Noncommutative geometry, quantum fields and motives. In the rst part we formulate a version of the conjectural local langlands corre. Then again, with the more abstract topics in algebraic geometry, ncategories, stacks and all that stuff, these developments could be carried over to noncommutative geometry, and since ncg is at the heart of many developments in physics, it might give wonderful applications to string theory etc. Poisson geometry, quantization, and noncommutative geometry integrable systems, frobenius manifolds, and the geometric langlands program. Product of the continuum by the discrete and the symmetry breaking mechanism 574 4. Langlands reciprocity for the even dimensional noncommutative tori article pdf available in proceedings of the american mathematical society 912 april 2010 with 34 reads how we measure. In this lecture ill take both of these discoveries as a pretext and point towards the extension of our familiar geometrical concepts beyond the classical, commutative case. Very useful for those intereseted in the langlands program. Noncommutative geometry has as its limiting case the classical geometry, but geometry expressed in algebraic terms. The geometric langlands correspondence can be formulated as follows.
Noncommutative geometry and arithmetics springerlink. My aim is to discuss the foundation of noncommutative geometry. Concepts will be emphasized and technical details avoided contents 1. Arithmetic geometry and noncommutative geometry scedule. Geometric langlands beilinson, drinfeld, lauman, vilonen, and others. We discuss the representation theory of reductive padic groups from two di erent viewpoints. Thus to understand its relation with classical geometry one should. Langlands duality and global springer theory, pdf, compositio. Robert langlands at his office at the institute for advanced study, 1999. The langlands program has classically been rather unfunctorial despite the name of langlands functoriality and ungeometric.
Connes, noncommutative geometry and the riemann zeta function, mathematics. This is the subject of non commutative differential geometry and is believed. The main ideas of the geometric langlands correspondence is now summerized and compaired with our construction in order to show that the same automorphic representations are obtained by the both methods. Ron donagi, tony pantev, lectures on the geometric langlands. We describe basic concepts of noncommutative geometry and a general construction extending the familiar duality between ordinary spaces and commutative. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not. Noncommutative geometry builds on, and vastly extends, this fundamental duality between classical geometry and commutative algebras. Dimensional regularization and noncommutative geometry. Trends in noncommutative geometry, northwestern university, 5185252007. Notes on noncommutative geometry igor nikolaev email. The law of quadratic reciprocity was proven in 1801 by gauss in.
It has has seen two notable advances recently, both tying it to geometry. In any case, the kapustinwitten approach to the geometric langlands program has had a tremendous and powerful in. The very notion of noncommutative geometry comes from the identi. The noncommutative geometry of yangmills elds 9 10. So the origin of the langlands program is in number theory. Applications of noncommutative geometry mathoverflow. Geometric langlands and nonabelian hodge theory 87 the work surveyed here is based on an abelianization of the geometric langlands conjecture in terms of higgs bundles.
Another program i am working on applies noncommutative geometry bne2, bne5, bne6, in particular the geometry of di. Msri program geometric representation theory, berkeley, 81812192014. Conjectures about padic groups and their noncommutative geometry. As the first event in the semester, this workshop will feature a tapas menu of current research and open questions. Very basic noncommutative geometry math department. Noncommutative algebraic geometry is a branch of mathematics, and more specifically a direction in noncommutative geometry, that studies the geometric properties of formal duals of noncommutative algebraic objects such as rings as well as geometric objects derived from them e. Local langlands program generalizations of the springer correspondence and cuspidal langlands parameters joint with annemarie aubert and ahmed moussaoui manus. This is a belated response to a mathoverflow exchange between thomas riepe and chandan singh dalawat asking for a possible connection between connes noncommutative geometry approach to the riemann hypothesis and the langlands program heres the punchline. This is a subject where geometry illuminates representation theory and vice versa. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the.
The discovery of the higgs boson at the lhc in 2012 constitutes a milestone in particle physics, since it completes the theoretical framework of the extremely successful standard model of elementary particle physics. Edward frenkel, lectures on the langlands program and conformal field theory, in frontiers in number theory, physics, and geometry ii, springer berlin heidelberg, 2007. The langlands program and noncommutative geometry published january 30, 2018 by lievenlb the bulletin of the ams just made this paper by julia mueller available online. Lectures on february 9, 16, march 2, 16, 30, april, 27, may 18 2010 exercise classes on february 23, march 9, 23, april 6, 20, may 11, 25 2010. He is a professor of mathematics at the institute for advanced study in princeton. Pdf the structure of a manifold can be encoded in the commutative algebra of. A quantization procedure of fields based on geometric. The notion of manifold in noncommutative geometry 598 5.
Moduli space of dirac operators and yukawa parameters 252. Noncommutative geometry ncg is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions possibly in some generalized sense. Chapter 1 deals with the simplest examples of functors arising in algebraic geometry, number the. Mathematics, cambridge university, england recipient of a 5year advanced fellowship, epsrc, at edinburgh, scotland.
His main contribution to mathematics is a program that predicts hidden relations between algebraic geometry and the theory of representations of lie groups. The noncommutative twotorus bundle in our sense is the actual torus bundle, not in the sense of stacks. More on geometric langlands a grand unified theory of. The article langlands reciprocity for the evendimensional noncommutative tori by igor nikolaev proceedings of the american mathematical society 9 2011, no. Geometry and topology, low dimensional topology, topological quantum field theory ph. Positivity in hochschild cohomology and inequalities for the yangmills action 569 3. We intend to illustrate how the methods of noncommutative geometry are currently used to tackle problems in class field theory. The geometry of noncommutative spaces has been intensively studied during recent decades by connes and many others, and most of the tools of di erential geometry have been generalized to the noncommutative framework. Pdf langlands reciprocity for the even dimensional. Langlands reciprocity for the even dimensional noncommutative tori article pdf available in proceedings of the american mathematical society 912. After all, geometric langlands is a very general categorification program in representation theory, so one would expect it to. Topological quantum field theory and the geometric langlands correspondence.
We show that any compact riemannian spin manifold whose isometry group. Such noncommutative spaces must necessarily be delocalized, in the sense that there are not enough points homomorphisms to c to determine the coordinates. Spectral triple obtained from an algebra bundle 9 10. Grassmann variables, grassmann integration and pfa ans 7 notes 8 chapter 10. The noncommutative geometry of electrodynamics 127 9. The metric aspect of noncommutative geometry 552 1. There is a growing evidence that noncommutative geometry may have a lasting impact on the unsolved classical problems of number theory. Cautis and kamnitzer also have very interesting work in progress on such a relation.
Ams proceedings of the american mathematical society. Illusie on grothendiecks seminar at ihes notes in pdf. Topological quantum field theory and the geometric. Geometric langlands seminar page university of chicago. The derived category of finitedimensional representations of a quiver is an important and somewhat unusual example of a smooth proper dg category. Langlands program, field theory, and mirror symmetry. Algorithmsandarithmeticdms0901971,july2009august2011 andcareer. We end with a tentalizing connection between the renormalization group and the missing galois theory at archimedian places. Notes on noncommutative geometry accompanying dirac operators and spectral geometry by joseph c. Geometry, representation theory, and the langlands program. Noncommutative geometry and the standard model 230. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Noncommutative geometry of the standard model and beyond.
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