalent to an assistant professor) at the Hebrew University of Jerusalem. I am interested in stable homotopy theory and higher categories, and, in particular, in chromatic homotopy theory, algebraic K-theory, and topological Hochschild homology. 

A lot of my research focuses on the notion of higher semiadditivity (a.k.a ambidexterity) and its applications. I am currently mostly interested in redshift phenomena relating categorification and chromatic height. 

General info

Papers and preprints


We reprove Lurie's result (from Elliptic Chomology III) that the semiadditive cardinality of a p-typical pi-finite space A in pi_0(E_n) is an integer given by the Baez-Dolan homotopy cardinality of the n-fold free loop space L^n A. Our method differs drastically from Lurie's, which relies on tempered cohomology, and employs the higher descent results on chromatically localized algebraic K-theory from our previous paper. 

We show that T(n+1)-localized algebraic K-theory of stable oo-categories of chromatic height up to n satisfies descent with respect to p-local pi-finite group actions (aka higher descent). We deduce from this that it also takes height n cyclotomic extensions to height n+1 cyclotomic extensions (aka cyclotomic redshift).   We deduce that K(n+1)-locally algebraic K-theory satisfies hyperdescent along the cyclotomic tower. The interest in this example stems from the fact that T(n+1)-locally this fails by the work of Burklund, Hahn, Levy, and Schlank, which therefore disproves the telescope conjecture at all primes and all heights n > 1. 

This paper partially answers the question of Baez on formalizing the relationship between classical Euler characteristic and the Baez-Dolan homotopy cardinality, suggested by many heuristics. The answer is given by a common p-typical generalization for every odd prime p. This generalization is obtained by an l-adic extrapolation of  Euler characteristics with respect to the Morava K(n)-s to n = -1.

We develop a general character theory for pointwise dualizable local systems on suitably finite spaces, using the higher categorical perspective on topological Hochschild homology. A key ingredient is a very general form of the induced character formula.  As an application, we reprove and sharpen results on Becker-Gottlieb transfers, the Hochschild homology of Thom spectra, and the additivity of traces in stable oo-categories.

We extend the classical Discrete Fourier Transform (DFT) to higher chromatic heights generalizing the work of Hopkins and Lurie. Some of the applications include affineness and Eilenberg-Moore type results in T(n)-local spectra and the computation of the discrepancy spectrum of  Morava E-theory. 

The number of maximal (with respect to inclusion) abelian subgroups of a given finite p-group is shown to be congruent to 1 modulo p. This elementary observation has some relevance to the computation of Euler characteristics of classifying spaces of finite groups with respect to the Morava K-theories,   

We lift all the abelian Galois extensions of the K(n)-local sphere to Galois extensions of the T(n)-local sphere,  by realizing them as higher semiadditive analogues of cyclotomic extensions. This relies on the oo-semiadditivity of the T(n)-local categories and the theory of semiadditive height from our previous work. We also use this to construct some non-trivial elements of the T(n)-local Picard group, by a general form of Kummer theory and discrete Fourier transform, which we also develop in this paper.  

For every right adjoint functor of oo-categories, we have a factorization through a transfinite tower of monadic functors.  Under presentability assumptions, this tower stabilizes on a coreflection onto a full subcategory of the source.  A dual construction (which inspired this work) recovers the "long homology localization tower" of Dwyer and Farjoun. 

We prove that the oo-categories of T(n)-local spectra are oo-semiadditive for all n, answering a question of Hopkins. This extends the results of Hopkins-Lurie on the higher semiadditivity of the oo-categories of K(n)-local spectra and the results of Kuhn on the 1-semiadditivity of T(n)-local spectra. We further classify all higher semiadditive localizations of spectra with respect to a homotopy ring and show that the T(n)-localizations are in a precise sense the maximal ones.

We define and study the notion of semiadditive height for higher semiadditive oo-categories, which generalizes the chromatic height in stable homotopy theory.  We show that a stable higher semiadditive oo-category decomposes into a product according to height and that local systems on an n-connected pi-finite space, valued in a height n higher semiadditive oo-category, exhibit a form of "semicmplicity". We also show that this notion of height exhibits a form of a red-shift principle, similar to the Ausoni-Rognes chromatic red-shift phenomenon in algebraic K-theory.  

A rather technical note on models of oo-categories and oo-operads with (d-1)-truncated (multi-)mapping spaces. The main (model-independent) application is that these form reflective subcategories of the oo-categories of oo-categories and oo-operads respectively, and that the reflection is given by truncating the (multi-)mapping spaces.  

For two reduced oo-operads P and Q, we show that if all the multi-mapping spaces of P are d1-connected and those of Q are d2-connected, then the multi-mapping spaces of the Boardman-Vogt tensor product of P and Q are (d1+d2+2)-connected. We deduce this from a relative version, which might be of interest on its own.  Along the way, we extend some of the truncated/connected calculus of oo-topoi to arbitrary presentable oo-categories. 

We give an abstract oo-categorical treatment of "descent along adjoints",  which allows us to neatly unify seemingly different formulas for counting "conjugate objects".  Among the examples: Galois cohomogloy for classifying Galois forms, lim^1 of a tower of groups for classifying Postnikov conjugates of a space, and the double cost formula for the Mislin genus of a nilpotent group. 

Provides a streamlined and detailed exposition of the work of Kazhdan and Lusztig on the algebro-geometric properties of affine Springer fibers for a connected semisimple algebraic group. 


Some talks

A seminar talk about my joint work with Shachar and Tomer on constructing a chromatic analogue of the discrete Fourier transform and its applications for affiness and Eilenberg-Moore type results in the T(n)-local category. 

A seminar talk about my joint work with Shachar and Tomer on constructing Galois extensions of the T(n)-local sphere using higher semiadditivity.  

A talk at the HHH program about my joint work with Shachar and Tomer on the higher semiadditivity of the T(n)-local categories. 

A talk at YTM on my joint work with Tomer regarding the connectivity of the Boardman-Vogt tensor product of reduced oo-operads.

A colloquium talk about an old project with Tomer (waiting to be written). We use l-adic extrapolation of Euler characteristics for Morava K-theories of different height to construct a generalized homotopy cardinality function answering a question of Baez. 

Expository notes

Review of various trace maps in algebraic K-theory. 

A rough overview of the Goerss-Hopkins-Miller obstruction theory for realizing E_oo ring structures, using the theory of synthetic spectra of Pstragowski.  

An introduction to factorization homology and tensor-excision, for a workshop on the cobordism hypothesis given by David Ayala and John Francis. at Caesaria 2018.

From the report of the Oberwolfach Arbeitsgemeinschaft: Elliptic Cohomology according to Lurie .

Notes for my talks at the workshop on super-strong approximation. Sketches Nori's proof of the "quantitative strong approximation theorem" with an introduction to algebraic geometry and the theory of algebraic groups. 

Descent in Algebraic K-theory - notes




"Good artists borrow, great artists steal"  -  Lior Yanovski.