# 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

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

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

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.