# MSCS Seminars Today

## Calendar for Thursday September 21, 2017

Thursday September 21, 2017

**Louise Hay Logic Seminar**

VC Dimension, VC Density, and the Sauer-Shelah Dichotomy - Part I

Roland Walker (UIC)

1:00 PM in SEO 427

Vapnik-Chervonenkis dimension and density are two measures of combinatorial complexity which arose from the study of probability theory. During this two-part talk, we will discuss these measures and their duals both in the classical and model-theoretic contexts, prove the famous Sauer-Shelah Lemma, discuss the relationship between VC dimension and NIP, and time permitting discuss some recent applications and open questions.

**Quantum Topology / Hopf Algebra Seminar**

What is a Rotweldid Knot?

Jonathan Schneider (UIC)

3:00 PM in SEO 612

This series of talks represents joint work with Eiji Ogasa and Lou Kauffman.
Shin Satoh provided a topological model for the diagrammatic theory of
Welded knots. Specifically, he described a map from knot diagrams to
toral surfaces which respects Welded-equivalence, so equivalent diagrams
are modeled by isotopic toral surfaces.
However, Satoh was unable to determine whether this model
was classifying for Welded knot theory— whether inequivalent
Welded knot diagrams might nonetheless be modeled by isotopic toral surfaces. The question of whether Satoh’s model is classifying remains open as of 2017.
Colin Rourke claimed to provide another model for Welded knot theory,
based on Satoh’s model but “strengthened”. In his proposal, Rourke
adorned Satoh’s toral surfaces with a fiber-structure. Under this construction, inequiv- alent knot diagrams never map to fiberwise-isotopic toral surfaces, so Rourke’s model— if it is a valid model at all— is a classifying model for the theory.
Unfortunately, Rourke’s construction is not invariant of Welded knot type
, thus it is not a valid model for Welded knot theory. This is Main Theorem 1 of the present paper. We prove the theorem by exhibiting two knot diagrams which are Welded-equivalent, but whose Rourke-models are not fiberwise-isotopic.
1. Rourke’s construction is thus useless as a model for Welded knot theory. However, the theory can be modified into a new diagrammatic theory, which we call Rotweldid knot theory, by omitting the single Welded Reidemeister move where Rourke’s invariance proof failed. The invariance of Rourke’s construction as a model for Rotweldid knot theory is our Main Theorem 2. The proof is the same as Rourke’s erroneous proof for Weldid, with the bad part excised.
Not only is Rourke’s construction a valid model for Rotweldid knot theory,
it also happens to be a classifying model. This is our Main Theorem 3.

**Logic Seminar**

Constructing Analyzable Types in Differentially Closed Fields with Logarithmic Derivatives

Ruizhang Jin (Waterloo)

4:00 PM in SEO 427

We generalize the well-known fact that the equation $\delta(\mathrm {log}\delta x)=0$ is analyzable in but not internal to the constants. We use the logarithmic derivative as a building block to construct analyzable types with a unique analysis of minimal length (up to interalgebraicity). We also look for criteria for a given definable set such that its pre-image under the logarithmic derivative is analyzable in but not internal to the constants.

We meet for lunch at noon on the first floor of SEO.