This will be a discussion of the paper https://arxiv.org/abs/1604.02618
This will be a quick introduction to tropical algebra and the main results from the paper https://arxiv.org/pdf/1604.00113.pdf
Finding implementable descriptions of the possible configurations of a knotted DNA molecule has remarkable importance from a biological point of view, and it is a hard and well studied problem in mathematics.
Here we present two newly developed mathematical tools that describe the configuration space of knots and model the action of Type I and II Topoisomerases on a covalently closed circular DNA molecule: the Reidemeister graphs.
We determine some local and global properties of these graphs and prove that in one case the graph-isomorphism type is a complete knot invariant up to mirroring.
Finally, we indicate how the Reidemeister graphs can be used to infer information about the proteins' action.
In this talk I will first briefly introduce 1-parameter persistent homology, and discuss some applications and the theoretical challenges in the multiparameter case. If time remains I will explain how tools from commutative algebra give invariants suitable for the study of data. This last part is based on the preprint https://arxiv.org/abs/1708.07390.
Topological field theories (TFT's) are physical theories depending only on the topological properties of spacetime as opposed to also depending on the metric of spacetime. This talk will introduce topological field theories, and the work of Freed and Hopkins on how a class of TFT's called "invertible" TFT's describe certain states of matter, and are classified by maps of spectra. Constructions of field theories corresponding to specific maps of spectra will be described.
A main part of the proof uses forcing to establish a Ramsey theorem on a new type of tree, though the result holds in ZFC. The space of such trees almost forms a topological Ramsey space.