nLab Urs Schreiber

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Urs Schreiber

contents

animated logo of CQTS

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).


Contact


Research

I am researching mathematical and fundamental quantum physics, with a focus on using tools of algebraic topology and geometric homotopy theory for understanding elusive strongly-coupled quantum systems with applications to future topological quantum technology.

Projects:

  1. Hypothesis H on flux quantization in M-theory.

  2. Geometric engineering of anyons on M5-branes.

  3. Quantum language via linear homotopy types.


Selected talks

complete list of invited talks: here.



Publications

Research articles — see the lists:


Edited collection:

Cover blurb: Conceptual progress in fundamental theoretical physics is linked with the search for the suitable mathematical structures that model the physical systems. Quantum field theory (QFT) has proven to be a rich source of ideas for mathematics for a long time. However, fundamental questions such as “What is a QFT?” did not have satisfactory mathematical answers, especially on spaces with arbitrary topology, fundamental for the formulation of perturbative string theory.

This book contains a collection of papers highlighting the mathematical foundations of QFT and its relevance to perturbative string theory as well as the deep techniques that have been emerging in the last few years.

The papers are organized under three main chapters: Foundations for Quantum Field Theory, Quantization of Field Theories, and Two-Dimensional Quantum Field Theories. An introduction, written by the editors, provides an overview of the main underlying themes that bind together the papers in the volume.


Research monographs:

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 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.


In this book we prove classification results for stable equivariant Γ \Gamma -principal bundles in the case when the underlying homotopy type ʃΓ\esh \, \Gamma of the topological structure group Γ\Gamma is truncated, meaning that its homotopy groups vanish in and above some degree nn. We discuss how this coincides with the classification of equivariant higher non-abelian gerbes and generally of equivariant principal ∞ \infty -bundles with structure n n -group ʃΓ\esh \Gamma; and we show how the equivariant homotopy groups of the respective classifying GG-spaces are given by the non-abelian group cohomology of the equivariance group with coefficients in ʃΓ\esh \Gamma.

The result is proven in a conceptually transparent manner as a consequence of a smooth Oka principle which becomes available after faithfully embedding traditional equivariant topology into the singular-cohesive homotopy theory of globally equivariant higher stacks. This works for discrete equivariance groups GG acting properly on smooth manifolds (“proper equivariance”) with resolvable singularities whence we are equivalently describing principal bundles on good orbifolds.

In setting up this proof, we re-develop the theory of equivariant principal bundles from scratch by systematic use of Grothendieck‘s internalization. In particular we prove that all the intricate equivariant local triviality conditions considered in the literature are automatically implied by regarding GG-equivariant principal bundles as principal bundles internal to the B G B G -slice of the ambient cohesive ∞ \infty -topos. We also show that these conditions are all equivalent. Generally we find that the characteristic subtle phenomena of equivariant classifying theory all reflect basic modal properties of singular-cohesive homotopy theory (hence of cohesive global equivariant homotopy theory).



Introductory monographs on the nLab:


Other:


Research wiki

nLab banner

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

with edit logs and discussion being had on:

For more on the nnLab, see:


Web logs

For logs of further activity see:

and last not least:


Teaching

See behind these links for lecture notes that I wrote:

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

on perturbative quantum field theory

on string 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:


Copyright statement

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





category: people

Last revised on April 23, 2026 at 13:10:09. See the history of this page for a list of all contributions to it.

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