TTT88: Manchester

University of Manchester.

Supported by the London Mathematical Society.

Date

17 May 2013

Speakers

Ramón Vera (University of Durham)
Near-symplectic 2n-manifolds

In this talk, I will explain the generalisation of two concepts from low-dimensional topology in higher dimensions: near-symplectic manifolds and overtwisted contact structures. By near-symplectic it is meant a closed 2-form that is non-degenerate, or symplectic, outisde a submanifold where it is singular.

This approach uses some simple mappings coming from singularity theory called broken Lefschetz fibrations. It seems that a similar phenomena occurring in low dimensions appears also in higher dimensions: a near-symplectic 2n-manifold induces a contact structure on a codimension 1 submanifold, which is PS-overtwisted.

Nick Gurski (University of Sheffield)
Semi-strict higher categories

Joint with John Bourke.

Strict higher categories admit a very simple inductive definition, with n-categories just being categories enriched over (n-1)-categories. These are not general enough for many purposes, for example it is known that strict n-groupoids do not model homotopy n-types. Many definitions of weak higher categories have been given, but very few definitions of semi-strict higher categories (ie, not strict, but not entirely weak either) are known.

I will discuss an inductive approach to semi-strict higher categories using machinery very familiar to topologists: the cofibrantly generated factorisation system given by the inclusion of boundary spheres into disks.

Alexander Vishik (University of Nottingham)
Unstable operations in algebraic cobordism

Algebraic cobordism of Levine-Morel is an algebraic analogue of the complex oriented cobordism theory in topology. In particular, it is the universal oriented theory, and has the same coefficient ring as MU. It provides an important invariant of algebraic varieties which is much richer than the classical Chow groups or K_0.

The structure (as in topology) is provided here by cohomological operations. The stable ones among them are Landweber-Novikov operations. These can be constructed using universality, and have various applications in algebraic geometry. But for some time it was observed that to get sharp results on rationality of algebraic cycles one needs unstable operations.

Unfortunately, no general methods to construct such operations were known up to recently. The new technique permits to describe and construct all (unstable) additive operations from any theory obtained from algebraic cobordism by change of coefficients to any other theory.

As applications we get:

1. The description of multiplicative operations as morphisms of the respective formal group laws.

2. The construction of T.tom Dieck style Steenrod operations in algebraic cobordism.

3. The construction of integral (!) Adams operations in the mentioned theories, which specialise to classical Adams operations in K_0.

4. The construction of symmetric operations for all primes (have applications to rationality of cycles).