AGNT Ical Atom

tbd

In the talk, I will discuss the Turner-Turaev formalism for unoriented Topological Quantum Field Theory (TQFT). Building upon this formalism, I will introduce an analogous version for (d+1)-dimensional TQFT for pro-p Poincare duality groups. In the case of d = 1, this enables us to study cobordisms and TQFTs for both the maximal pro-p quotient of absolute Galois groups of p-adic fields and pi_1(X)^p, the pro-p completions of fundamental groups of surfaces. This generalisation gives a framework for arithmetic TQFTs and strengthens the analogies within arithmetic topology, which relates p-adic fields to surfaces (oriented mod p^r). I will explain the classification of TQFTS for the (1+1)-dimensional case, in terms of Frobenius algebras with some extra structure.

If time permits, I will explain how we define a Dijkgraaf Witten like theory, to get formulas for counting G-covers of X, where X is either a surface, or a p adic field, and G is a p-group (these formulas are similar to the ones given by Mednykh for surfaces using TQFTs, and by Masakazu Yamagishi using a more algebraic approach). I will also try to outline how we plan to also get similar formulas for Hom(\pi_1(X)^p,G), where G=GL_n(k) for k=F_{p^r} or Z/p^rZ.

The talk is based on joint work with Oren Ben-Bassat.

For a number field $K$ admitting an embedding into the field of real numbers $\mathbb{R}$, it is impossible to construct a functorial in $G$ group structure in the Galois cohomology pointed set $H^1(K,G)$ for all connected reductive $K$-groups $G$. However, over an arbitrary number field $K$, we define a diamond (or power) operation of raising to power $n$

We show that this operation has many functorial properties. When $G$ is a torus, the set $H^1(K,G)$ has a natural group structure, and $x^{\Diamond n}$ coincides with the $n$-th power of $x$ in this group.

For a cohomology class $x \in H^1(K,G)$, we define the period $\operatorname{per}(x)$ to be the greatest common divisor of $n>0$ such that $x^{\Diamond n}=1$, and the index $\operatorname{ind}(x)$ to be the greatest common divisor of the degrees $[L:K]$ of finite separable extensions $L/K$ splitting $x$. These period and index generalize the period and index of a central simple algebra over $K$ (in the special case where $G$ is the projective linear group $\operatorname{PGL}_n$, the elements of $H^1(K, G)$ can be represented by central simple algebras). For an arbitrary reductive group $G$ defined over a local or global field $K$, we show that $\operatorname{per}(x)$ divides $\operatorname{ind}(x)$, that $\operatorname{per}(x)$ and $\operatorname{ind}(x)$ have the same prime factors, but the equality $\operatorname{per}(x) = \operatorname{ind}(x)$ may not hold.

The talk is based on joint work with Zinovy Reichstein. All necessary definitions will be given, including the definition of the Galois cohomology set $H^1(K,G)$.