This is a survey (with some proofs) of chapter 2 of the notes http://renyi.mta.hu/~szamuely/beilintronew.pdf of T. Szamuely and G. Zabradi on Beilinson's approach to the p-adic Hodge decomposition theorem.
This is the first talk of the workshop organised by F. Brown, M. Kim and D. Rössler on Beilinson's approach to p-adic Hodge theory.
In this talk, we shall give the definition and recall various properties of the cotangent complex, which was originally defined by L. Illusie in his monograph "Complexe cotangent et déformations" (Springer LNM 239, 1971).
This is an introduction to the article
A. Beilinson, p-adic periods and derived de Rham cohomology, J. Amer. Math. Soc. 25 (2012), no. 3, 715--738.
It is quite easy to see that the sobrification of a
topological space is a dcpo with respect to its specialisation order
and that the topology is contained in the Scott topology wrt this
order. It is also known that many classes of dcpo's are sober when
considered as topological spaces via their Scott topology. In 1982,
Peter Johnstone showed that, however, not every dcpo has this
property in a delightful short note entitled "Scott is not always
sober".
Weng Kin Ho and Dongsheng Zhao observed in the early 2000s that the
Scott topology of the sobrification of a dcpo is typically different
from the Scott topology of the original dcpo, and they wondered
whether there is a way to recover the original dcpo from its
sobrification. They showed that for large classes of dcpos this is
possible but were not able to establish it for all of them. The
question became known as the Ho-Zhao Problem. In a recent
collaboration, Ho, Xiaoyong Xi, and I were able to construct a
counterexample.
In this talk I want to present the positive results that we have about
the Ho-Zhao problem as well as our counterexample.
We describe the representation theory of loop groups in
terms of K-theory and noncommutative geometry. For any simply
connected compact Lie group G and any positive integer level l we
consider a natural noncommutative universal algebra whose 0th K-group
can be identified with abelian group generated by the level l
positive-energy representations of the loop group LG.
Moreover, for any of these representations, we define a spectral
triple in the sense of A. Connes and compute the corresponding index
pairing with K-theory. As a result, these spectral triples give rise
to a complete noncommutative geometric invariant for the
representation theory of LG at fixed level l. The construction is
based on the supersymmetric conformal field theory models associated
with LG and it can be generalized, in the setting of conformal nets,
to many other rational chiral conformal field theory models including
loop groups model associated to non-simply connected compact Lie
groups, coset models and the moonshine conformal field theory. (Based
on a joint work with Robin Hillier)
"In a paper from 2001, Diestel and Leader characterised uncountable graphs with normal spanning trees through a class of forbidden minors. In this talk we investigate under which circumstances this class of forbidden minors can be made nice. In particular, we will see that there is a nice solution to this problem under Martin’s Axiom. Also, some connections to the Stone-Chech remainder of the integers, and almost disjoint families are uncovered.”
Point-free topology can often seem like an algebraic almost-topology,
> not quite the same but still interesting to those with an interest in
> it. There is also a tradition of it in computer science, traceable back
> to Scott's topological model of the untyped lambda-calculus, and
> developing through Abramsky's 1987 thesis. There the point-free approach
> can be seen as giving new insights (from a logic of observations),
> albeit in a context where it is equivalent to point-set topology. It was
> in that tradition that I wrote my own book "Topology via Logic".
>
> Absent from my book, however, was a rather deeper connection with logic
> that was already known from topos theory: if one respects certain
> logical constraints (of geometric logic), then the maps one constructs
> are automatically continuous. Given a generic point x of X, if one
> geometrically constructs a point of Y, then one has constructed a
> continuous map from X to Y. This is in fact a point-free result, even
> though it unashamedly uses points.
>
> This "continuity via logic", continuity as geometricity, takes one
> rather further than simple continuity of maps. Sheaves and bundles can
> be understood as continuous set-valued or space-valued maps, and topos
> theory makes this meaningful - with the proviso that, to make it run
> cleanly, all spaces have to be point-free. In the resulting fibrewise
> topology via logic, every geometric construction of spaces (example:
> point-free hyperspaces, or powerlocales) leads automatically to a
> fibrewise construction on bundles.
>
> I shall present an overview of this framework, as well as touching on
> recent work using Joyal's Arithmetic Universes. This bears on the logic
> itself, and aims to replace the geometric logic, with its infinitary
> disjunctions, by a finitary "arithmetic type theory" that still has the
> intrinsic continuity, and is strong enough to encompass significant
> amounts of real analysis.
Starting from Hilbert's 10th problem, I will explain how to characterise the set of integers by non-solubility of a set of polynomial equations and discuss related challenges. The methods needed are almost entirely elementary; ingredients from algebraic number theory will be explained as we go along. No knowledge of first-order logic is necessary.