$\,$
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).
I am researching mathematical and fundamental physics (e.g. on Hypothesis H and Quantum Systems).
For latest talk notes see here.
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
video: YT
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)
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:
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$]$
For making notes on math/phys I had started (in Nov 2008)
with edit logs and discussion being had on:
For more on the $n$Lab, see:
me on: What is… the nLab?
(presentation at: Big Data in Pure Mathematics 2022)
Wikipedia on: nLab.
For logs of further activity see:
my feed at X/Twitter (new)
my old feed at X/Twitter (lost my password…)
while $n$Lab-edits are announced
For links to technical discussions about math and physics see my pages:
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.
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 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
on string theory
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
To the extent that it matters, my contributions to the nLab are copyrighted according to CC BY-SA 3.0.
Last revised on October 15, 2024 at 18:56:00. See the history of this page for a list of all contributions to it.