Seminars
Helge Glöckner
Direct limits of topological groups
Christine Stevens
Character Groups for Unusual Topologies for $\mathbb R^n$
Group topologies for $\mathbb R^n$ that are weaker than the usual topology arise in a variety of contexts, including the study of Lie groups of transformations. We will focus on metrizable topologies that are defined by choosing a sequence in $\mathbb R^n$ and forcing it to converge to zero at (approximately) a specified rate. One technique for understanding such a topological group is to study its group of continuous characters. Although it is known that the character groups are uncountable, the proof of this fact does not provide a practical way of finding the dual group. There are simple examples in which it is easy to find a countable set of characters, but it is not obvious what the additional characters might be, even though there are uncountably many of them.
We will remedy that situation for a significant subset of these topologies, namely, topologies for the real numbers in which the "converging" sequence is a strictly increasing sequence of natural numbers with the property that each term divides the next. Our analysis will focus on a particular subgroup of the character group comprising characters that satisfy a Lipschitz-like condition.
Slides of the talk can be downloaded
here.
Vaja Tarieladze
Compatible locally convex topologies for topological vector groups
In P. S. Kenderov, On topological vector groups, Mat. Sb. (N.S.), 1970, Volume 81(123), Number 4, 580–599 it is shown that a (real or complex) vector space duality always admits the finest compatible locally convex topological vector group topology.
We plan to discuss Kenderov's proof of this statement.
Invited Lectures
Sergio Ardanza
Some applications of topological data analysis
Persistent homology is a tool derived from algebraic topology that can be applied to analize data in very different contexts. We will introduce this technique and show two applications firt to indentify different physical states in vibrated granular materials (joint work with R. Arevalo D. Maza and I.Zuriguel),
and second to find relevant regions on the genome for classifications/prognosis in cancer patients (joint work with T. Borman, F.J Arsuaga, G.González).
Xabier Domínguez
The Baire property on precompact abelian groups (joint work with M. J. Chasco and M. Tkachenko)
We give different necessary and/or sufficient conditions on a separated duality of abelian groups $\langle G, H \rangle$ for the topological group $(G,\sigma(G,H))$ to be a Baire space, where $\sigma(G,H)$ denotes the corresponding weak topology on $G$.
Slides of the talk can be downloaded
here.
Salvador Hernández
On some types of subgroups of compact groups (joint work with K. Hofmann y S. Morris)
The following questions will be discussed:
- Question 1. Does every infinite compact (Hausdorff) group $G$ have
an infinite metric subgroup $H$?
- Question 2. Does every compact (Hausdorff) group $G$ have
a subgroup $H$ which is not (Haar) measurable?
Short Talks
Daniel de la Barrera
Groups and topologies related to D-sequences
$D$-sequences are sequences of natural numbers which characterize linear topologies in the group of the integers. Since $(p^n)$ is a $D$-sequence, Prüfer groups and $p$-adic integers are generalized by means of $D$-sequences. This talk intends to be a survey of these groups and some topologies on them.
Slides of the talk can be downloaded
here.
Hugo J. Bello
Cross-sections and extensions of topological groups
(joint work with M. J. Chasco, M. Tkachenko and X. Domínguez)
This talk is devoted to the splitting of extensions of topological abelian groups.
We prove that every extension $0\to K\to X \to A(Y)\to 0$ of a free abelian topological group $A(Y)$ over a zero-dimensional $k_{\omega}$-space $Y$ by a compact abelian group $K$, splits. Moreover we show that if $K$ is a compact subgroup of a topological abelian group $X$ such that the quotient group $X/K$ is a zero-dimensional $k_{\omega}$-space, then there exists a continuous cross section from $X/K$ to $X$.
Slides of the talk can be downloaded
here.
Luis Tárrega
Equicontinuity criteria for metric-valued sets of continuous functions
(joint work with M. Ferrer and S. Hernández)
Combining ideas of Troallic and Cascales, Namioka, and Vera,
we prove several characterizations of almost equicontinuity and
hereditary almost equicontinuity for subsets of metric-valued continuous functions
when they are defined on a Cech-complete space. We also obtain some applications
of these results to topological groups and dynamical systems.
Slides of the talk can be downloaded
here.