TTT102: Sheffield

The University of Sheffield.

Supported by the London Mathematical Society.

Date

27 February 2017

Speakers

Nick Kuhn (University of Virginia)
Hurewicz maps for infinite loopspaces

Abstract: In a 1958 paper, Milnor observed that then new work by Bott allowed him to show that the n-sphere admits a vector bundle with non-trivial top Stiefel-Whitney class precisely when n=1, 2, 4, 8.  This can be interpreted as a calculation of the mod 2 Hurewicz map for the classifying space BO, which has the structure of an infinite loopspace. 

I have been studying such Hurewicz maps for generalised homology theories by relating the Adams filtration of the domain to a filtration of the range coming from Andre-Quillen homotopy calculus. When specialised to ordinary mod p homology, my general results have some tidy consequences, including Milnor's theorem and a variant with ko replaced by tmf.

Dirk Schuetz (Durham)
Computing Steenrod squares in Khovanov cohomology

Abstract: In recent work Lipshitz and Sarkar constructed a stable homotopy type for Khovanov cohomology, thus introducing cohomology operations to it. They showed that the second Steenrod square is non-trivial for many non-alternating knots, and used it to refine the Rasmussen invariant.

Nevertheless, computations become very time-intensive once a knot has more than 15 crossings. We will discuss techniques to improve computations for more complicated knots and how they can be used to identify the resulting stable homotopy types.

Jeff Giansiracusa (Swansea)
Exterior algebra and Plucker coordinates in tropical geometry

Abstract: Any Grassmannian Gr(d,n) admits an embedding in projective space called the Plucker embedding, and exterior algebra provides an elegant description of this embedding. In this talk I will present an analogue of this picture for the combinatorial objects called (valuated) matroids, which are the basic building blocks of tropical geometry.

A valuated matroid can be thought of as an object defined over the tropical idempotent semiring T by a tropical analogue of Plucker coordinates. I will describe how to define a tropical analogue of exterior algebra and use this to give a new cryptomorphic description of valuated matroids.

The main result is that a d-multivector w is a valuated matroid if and only if the quotient of T^n that is dual to the kernel of wedging with w has d-th exterior power free of rank 1. This gives a projective embedding of the Dressian (the space of all valuated matroids) in a tropical projective space and also provides it with a modular interpretation.