nLab
New Spaces for Mathematics and Physics

Contents

Context

Mathematics

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Geometry

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

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

Contents

Book

Volume I: New Spaces in Mathematics

I.1. Differential geometry

On differential geometry.

I.1.1 Diffeologies (Patrick Iglesias-Zemmour)

Patrick Iglesias-Zemmour on diffeological spaces.

I.1.2 New methods for old spaces: synthetic differential geometry (Anders Kock)
I.1.3 Microlocalanalysis and beyond (Pierre Schapira)

Pierre Schapira on microlocal analysis.

I.2. Topology and algebraic topology

On topology and algebraic topology.

I.2.1 Topo-logie (Mathieu Anel & André Joyal)

On Grothendieck toposes regarded as “logoses”:

I.2.2 Spaces as infinity-groupoids (Timothy Porter)
I.2.3 Homotopy type theory: the logic of space (Mike Shulman)

On homotopy type theory:

I.3. Algebraic geometry

On algebraic geometry.

I.3.1 Sheaves and functors of points (Michel Vaquié)

Michel Vaquié on sheaves and functorial geometry.

I.3.2 Stacks (Nicole Mestrano & Carlos Simpson)

Carlos Simpson on algebraic stacks.

I.3.3 The geometry of ambiguity – An introduction to derived geometry (Mathieu Anel)

On derived geometry.

I.3.4 Geometry in dg-categories (Maxim Kontsevich)

Maxim Kontsevich on derived noncommutative geometry in terms of formal duals of dg-categories (stable infinity-categories, enhanced triangulated categories, see also at spectrum of a triangulated category).

Volume II: New Spaces in Physics

II.1. Non-commutative and super-commutative geometries

On noncommutative geometry and supergeometry.

II.1.1 Noncommutative Geometry, the spectral standpoint (Alain Connes)

Alain Connes on spectral geometry.

II.1.2 Topos quantum theory (Klaas Landsman)

Klaas Landsman on Bohr toposes.

II.1.3 Super-geometry (Mikhail Kapranov)

Mikhail Kapranov on superalgebra as sphere spectrum-graded algebra (see also at spectral super-scheme):

II.2. Symplectic geometry

On symplectic geometry:

II.2.1 Derived stacks in symplectic geometry (Damien Calaque)

On derived symplectic geometry:

II.2.2 Higher pre-quantized geometry (Urs Schreiber)

On higher prequantum geometry:

II.3. Space-time

On spacetime.

II.3.1 Struggles with the continuum (John Baez)

On the continuum:

II.3.2 Twistor theory (Roger Penrose)

On twistors:

II.3.3 Loop quantum gravity (Muxin Han)
II.3.4 Stringy geometry and emergent space (Marcos Mariño)

Marcos Mariño on AdS-CFT duality.


Conference

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 Pradines)

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, see also at spectrum of a triangulated category).

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)

On homotopy type theory

Spaces in classical & quantum mechanics

Derived stacks in symplectic geometry (Damien Calaque)

Higher pre-quantized geometry (Urs Schreiber)

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 (see also at spectral super-scheme).

Twistor theory (Roger Penrose)

Roger Penrose on twistors.

(video recording)

Stringy geometry and emergent space (Marcos Mariño)

Marcos Mariño on AdS-CFT

(video recording)

category: reference

Last revised on July 19, 2021 at 06:55:29. See the history of this page for a list of all contributions to it.