Physical applications of non-commutative Geometry ( by Fedele Lizzi)

 La setmana del 19 tindrem un workshop de Geometria no-commutativa que constarà d'un curs a càrrec del professor Fedele Lizzi (Università di Napoli Federico II, i 3 xerrades convidades.

20 setembre

16h00 to 17h30 LIZZI I
17h30 to 17h45 break
17h45 to 18h45 PLAZAS

21 setembre

16h00 to 17h30 LIZZI II
17h30 to 17h45 break
17h45 to 18h45 EBRAHIMI-FARD

22 setembre
16h00 to 17h00 LIZZI III
17h00 to 18h00 LIZZI IV
18h00 to 18h15 break
18h15 to 19h15 GALVEZ

Lloc: Aula S03 de la FME ( .

F. LIZZI lecture

Part I: Spectral Geometry and A. Connes’ approach to the Standard Model
I will introduce Connes approach to the description of noncommutative spaces and their geometric properties. This is done by giving an intrinsic algebraic description of ordinary
geometry in terms of commutative C*-algebras, and then generalize it to the noncommutative case. I will describe the metric properties of these spaces via the generalized Dirac
operator, and then apply these ideas to the spectral action to describe the standard model of fundamental particle interactions coupled with gravity. I will take a renormalization
approach and discuss anomalies. Time permitting I will discuss other noncommutative spaces like the noncommutative torus or the fuzzy sphere and disc.

1.1 Intro and commmutative geometry. Gelfand Naimark, GNS.
1.2 Spectral Geometry. Dirac Operator and metric properties.
1.3 Spectral action.
1.4 Standard model.
1.5 Noncommutative Torus and Fuzzy spaces.

Tuesday, Sept. 20, 17:45-18:45
Jorge PLAZAS (Univ. De Granada)
Title: From Quantum Statistical Mechanics to Number Theory: The Arithmetic of Noncommutative Spaces

Abstract: Noncommutative geometry provides a natural framework for various unexpected interactions between physics and number theory, which have arisen in recent years. In this talk we explore the role and relevance of quantum statistical mechanical systems of arithmetic nature. After introducing the basics of this formalism we will discuss some of the central examples (Bost-Connes and Connes-Marcolli systems). Potential applications of this framework and work in progress towards some of these applications will also be discussed.

Wednesday, Sept. 21, 17:45-18:45


Title: Feynman graphs, Green functions and related algebraic structures

Abstract: Based on Dirk Kreimer's seminal findings, Alain Connes and Kreimer developed an algebraic setting encoding essential combinatorial and algebraic aspects underlying the so-called BPHZ renormalization method in perturbative quantum field theory. Key notions are pre-Lie and Hopf algebraic structures of Feynman graphs, and hence Green functions. In this talk we will review these structures, i.e. Connes-Kreimer Hopf algebra of Feynman graphs. Also, we will briefly report on recent joint work with F. Patras (CNRS, Univ. of Nice, France) on the so-called exponential renormalization method. Using Dyson's identity for Green's functions as well as the link between the Faà di Bruno Hopf algebra and Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed. Eventually we analyze the role of the Rota-Baxter
property for renormalization scheme maps.

Thursday, Sept. 22, 18:15-19:15
Imma GALVEZ (Universitat Politècnica de Catalunya)
Title: Groupoids, and Faà di Bruno formulae for Green functions
Abstract: The Connes-Kreimer Hopf algebra of trees (or of Feynman graphs) encodes the combinatorics of the BPHZ renormalisation procedure in pQFT. The comultiplication of a tree returns all the ways of "cutting" the tree. However, the individual trees (or graphs) do not have direct physical interpretation; rather certain infinite sums, the so-called Green functions, carry the physical meaning. Van Suijlekom recently discovered that the Green functions satisfy a version of the classical Faà di Bruno formula for substitution of power series. In this talk we will show how thetheory of groupoids can be used to give a very conceptual proof of the Faà di Bruno formulae for Green functions in the bialgebra of trees. In this framework a Green function is (the cardinality of) a groupoid and the Faà di Bruno formula is shown to be essentially an equivalence of groupoids (joint work with J Kock, A Tonks).
Eva Miranda,
7 set. 2011 , 6:28