F_un days three and fourth: the epilogue....

By AC and K. Consani

The second part of the workshop on the interaction between NCG & the field with one element held at Vanderbilt University this past May 15-18, included the presentation of several new results as well as some speculative discussions with topics ranging from number theory and noncommutative/arithmetic geometry to the most recent developments in Hopf-cyclic theory.

Within the broad area of number-theory/arithmetic geometry we shall review two talks both presented on Saturday May 17th.

In the first talk, David Goss presented his new ideas on ``Zeta-phenomenology in characteristic p''. This great talk has been inspired by the fabulous insights of L. Euler, the first zeta-phenomenologist, as David likes to say. The talk of Goss focussed on how to set up analogs of Bernoulli-Carlitz elements in complete generality. Click here to access his summary in pdf form.

The second talk with a number-theoretic flavor was given on Saturday by Cristian Popescu. Cristian spoke on "1-Motives and Equivariant Iwasawa Theory". The main conjecture in classical Iwasawa theory makes a link between special values of l-adic L-functions and certain Iwasawa modules which are the characteristic zero analogues of the l-adic realizations (Tate modules) of Jacobians of smooth projective curves defined over finite fields of characteristic different from l. From this vantage point, the classical main conjecture is the exact analogue of Weil's Theorem expressing global L-functions in characteristic p (essentially) as characteristic polynomials of the geometric Frobenius morphism acting on the l-adic realizations of the corresponding Jacobian varieties. In the talk, Cristian explained how to extend this analogy in two directions: where on one hand Jacobians are replaced by 1-motives and on the other L-functions are replaced by G-equivariant L-functions.

Among the talks in noncommutative-arithmetic geometry, we will review the talk given by Jorge Plazas-Vargas on "Endomotives, abelian varieties and real multiplication". Jorge explained his new insights on the construction of an algebraic endomotive closely related to the Bost-Connes system which arises by considering iterated powers of a Morita type endomorphism on a real multiplication noncommutative torus. Click here to access his summary in pdf form.

Abhishek Banerjee spoke on "Periodicity in Cyclic Cohomology and Monodromy at Archimedean Infinity". In the talk, Abhishek exposed his recent results on an interpretation of the local monodromy operator for degenerations of arithmetic varieties (both over a disk and at archimedean infinity) in terms of Connes's periodicity operator in cyclic theory. Click here to access his summary in pdf form.

Still on Saturday, Snigdhayan Mahanta gave a speculative interesting talk with the goal to convey his recent ideas that simplicial/cyclic topology should capture the combinatorial aspects of the geometry over the field with one element. In the talk Sniggy discussed some general features of the geometry over F1, mostly highlighting the simplicial structures. Click here to access his summary in pdf form.

The talks of Nadia Larsen "Phase transition in the Bost-Connes C*-dynamical systems from number fields" and Sergey Neshveyev "On von Neumann algebras arising from Bost-Connes type systems" (pdf file here) presented their recent results (in collaboration with Marcelo Laca) on a phase-transition theorem for a quantum statistical mechanical system (due to Ha-Paugam) which generalizes the Bost-Connes system to an arbitrary number field K. Here is the pdf.

Most of the talks on Sunday concentrated on several very recent results in Hopf-cyclic theory.
Masoud Khalkhali talked on "Hopf cyclic Cohomology in Braided Monoidal Categories". In the talk he explained his insights, described by working out several specific examples, on how to develop a Hopf cyclic theory for Hopf algebra objects in an abelian braided monoidal category. Here is his summary in pdf form.

Bahram Rangipour then spoke on "Hopf algebras arising from formal vector fields on the real line and their Hopf cyclic cohomology". Bahram's talk reviewed Hopf cyclic cohomology with coefficients and the powerful results of his recent collaboration with Henri Moscovici. Here is the summary in pdf form.

Finally, Atabey Kaygun spoke on "Products in Hopf-cyclic (co)homology". Atabey derived the structure maps of the cyclic module which defines the Hopf-cyclic cohomology using very basic principles. Here is the pdf of his summary.

As an epilogue of this long-overview articles on the Workshop at Vanderbilt University we would like to thank all the speakers for their spontaneous and generous participation and for sharing their ideas with us about the field with one element and the new connection with NCG. We also would like to thank all the participants for coming to the talks and patiently listening to the discussions which were at times intense and certainly "very alive" and stimulating...