Given a map $g:Y\to Y$ that is continuous and onto, one can construct a solenoid, which is the stationary inverse limit associated to g. This process leads to a space X and a homeomorphism $v\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}p\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}i$. The dynamics of g and $v\phantom{\rule{0}{0ex}}a\phantom{\rule{0}{0ex}}r\phantom{\rule{0}{0ex}}p\phantom{\rule{0}{0ex}}h\phantom{\rule{0}{0ex}}i$ are very much related. I will discuss various examples of this process and the C*-algebras associated with solenoids. In particular, we will see examples where $Y$ is non-Hausdorff, but the associated solenoid $X$ is Hausdorff.

Solenoids and their C*-algebras

Mar. 02, 2023 3:35pm (MATH 3…

Probability

Manuel Lladser (CU Boulder)

X

Resolvability is a generalization of the notion of trilateration of the plane to an arbitrary metric space. Given a finite metric space $X,d)$, a subset $R=\{{r}_{1},\dots ,{r}_{k}\}\subset X$ is said to resolve $X,d)$ when the transformation $x\to (d(x,{r}_{1}),\dots ,d(x,{r}_{k}))$, with $x\in X$, is one-to-one. In particular, this transformation can be used to represent points in $X$ as Euclidean vectors and, due to the triangular inequality, points close by in $X$ are represented as vectors with similar coordinates in $\mathbb{R}}^{k$. Resolving sets can therefore produce feature vectors for points in abstract metric spaces, which may find applications in classification problems of symbolic objects under suitably chosen metrics. Unfortunately, the problem of resolving optimally large metric spaces (i.e., minimizing the cardinality of $R$) is NP-complete. In fact, the state of the art algorithm, the Information Content Heuristic (ICH), has complexity $O(|X{|}^{3})$ and is only guaranteed to find a resolving set of cardinality at most $1+(1+o(1))\cdot \mathrm{ln}|X|$ times the optimal one.

In this talk, I will present work in progress---which uses probabilistic arguments---regarding the multilateration of Jaccard spaces, i.e., metric spaces of the form ${2}^{X},d)$, where $2}^{X$ is the power set of a finite set $X$, and $d$ is the Jaccard distance. Namely, for all $A,B\subset X$, $d(A,B):=|A\phantom{\rule{0}{0ex}}\Delta \phantom{\rule{0}{0ex}}B|/|A\cup B|$, where $|\cdot |$ denotes cardinality and $\Delta$ the symmetric difference of sets. This work is partially funded by the NSF grant No. 1836914.

Resolvability of Jaccard Metric Spaces

Mar. 02, 2023 4pm (MATH 220)

Init Conditions

Stephanie Oh Representable functors and the Yoneda Lemma