Shai Keidar

Math homepage

I am a PhD student in the Hebrew University of Jerusalem supervised by Prof. Tomer Schlank . I am currently a visiting student at the University of Chicago.

My mathematical fields of interest include chromatic homotopy theory, higher semiadditivity, algebraic K-theory and trace methods, representation theory and higher (higher) categories.


Contact: shai.keidar (at) mail.huji.ac.il

Projects in preparation

On The Telescopic Picard Group - Draft

This is a joint work with Tomer Schlank.

In this work we define formally the notion of an "asymptotically defined natural transformation", and use the periodicity theorem to construct one on the category of compact spectra of type n. Inverting it, we get the category of compact T(n)-local spectra. We show that any category with an asymptotically defined natural isomorphism (in particular compact T(n)-local spectra) admits a coherent group action and use it to show there is an embedding of a large group in the automorphism of this category, which are identified with the telescopic Picard group.

Utilizing Kummer theory we construct a lift of a non-abelian Galois extensions from the K(n)-local category to the T(n)-local categry.

Higher Galois Theory

We show that in a presentably symmetric monoidal category C, if G is a C-dualizable group then faithful Galois extensions are co-representable: A faitfhul Galois extension in C is the same data as a symmetric monoidal functor from Spc^{BG} to C.

Using this we show that for nice semiadditive categories of height n there is an pro-(n+1)-finite Galois space classifiying n-finite Galois extensions, extending a result of Mathew for stable categories. We also extend Kummer theory to this setting, and discuss Galois-closure of categories.

Cyclic Azumaya Algebras

Using the co-representability of Galois extensions (see above), we prove that for a semiadditive category C, the data of a cyclic Galois extension and a strict unit is equivalent to a symmetric monoidal, colimit preserving functor from Mod_{Z}(C)^{B Z/m} to C.

Given such a symmetric monoidal functor F, we define its corresponding cyclic Azumaya algebra as F(Z/m⋉Z) extending the constructions of Baker-Richter-Syzmik and Mor.

We use it to construct a cyclic subgroup of order p-1 in the telescopic Brauer group.

Semiadditive Alternating Powers

This is a joint work with Shauly Ragimov.

In this work we extend the construction of the alternating power of a module to the higher semiadditive settings, where the symmetric group admits many more characters (even when p=2). Similarly to how alternating and symmetric powers decategorify to power operations on KU, we show that our alternating powers decategorify to "twisted semiadditive" power operations.

For categories with cyclotomically closed unit of height n, characters of the (n+1)-st stable stem induce a character on all symmetric groups. We show that for these characters all alternating powers assemble to an algebra, generalizing the classical symmetric and exterior algebras.

Using iterative application of the induced character formula of Carmeli-Cnossen-Ramzi-Yanvoski we prove a height induction formula for two categories of interest and use it to compute the monoidal dimensions of alternating powers in low heights.

Full-Dualizability and Lax Tensor Products

This is a joint work with Leor Neuhause, Tomer Schlank, Lior Yanvoski.

We define, using the Gray tensor product, the lax external product of a d-monoidal (∞,∞)-category and an e-monoidal (∞,∞)-category, returning and (d+e)-monoidal (∞,∞)-category.
Denoting by FD(n,d) the free d-monoidal category generated by an n-dualizable object we show that the lax external product of FD(n,d) and FD(m,e) identifies with FD(n+m,d+e).

In particular we show that FD(∞,∞) is an idempotent algebra with respect to a lax tensor product and assuming the cobordism hypothesis get a multiplicative-structure Pontryagin-Thom isomorphism.

In a work in progress we construct maps from the lax external product of framed tangle categories to an appropriate tangle category. Thus reformulating the Tangle hypothesis to proving that these maps are isomorphisms.

Organization

Homotopy Theory: Revisiting Localizations and Beyond

Yonatan Harpaz, Leor Neuhauser, Tomer Schlank, Lior Yanovski and myself organized the Homotopy Theory: Revisiting Localizations and Beyond in June 2024

Caesarea Workshop on Topological Cyclic Homology

Lior Yanovski, Shay Ben-Moshe and myself organized the Caesarea workshop on topological cyclic homology in May 2023. Lecture notes from all talks are available.

Sea of Galilee workshop on algebraic models for spaces

Tomer Schlank, Segev Cohen and myself organized the Sea of Galilee workshop on algebraic models for spaces in September 2021.

Research Talks

Telescopic Picard, Brauer and Galois Groups
UIUC Topology Seminar

Higher Galois Theory in Chromatic Homotopy Theory
MIT Topology Seminar

Higher Galois Theory
Young topologists meeting 2024, Münster, Germany
Slides

The chromatic Hecke algebra
Poster, Young topologists meeting 2023, EPFL, Switzerland
Poster

The Telescopic Galois, Picard and Brauer Groups
Poster, Young topologists meeting 2022, Copnehagen, Denmark
Poster

The telescopic Galois, Picard, and Brauer Groups
Jerusalem-Münster Homotopy Fridays 2021, held over Zoom
Slides

Conference and Workshop talks

Ambidexterity and chromatic cyclotomic extensions
Arbeitsgemeinschaft: Algebraic K-Theory and the Telescope Conjecture, MFO Oberwolfach 2024
Notes

The Prismatic cohomology of Z and
Talbot 2024, Texas

Polygonic spectra
Caesarea workshop on topological cyclic homology 2023
Notes

The Bousfield-Kuhn Functor
Sea of Galilee workshop on algebraic models for spaces 2021, Sea of Galilee, Israel
Notes

Seminar talks and notes

Equivalent definitions of dualizable categories
Dualizable categories and continouous K-theory seminar, University of Chicao 2024
Notes

Thom spectra and orientations of E  theory
Chromatic Nullstellensatz seminar, Jerusalem 2024
Notes

Lichtenbaum-Quillen for ℓ^hZ
Telescope conjecture seminar, Jerusalem 2024
Notes

Boundedness in cyclotomic spectra
Telescope conjecture seminar, Jerusalem 2024
Notes

The cyclic category and THH
Münster-Jerusalem seminar about TC, 2023
Notes

The spectrum of a T(n)-local commutative algebra
Chromatic Nullstellensatz seminar organized by Paul VanKoughnett, Bonn 2022
Notes

Modular forms
Algebraic geometry and number theory lunch seminar, Jerusalem 2021
Video

Spivak normal structure
Differential topology and the h-cobordism seminar, Jerusalem 2021
Notes

Morse theory
Differential topology and the h-cobordism seminar, Jerusalem 2020
Notes

Master's thesis

I completed my master's thesis in geometric representation theory under the guidance of Prof. Dmitry Gourevitch. You can find it here.