Noncommutative geometry and algebra

Algebra and Combinatorics

Algebra has undergone significant developments in the past decade. It is a very active subfield, with significant interaction with topology, geometry, and theoretical physics. The United States is regarded as the leader, with Western Europe a close second; both have benefited from the emigration of mathematicians from the former Soviet Union. There are also major centers of activity in Russia, Japan, Israel, and Australia.

In algebraic representation theory, research is enhanced by interactions with geometry, combinatorial methods, and theoretical physics. The United States is a leader, and Western Europe and Japan have significant strengths. U.S. leadership also holds in finite and combinatorial group theory, where the infusion of geometric ideas is leading to novel approaches and results. There are also strong centers of activity in Western Europe, Russia, and Israel. Rather exciting developments are underway in noncommutative geometry and Lie theory, with connections to algebraic geometry and to theoretical physics (quantum groups). The United States and Western Europe share research leadership. The United States is the clear leader in ring theory, where a breakthrough is needed for further major advances. Computational algebra, although still in its infancy, holds great promise. Europe has decidedly more depth and breath than the United States, and Australia is a strong participant.

Reviews of Conne's book from the Notices of the AMS (in pdf format):
  Review of Noncommutative Geometry by Alain Connes by Vaughan Jones and Henri Moscovici
  Review of Noncommutative Geometry by Alain Connes by Andrew Lesniewski
Colloquium announcement, for a talk given by Connes at UCLA
Biography of Connes from St. Andrews
Information about a course Noncommutative Geometry and the Riemann Zeta Function to be given this fall at Ohio State by Alain Connes, and his expository paper giving some background

Books on Noncommutative Geometry (local copy) or original page (with pictures)

Conferences:   |   Oberwolfach 2002   |   Almeria 2002   |   Quantum Field Theory, Noncommutative Geometry, and Quantum Probability , Trieste, March, 2001   |   Noncommutative Geometry and Quantum Groups, Warsaw, September, 2001

Here are some articles that might be of interest (collected by Paul Smith, Univ. of Washington):

Noncommutative curves and noncommutative surfaces by Toby Stafford and Michel Van den Bergh
A Mad Day's Work: From Grothendieck to Connes and Kontsevich: evolution of the notions of space and symmetry by Pierre Cartier
Noncommutative Geometry for Pedestrians by J. Madore
Deformation quantization of algebraic varieties by Maxim Kontsevich
Two Lectures on D-Geometry and Noncommutative Geometry by Michael Douglas
A Point's Point of View of Stringy Geometry by Paul Aspinwall
D-branes, Discrete Torsion and the McKay Correspondence by Paul Aspinwall and M. Ronen Plesser
Stacks for Everybody by Barbara Fantechi
Algebraic Stacks by Tomas Gomez
What is a stack? by Bill Fulton
Notes on Stacks by Herb Clemens, Aaron Bertram, et al.
Topological Stacks by Angelo Vistoli
Discrete Torsion and Gerbes II by Eric Sharpe
Algebraic orbifold quantum products by Dan Abramovich, Tom Graber, and Angelo Vistoli
Gerbes over Orbifolds and Twisted K-theory by Ernesto Lupercio and Bernardo Uribe
Orbifolds as Groupoids: An introduction by Ieke Moerdijk
Differential Geometry of Gerbes by Lawrence Breen and William Messing
An overview of the search for minimal models of algebraic threefolds by Sinan Sertoz
Weak Hopf algebras and quantum groupoids by Peter Schauenburg
Quantum Groupoids by Ping Xu
Finite Quantum Groupoids and their applications by Dmitri Nikshych and Leonid Vainerman
Functoriality and Morita equivalence of operator algebras and Poisson manifolds associated to groupoids by N.P. Landsman

Non-commutative geometry and topology from Connes's perspective.

Noncommutative Geometry, Year 2000 by Alain Connes
A short survey of Noncommutative Geometry by Alain Connes
Quantum Spaces and Their Noncommutative Topology by Joachim Cuntz
Applications of Non-commutative geometry to topology J. Rosenberg's minicourse at the
Clay Institute Symposium on Non-commutative Geometry
Review of Noncommutative Geometry by Alain Connes by Vaughan Jones and Henri Moscovici
Review of Noncommutative Geometry by Alain Connes by Andrew Lesniewski
Review of Noncommutative Geometry by Alain Connes by John Roe

Some sources for an introduction: