17:00
``An analogue of the Conjecture of Dixmier is true for the algebra of polynomial integro-differential operators''
Abstract
In 1968, Dixmier posed six problems for the algebra of polynomial
differential operators, i.e. the Weyl algebra. In 1975, Joseph
solved the third and sixth problems and, in 2005, I solved the
fifth problem and gave a positive solution to the fourth problem
but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra `behaves'
like a finite field. The first problem/conjecture of
Dixmier: is it true that an algebra endomorphism of the Weyl
algebra an automorphism? In 2010, I proved that this question has
an affirmative answer for the algebra of polynomial
integro-differential operators. In my talk, I will explain the main
ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.
11:00
"Model theoretic properties of S-acts and S-poset".
Abstract
An S-act over a monoid S is a representation of a monoid by tranformations of a set, analogous to the notion of a G-act over a group G being a representation of G by bijections of a set. An S-poset is the corresponding notion for an ordered monoid S.
17:00
11:00
Revisiting the image of J
Abstract
Some features
that I would like to have are as follows:
1) Most of the spectra involved in the story should be E_\infty (or strictly
commutative)
ring spectra, and most of the maps involved should respect this structure. New
machinery for dealing with E_\infty rings should be used systematically.
2) As far as possible the constructions used should not depend on arbitrary choices
or on gratuitous localisation.
3) The Bernoulli numbers should enter via their primary definition as coefficients of a
certain power series.
4) The image of J spectrum should be defined as the Bousfield localisation of S^0 with
respect to KO, and other properties or descriptions should be deduced from this one.
5) There should be a clear conceptual explanation for the parallel appearance of
Bernoulli numbers in the homotopy groups of J, K(Z) and in spectra related to
surgery theory.
15:45
Chromatic phenomena in equivariant stable homotopy
Abstract
There is a well-known relationship between the theory of formal group schemes and stable homotopy theory, with Ravenel's chromatic filtration and the nilpotence theorem of Hopkins, Devinatz and Smith playing a central role. It is also familiar that one can sometimes get a more geometric understanding of homotopical phenomena by examining how they interact with group actions. In this talk we will explore this interaction from the chromatic point of view.
14:45
A preferential duplication random graph and links to vertex reinforced random walks
An introduction to higher-dimensional category theory
Abstract
Category theory is used to study structures in various branches of
mathematics, and higher-dimensional category theory is being developed to
study higher-dimensional versions of those structures. Examples include
higher homotopy theory, higher stacks and gerbes, extended TQFTs,
concurrency, type theory, and higher-dimensional representation theory. In
this talk we will present two general methods for "categorifying" things,
that is, for adding extra dimensions: enrichment and internalisation. We
will show how these have been applied to the definition and study of
2-vector spaces, with 2-representation theory in mind. This talk will be
introductory; in particular it should not be necessary to be familiar with
any category theory other than the basic idea of categories and functors.