nLab Urs Schreiber

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position

I work in
the Mathematics Division of Science
of New York University in Abu Dhabi
in the group of Prof. Hisham Sati leading the
Research Center for Quantum and Topological Systems
(see at CQTS for latest activity).

• page at NYU AD

contact

• [e-mail]

• [e-mail]

research

I am researching mathematical and fundamental physics (e.g. on Hypothesis H and Quantum Systems).

For latest talk notes see here.

selected talks

• Higher Topos Theory in Physics

  talk in: Zulip Category Theory Seminar (Jun 2024)

  video: YT

• Topological Quantum Programming with Linear Homotopy Types

  talk at: Homotopy Type Theory Electronic Seminar,, 02 Feb 2024

  video: YT

• Introduction to Quantum Hypothesis H

  talk at: M-Theory and Mathematics 2024, Jan 2024

  video: kt

• Higher Topos Theory in Physics

  talk at: Wolfram Science Winter School, Jan 2024

  video: YT

• Topological Quantum Gates in Homotopy Type Theory

  talk at: QFT and Cobordism 2023

  video: YT

• Higher Topos Theory in Physics

  talk at: Workshop on Homotopy Theory and Applications

  video: YT

• Topological Quantum Gates from M-Theory

  talk at: M-Theory and Mathematics 2023

  video: YT

• Topological Quantum Programming in TED-K

  talk at: PlanQC 2022

  video: YT

publications:

• see at GoogleScholar

• & on the arXiv preprint server

• & those about Hypothesis H

recent monograph:

• The Character Map in Non-Abelian Cohomology

  (with D. Fiorenza & H. Sati)

  World Scientific (2023)

  [doi:10.1142/13422]

Cover blurb: This book presents a novel development of fundamental and fascinating aspects of algebraic topology and mathematical physics: “extra-ordinary” and further generalized cohomology theories enhanced to “twisted” and differential-geometric form with focus on their rational approximation by generalized Chern character maps and on the resulting charge quantization laws in higher $n$-form gauge field theories appearing in string theory and in the classification of topological quantum materials.

Motivation for the conceptual re-development is the observation, laid out in the introductory chapter, that famous and famously elusive effects in strongly interacting (“non-perturbative”) physics demand “non-abelian” generalization of much of established generalized cohomology theory. But the relevant higher non-abelian cohomology theory (”higher gerbes“) has an esoteric reputation and has remained underdeveloped.

The book’s theme is that variously generalized cohomology theories are best viewed through their classifying spaces – possibly but not necessarily infinite-loop spaces – from which perspective the character map is really an incarnation of the fundamental theorem of rational homotopy theory, thereby uniformly subsuming not only the classical Chern character and a multitude of scattered variants that have been proposed, but now seamlessly applying in the previously elusive generality of (twisted, differential and) non-abelian cohomology.

In laying out this result with plenty of examples, we provide modernized introduction and review of fundamental classical topics: 1. abstract homotopy theory via model categories, 2. generalized cohomology in homotopical incarnation, 3. dg-algebraic rational homotopy theory, whose fundamental theorem we re-cast as a (twisted) non-abelian de Rham theorem which naturally induces the (twisted) non-abelian character map.

• further monographs on the nLab:

  • Introduction to Homological algebra

  • Introduction to Stable homotopy theory

  • geometry of physics – categories and toposes

• other:

  • Higher Prequantum Geometry, in: New Spaces for Mathematics and Physics, Cambr. Univ. Press (2021) $[$doi:10.1017/9781108854429$]$

  • Mathematical Foundations of Quantum Field and Perturbative String Theory, Proceedings of Symposia in Pure Mathematics 83 Amer. Math. Soc (2011) $[$ISBN:978-0-8218-5195-1$]$

research wiki

For making notes on math/phys I had started (in Nov 2008)

• the $n$Lab research wiki project

with edit logs and discussion being had on:

• the nForum discussion board.

For more on the $n$Lab, see:

• me on: What is… the nLab?

  (presentation at: Big Data in Pure Mathematics 2022)

• Wikipedia on: nLab.

more web logs

For logs of further activity see:

• my feed at X/Twitter (new)

• my old feed at X/Twitter (lost my password…)

• my channel on YouTube

while $n$Lab-edits are announced

• on the $n$Forum.

For links to technical discussions about math and physics see my pages:

• at MathOverflow

• at PhysicsOverflow

• at Physics.SE

• at QuantumComputing.SE.

teaching

See behind these links for lecture notes that I wrote:

date	lecture notes
winter $\,\,$ 2017	Mathematical Quantum Field Theory
summer 2017	Topological K-Theory
summer 2017	Topology
winter $\,\,$ 2016	Super Cartan Geometry
summer 2016	Complex oriented cohomology theory
summer 2016	Stable Homotopy Theory
summer 2015	Structure Theory for Higher WZW Terms
summer 2015	Higher Cartan Geometry
summer 2014	Homological Algebra

A list of further teaching in the past is here.

column

I used to write an irregular column at PhysicsForums Insights. Articles in the series include these:

on pre-quantum field theory

• Higher prequantum geometry I: The need for prequantum geometry

• Higher prequantum geometry II: The principle of extremal action – Comonadically

• Higher prequantum geometry III: The global action functional – Cohomologically

• Higher prequantum geometry IV: The covariant phase space – Transgressively

• Higher prequantum geometry V: The local observables – Lie theoretically

• Examples of Prequantum field theories I: Gauge fields

• Examples of Prequantum field theories II: Higher gauge fields

• Examples of Prequantum field theories III: Chern-Simons-type theories

• Examples of Prequantum field theories IV: Wess-Zumino-Witten-type theories

on perturbative quantum field theory

• Introduction to Perturbative Quantum Field Theory

on string theory

• It was 20 years ago today – the M-theory conjecture

• Homotopy Lie n-algebras in Supergravity

• Emergence from the superpoint

• Why supersymmetry? Because of Deligne’s theorem

• 11d Gravity from just the Torsion Constraint

• Spectral Standard Model and String Compactifications

• Why Higher Category Theory in Physics?

• Super p-Brane Theory from Super Homotopy Theory

joining our research

If you are a graduate student or young postdoc thinking of entering the Sati-Schreiber program, here is some basic reading to start with:

• for relevant basics of general homotopy theory and $\infty$-stacks/smooth $\infty$-groupoids:

  see the chapter Model categories in The Character Map in Non-Abelian Cohomology (chapter 1 in the printed version, appendix in the arXiv version)

  and for more details the nLab entries Introduction to Homotopy Theory or for yet more details geometry of physics – categories and toposes

• for relevant basics of rational homotopy theory and $L_\infty$-Lie algebras:

  see the part Non-abelian de Rham cohomology in The Character Map in Non-Abelian Cohomology

• for relevant basics of supergravity and super $p$-branes:

  see the articles Flux Quantization on 11d Superspace and Flux Quantization on M5-Branes

  and/or the nLab entry geometry of physics – supergeometry and superphysics

copyright statement

To the extent that it matters, my contributions to the nLab are copyrighted according to CC BY-SA 3.0.

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category: people