nLab New Spaces for Mathematics and Physics

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

• Gabriel Catren, Mathieu Anel, (eds.)

  
  

  New Spaces for Mathematics and Physics

  
  

  conference (IHP Paris, Oct-Sept 2015)

  and

  book project:

  
  

  New spaces in mathematics: formal and conceptual reflections,

  Cambridge University Press (2021)

  ISBN:9781108854429

  doi:10.1017/9781108854429

  gBooks

  
  

  New spaces in physics: formal and conceptual reflections,

  Cambridge University Press (2021)

  ISBN:9781108854399

  doi:10.1017/9781108854439

toc pdf

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

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)

• Anders Kock, New methods for old spaces: synthetic differential geometry

  (arXiv:1610.00286)

I.1.3 Microlocalanalysis and beyond (Pierre Schapira)

Pierre Schapira on microlocal analysis.

• arXiv:1701.08955

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”:

• Mathieu Anel, Andre Joyal, Topo-logie (pdf)

I.2.2 Spaces as infinity-groupoids (Timothy Porter)

• Timothy Porter, Spaces as infinity-groupoids (preprint pdf): on homotopy types as infinity-groupoids.

I.2.3 Homotopy type theory: the logic of space (Mike Shulman)

On homotopy type theory:

• Mike Shulman, Homotopy type theory: the logic of space (arXiv:1703.03007)

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):

• Mikhail Kapranov, Supergeometry in mathematics and physics, in Gabriel Catren, Mathieu Anel, (eds.) New Spaces for Mathematics and Physics (arXiv:1512.07042)

• Mikhail Kapranov, Super-geometry, talk at New Spaces for Mathematics & Physics, IHP Paris, Oct-Sept 2015 (video recording)

II.2. Symplectic geometry

On symplectic geometry:

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

On derived symplectic geometry:

• Damien Calaque, Derived stacks in symplectic geometry (arXiv:1802.09643)

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

On higher prequantum geometry:

• Urs Schreiber, Higher Prequantum Geometry,(arXiv:1601.05956, video recording)

II.3. Space-time

On spacetime.

II.3.1 Struggles with the continuum (John Baez)

On the continuum:

• John Baez, Struggles with the continuum (arXiv:1609.01421)

II.3.2 Twistor theory (Roger Penrose)

On twistors:

• Roger Penrose, Twistor Theory: A Geometric Perspective for Describing the Physical World (doi:10.1017/9781108854399.011)

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).

• Maxim Kontsevich, Geometry in triangulated categories, talk at New Spaces for Mathematics and Physics, IHP Paris, Oct-Sept 2015 (video recording)

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

• Mike Shulman, Homotopy type theory: the logic of space (arXiv:1703.03007)

Spaces in classical & quantum mechanics

Derived stacks in symplectic geometry (Damien Calaque)

• Damien Calaque on derived symplectic geometry:

• Damien Calaque, Derived stacks in symplectic geometry (arXiv:1802.09643)

Higher pre-quantized geometry (Urs Schreiber)

• Urs Schreiber, 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 (see also at spectral super-scheme).

• Mikhail Kapranov, Supergeometry in mathematics and physics, in Gabriel Catren, Mathieu Anel, (eds.) New Spaces for Mathematics and Physics (arXiv:1512.07042)

• Mikhail Kapranov, Super-geometry, talk at New Spaces for Mathematics & Physics, IHP Paris, Oct-Sept 2015 (video recording)

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