Time: 13:30
Room: MC 107
Speaker: Emre Coskun (Western)
Title: Gerbes from a Stack-Theoretic Perspective

In this talk, we will define a gerbe and examine the relation to the second cohomology with coefficients in an abelian group scheme. As examples, we will discuss relations to Azumaya algebras and Brauer-Severi schemes

Time: 13:00
Room: MC 107
Speaker: Emre Coskun (Western)
Title: Gerbes from a Stack-Theoretic Perspective - Part II

In this talk, we will discuss some of the properties of gerbes from a stack-theoretic perspective. We will talk about the universal property of the structure morphism of a gerbe, define a neutral gerbe and prove that a neutral gerbe is of the form BG, where G is a sheaf of groups.

Time: 14:00
Room: MC 107
Speaker: Bahram Rangipour (New Brunswick)
Title: Hopf algebras in Geometry without groups

It is now more than a decade that Hopf algebras established themselves as an integral part of Noncommutative Geometry via the work of Connes and Moscovici on the computation of the index of hypoelliptic operators on manifolds. The latest Hopf algebras constructed were those associated to Cartan-Lie pseudogroups. In this talk we canonically associate a Hopf algebra to any bicrossed sum Lie algebras. This construction covers all known cases in type II and also type III. The constructed Hopf algebra is naturally equipped with a modular pair in involution which is the coefficients for the Hopf cyclic cohomology of the Hopf algebra. At the end we show how to compute the Hopf cyclic cohomology of these Hopf algebras.

Time: 15:30
Room: MC 107
Speaker: Ivan Fesenko (Nottingham)
Title: Higher fields and adeles associated to arithmetic surfaces, and a translation invariant integration on higher local fields and adeles

This is part 1 of the series "A generalization of the adelic analysis theory of Tate and Iwasawa to arithmetic surfaces"

On arithmetic surfaces one can work with refined structures which the classical algebraic geometry does not really see. Some of them come as higher global, local-global and local fields. The latter fields include power series over usual local fields. Their topology is very unusual. Remarkably, one can integrate over them.

Unlike the classical case of dimension one, arithmetic surfaces have two quite different adelic spaces associated to them: one which is good for geometric applications and another which is good for zeta functions and integration.

The apparent separation between geometry and analysis is a typical phenomenon in dimension two; the BSD conjecture shows that they are not unrelated.

Time: 15:30
Room: MC 107
Speaker: Ivan Fesenko (Nottingham)
Title: K-delic structures on arithmetic surfaces and two-dimensional class field theory

This is part 2 of the series "A generalization of the adelic analysis theory of Tate and Iwasawa to arithmetic surfaces"

Abelian extensions of a two-dimensional local field can be described by open subgroups of the topological Milnor K₂-group of the field. Abelian extensions of the field of rational functions of an arithmetic surface can be described using a Milnor K₂-deles which generalize the K₁-group of adeles in dimension one, and its appropriate quotients. The K₁-groups of the two adelic spaces on the arithmetic surface are related via the K₂-delic groups.

Time: 15:30
Room: MC 107
Speaker: Ivan Fesenko (Nottingham)
Title: Zeta integral on a regular model of elliptic curve over global field and applications to three fundamental properties of its zeta function

This is part 3 of the series "A generalization of the adelic analysis theory of Tate and Iwasawa to arithmetic surfaces"

Using the integration on the two-dimensional objects we define the zeta integral on a relative arithmetic surface whose generic fibre is a smooth geometrically irreducible curve. The theory is the simplest when the genus of the curve is 1.

In dimension two mathematicians have been working with the L-function rather than with the zeta function; however, the existing methods are very restrictive: the base number field cannot be too far away from totally real fields. The zeta integral allows to study the zeta function of the surface directly, for the first time.

The additive duality in dimension two leads to a new theta formula and a two-dimensional version of the Tate thesis. Two-dimensional adelic analysis reduces the analytic properties of the zeta function to those of a so called boundary term given in its integral adelic representation. The boundary term related the geometric and analytic structures. In particular, two-dimensional adelic analysis includes a new powerful method to settle the BSD conjecture. Aspects of the meromorphic continuation and functional equation, location of poles and behavior at the central point become very closely interrelated with each other in the two-dimensional theory.