9-10 (Plenary talk)
Matilde Marcolli (Max-Planck-Institute, Bonn)
Renormalisation, Galois symmetries and motives
Abstract:
click here
13.25-14.05
Caterina Consani (Johns Hopkins, Baltimore)
Noncommutative geometry and motives
Abstract: Since A. Grothendieck proposed
the first definition of a "motif", in the early 60's, this notion has
undergone several interesting developments and enrichments. Very recently, some applications of a new theory of noncommutative
motives to number-theory and arithmetic geometry have been found with
the implementation of techniques supplied by Noncommutative Geometry and
Operator Algebras. The talk will shortly overview the central topic of motives in
algebraic geometry and will outline some of the most recent applications
of the new notion of "endomotive".
14.15-14.55
Yuri Manin (Max-Planck-Institute, Bonn and
Northwestern)
An update on real multiplication
Abstract: The Real
Multiplication program proposes to study arithmetics of real quadratic
fields and their extensions using non-commutative tori with real
multiplication and non-commutative modular tower in the same way as
elliptic curves with Complex Multiuplication are treated in the
classical theory. In the talk, I will review some recent developments
supplying, in particular, the necessary algebraic structures:
non-commutative projective models of RM quantum tori (A. Polishchuk, M.
Vlasenko); Hopfish structures (A. Weinstein et al.) I will start with
basic analogies and finish with open problems.
14.55-15.25
break
15.25-16.05
Pedro Resende (Instituto Superior Técnico, Lisboa)
Noncommutative geometry and Bohr's doctrine of classical
concepts
Abstract: Many examples of noncommutative spaces are based on convolution algebras of locally compact groupoids that are often etale, and in this talk I will show that for any etale groupoid G the algebra of compactly supported functions C_c(G) has a representation by global sections of a G-sheaf of 'internal matrices' and, furthermore, it can be reconstructed from the category BG of G-sheaves (the classifying topos of G). This suggests new ways in which to study toposes, namely using the tools of noncommutative geometry, and potentially opens up a wide new range of applications. On the other hand, noncommutative spaces represented by toposes automatically carry a logic that ties them up with recent topos theoretic developments in physics and, in particular, with the idea of providing a mathematical realization of Bohr's doctrine of classical concepts.
16.15-16.55
Walter van Suijlekom (Radboud Universiteit,
Nijmegen)
On the geometry of noncommutative gauge fields
Abstract: In this talk
we study the geometry of gauge fields on a noncommutative 4-dimensional
sphere. This example serves as a guidance in the search for a proper
algebraic setting for gauge theories on noncommutative manifolds. In
the case of the 4-sphere, we construct a special class of solutions to
Yang-Mills equations by giving connections with selfdual curvature,
also known as instantons. The construction involves a noncommutative
Hopf fibration whose symmetries are described by quantum groups.
The past decade has witnessed a sudden and sometimes quite unexpected growth of applications of non-commutative geometry (in the style of Connes) to other fields of mathematics and physics. To name a few, in physics we see relations to the Standard Model, quantisation, instantons and gauge theory. In number theory, the real multiplication programme of Manin led to surprising relations between non-commutative tori in differential- and algebraic geometry. There is the study of non-commutative boundaries of classical moduli spaces from number theory, and related work of Connes, Marcolli et. al. on thermodynamics and motives. Finally, links between uniformization and spectral triples are being investigated following original work of Consani and Marcolli. The symposium will highlight some of these applications. The list of proposed speakers is meant to reflect both the physical and the number theoretical / algebraic geometrical applications, with two speakers from each branch.