Alain Connes (born on April 1, 1947) is a French mathematician, Fields medalist (1982), Crafoord prize winner (2001), Professor at IHÉS, Professor at Collège de France and part-time working as a Professor at Vanderbilt University. His interests include geometry, topology, especially K-theory and index theory, operator algebras, the connections between noncommutative geometry and number theory, and more recently also the absolute geometry over a field with one element.
Most recently Connes is studying number theory including connections to the “thermodynamic aspect” via the KMS states, to the absolute geometry via the field with one element and a related approach to the Riemann hypothesis.
Alain Connes uses intuition from mathematical physics like the notion of KMS state and has introduced the noncommutative extensions of standard model of particle physics predicting a value for the mass of Higgs particle. He proposed elements of a unification of gravity and gauge theories via spectral action functionals on spaces of spectral triples.
Connes is most well known for introducing a dominant direction to noncommutative geometry where his contributions include, most remarkably, the introduction of cyclic homology and its connections to the K-theory of spaces and of operator algebras, study of fundamental examples of noncomutative spaces like the space of leaves of a foliation, noncommutative tori, groupoid operator algebras, spaces of Penrose tilings, introducing noncommutative motives into operator algebras (with Matilde Marcolli), Baum-Connes hypothesis, the local index formula?, combinatorial approach to Feynman diagrams (Connes-Kreimer Hopf algebra of renormalization) and analytic aspects connecting them to the Birkhoff decomposition, introducing Hopf-cyclic homology etc.
As an inspirative and energetic lecturer Connes also directly contibuted to the popularization of noncommutative geometry and its connections to physics.
Introducing the cyclic category and cyclic objects (cyclic sets, cyclic spaces) for cyclic homology:
On the equivariant Chern character:
Alain Connes, Noncommutative geometry, Academic Press 1994, 661 p. PDF
Alain Connes, Matilde Marcolli, Noncommutative geometry, quantum fields and motives, draft pdf
For more see Connes’ official website.
cyclic object, cycle category, cyclic cohomology?