TTT109: Leicester

Dedicated to the memory of Sir Michael Atiyah.

University of Leicester.

Supported by the London Mathematical Society.

Date

8 March 2019

Videos

John Greenlees (Warwick): Adelic models of tensor triangulated categories

Bernd Schroers (Heriot-Watt): Magnetic Skyrmions

Graeme Segal (Oxford): Michael Atiyah and algebraic topology

Speakers

Stephen Garrett (Head of Department of Mathematics)
Welcome 

John Greenlees (Warwick)
Adelic models of tensor triangulated categories (joint with Scott Balchin)

Watch a video of the talk

Abstract: Atiyah brought commutative algebra (or at least completion) into algebraic topology through his calculation of the K theory of classifying spaces, and this process was taken to its conclusion through the Atiyah-Segal completion theorem for equivariant K theory.

Certainly, this was how I was led to a close interest in commutative algebra, and this has been a two-way exchange between (homotopical) algebraic topology and (derived) commutative algebra.

My talk is about another instance of this. It began with the classification of equviariant cohomology theories (Greenlees-Shipley). It turns out that the structure of the argument applies in a much more general context, and even in the classical algebraic context it gives an interesting perspective. The talk will reverse the historical process and build the motivation from commutative algebra.

Abelian groups can be constructed from modules over the rationals and modules over the p-adic integers for all p. One way of making this precise is to start from the Hasse square:

Z ————> Q

 |                      |

V                    V

\prod Z_p—>Adeles

And then to build a model for abelian groups from the categories of modules over Q and Z_p. The same idea works for well behaved higher dimensional rings. Replacing Spec(Z) by the Balmer spectrum of a tensor triangulated category one can build a model of this type from modules over completed localised rings more generally. For example, for rational torus equivariant cohomology theories it leads to the algebraic model alluded to above.

Bernd Schroers (Heriot-Watt)
Magnetic skyrmions

Watch a video of the talk

Abstract: Magnetic skyrmions are topological defects in planar ferromagnets.  Mathematically they are described by maps from the 2-plane to the 2-sphere which are stationary points of an energy functional which combines the usual Dirichlet energy term with a term which is linear in derivatives. It turns out that, for a critical choice of coupling constants, the energy expression can naturally be written in terms of a covariant derivative and its stationary points be determined as solutions of a first order differential equation.

In terms of the standard complex structures of the plane and the sphere, solutions have a fixed anti-holomorphic part and an arbitrary holomorphic part. I will describe these solutions, their topological properties and their physical interpretation.

The talk is based on arXiv 1812.07268.

Frances Kirwan (Oxford)
Morse theory without nondegeneracy

Abstract: This talk will report on joint work with Geoff Penington on extending Morse theory to smooth proper functions without nondegeneracy assumptions.

Graeme Segal (Oxford)
Michael Atiyah and algebraic topology

Watch a video of the talk

Abstract: More than almost anyone else Michael Atiyah set the direction of the world’s mathematical research in his time. He believed passionately in the unity of mathematics, and his greatest hero was Herman Weyl. He resisted being assigned to a 'field', especially the field of algebraic topology. Nevertheless, I shall argue that his own work can best be seen as part of the history of algebraic topology, though he had a perspective on it that was all his own.