New Spaces for Mathematics and Physics





physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics


This page collects pointers related to the conference and book project:

on current developments regarding foundations of geometry in mathematics and physics.



Volume I: New Spaces in Mathematics

I.1. Differential geometry I.1.1 Diffeologies (Patrick Iglesias-Zemmour) I.1.2 New methods for old spaces: synthetic differential geometry (Anders Kock) I.1.3 Microlocalanalysis and beyond (Pierre Schapira)

I.2. Topology and algebraic topology I.2.1 Topo-logie (Mathieu Anel & André Joyal) (pdf) I.2.2 Spaces as infinity-groupoids (Timothy Porter) I.2.3 Homotopy type theory: the logic of space (Mike Shulman)

I.3. Algebraic geometry I.3.1 Sheaves and functors of points (Michel Vaquié) I.3.2 Stacks (Nicole Mestrano & Carlos Simpson) I.3.3 The geometry of ambiguity – An introduction to derived geometry (Mathieu Anel) I.3.4 Geometry in dg-categories (Maxim Kontsevich)

Volume II: New Spaces in Physics

II.1. Non-commutative and super-commutative geometries II.1.1 Noncommutative Geometry, the spectral standpoint (Alain Connes) II.1.2 Topos quantum theory (Klaas Landsman) II.1.3 Super-geometry (Mikhail Kapranov)

II.2. Symplectic geometry II.2.1 Derived stacks in symplectic geometry (Damien Calaque) II.2.2 Higher pre-quantized geometry (Urs Schreiber)

II.3. Space-time II.3.1 Struggles with the Continuum (John Baez) II.3.2 Twistor theory (Roger Penrose) II.3.3 Loop quantum gravity (Muxin Han) II.3.4 Stringy geometry and emergent space (Marcos Mariño)


Smoothness and singularities

Schemes (Pierre Cartier)

Pierre Cartier on schemes.

(video recording)

Synthetic differential geometry - new methods for old spaces (Anders Kock)

Anders Kock on synthetic differential geometry

(video recording)

Lie (or differentiable) groupoids (Jean Pradine)

Jean Pradines on Lie groupoids

(video recording)

Diffeologies (Patrick Iglesias-Zemmour)

Patrick Iglesias-Zemmour on diffeological spaces.

(video recording)

(lecture notes pdf)

Spaces with categories of points

Geometric aspects of topos theory in relation with logical doctrines (André Joyal)

André Joyal on topos theory.

(video recording)

Sheaves and functors of points (Michel Vaquié)

Michel Vaquié on sheaves and gros toposes.

Stacks and the Artin property (Carlos Simpson)

Carlos Simpson on algebraic stacks and the Artin representability theorem.

(video recording)

The Geometry of Ambiguity: An introduction to the ideas of Derived Geometry (Mathieu Anel)

Mathieu Anel on derived geometry (pdf)

(video recording)

Non-commutative geometry

Geometry in triangulated categories (Maxim Kontsevich)

Maxim Kontsevich on derived noncommutative geometry in terms of formal duals of stable infinity-categories (enhanced triangulated categories).

Quantum differential geometry (Shahn Majid)

Shahn Majid on noncommutative geometry

Spaces up to homotopy

Spaces as infinity-groupoids (Timothy Porter)

Timothy Porter on thinking of spaces as infinity groupoids and the relation with the homotopy hypothesis.

(video recording) (draft of article)

Homological decomposition and motives (Denis-Charles Cisinski)

Denis-Charles Cisinski on motives.

(video recording)

Homotopy type theory: the logic of space (Mike Shulman)

Mike ShulmanHomotopy type theory: the logic of space (arXiv:1703.03007)

(on homotopy type theory)

Spaces in classical & quantum mechanics

Derived stacks in symplectic geometry (Damien Calaque)

Damien Calaque on derived symplectic geometry (arXiv:1802.09643)

Higher pre-quantized geometry (Urs Schreiber)

Urs Schreiber on Higher Prequantum Geometry.

(arXiv:1601.05956, video recording)

Spaces in quantum gravity

Spin networks and spinfoams (Hanno Sahlmann)

Hanno Sahlmann on spin networks.

(video recording)

Super-geometry (Mikhail Kapranov)

Mikhail Kapranov on superalgebra as sphere spectrum-graded algebra.

Twistor theory (Roger Penrose)

Roger Penrose on twistors.

(video recording)

Stringy geometry and emergent space (Marcos Mariño)

Marcos Mariño on AdS-CFT

(video recording)

Open questions

What experiments tell us about space and time (John Baez)

John Baez on experiments.

category: reference

Last revised on February 8, 2020 at 12:48:12. See the history of this page for a list of all contributions to it.