Topology Seminars - Miami University
Spring 2012

Thursday, April 19, 2012      3:15 pm - Rm 118 BAC

Title: ON DISCRETELY GENERATED BOX PRODUCTS
Abstract:  This is a joint  work  with  R. Wilson.  We will prove that an
arbitrary box  product  of monotonically normal spaces is discretely  generated.
In  particular,  any  finite  product  of monotonically normal spaces is
discretely  generated.  Therefore there exist countable spaces not embeddable
into a box product of real lines.



Topology Seminar
Topology Seminars - Miami University
Spring 2012

Thursday, January 26, 2012      3:15 pm - Rm 118 BAC
Thursday, January 19, 2012      3:15 pm - Rm 118 BAC

Title: How much do discrete subspaces of a space characterize its topology?
Abstract: We will give a cycle of talks on topological properties determined to some
extent by discrete subspaces of a given space. The first property we are going to talk
about is discrete generability which is a convergence property with a flavor of a global one.
A space $X$ is discretely generated if for any $A\subset X$ and any point $x\in \overline A$
there exists a discrete subset $D\subset A$ such that $x\in \overline D$. It is evident that
discretely generated spaces form a class that contains both Frechet-Urysohn spaces and
scattered spaces so discrete generability is a local property with components of a global one.
We will present a survey of both recent and old results on discrete generability; the material
of the respective lectures is supposed to be sufficient to give a reasonably complete view of
the area.



Topology Seminars - Miami University
Fall 2011

Thursday, November 17, 2011      3:15 pm - Rm 118 BAC
Thursday, November 3, 2011      3:15 pm - Rm 118 BAC
Thursday, October 27, 2011      3:15 pm - Rm 118 BAC
Thursday, October 6, 2011      3:10 pm - Rm 118 BAC
Thursday, September 8, 2011      3:10 pm - Rm 118 BAC
Thursday, September 1, 2011      3:10 pm - Rm 118 BAC
Friday, August 26, 2011      3:10 pm - Rm 118 BAC

Title: How much do discrete subspaces of a space characterize its topology?
Abstract: We will give a cycle of talks on topological properties determined to some
extent by discrete subspaces of a given space. The first property we are going to talk
about is discrete generability which is a convergence property with a flavor of a global one.
A space $X$ is discretely generated if for any $A\subset X$ and any point $x\in \overline A$
there exists a discrete subset $D\subset A$ such that $x\in \overline D$. It is evident that
discretely generated spaces form a class that contains both Frechet-Urysohn spaces and
scattered spaces so discrete generability is a local property with components of a global one.
We will present a survey of both recent and old results on discrete generability; the material
of the respective lectures is supposed to be sufficient to give a reasonably complete view of
the area.



Thursday, October 20, 2011      3:15 pm - Rm 118 BAC

Thursday, October 13, 2011      3:15 pm - Rm 118 BAC
Tuesday, September 27, 2011      3:15 pm - Rm 118 BAC
Thursday, September 22, 2011      3:15 pm - Rm 118 BAC

Title: Noetherian type and other order-theoretic'' cardinal functions I, II, III, IV
Abstract:  A topological space has {\sl countable Noetherian type}'' if it
has a base where no element is contained in infinitely many elements of that base.
Study of this property goes back to the Russian School of General Topology of the
70-80s (Malykhin, Peregudov and Shapirovskii ${\ldots}$) and was taken up independently
by the American School in the 90s. Some fundamental questions about it remain open.
For example, if $X^2$ has countable Noetherian type, does $X$ need to have countable
Noetherian type too?  We will offer some partial answers to this question of Balogh,
Bennett, Burke, Gruenhage, Lutzer and Mashburn, both in the positive and in the
negative. This definition suggests introducing a cardinal function. Time permitting
we will show how this function is related to the Suslin Number (of homogeneous compacta)
and present an independence result regarding the value of this function on certain
countably supported box products.

This is joint work with Dave Milovich and Menachem Kojman.


Wednesday, October 12, 2011      3:15 pm - Rm 118 BAC

Friday, October 7, 2011      3:15 pm - Rm 118 BAC
Wednesday, September 28, 2011      3:15 pm - Rm 118 BAC
Friday, September 16, 2011      3:15 pm - Rm 118 BAC

Speaker: Daniel Farley Miami University
Title: Some iterated monodromy groups of intermediate growth I, II, III, IV
Abstract:  In a 1968 paper, John Milnor defined the growth function
of a finitely generated group, and asked whether every group has polynomial
or exponential growth. Grigorchuk produced the first examples of groups with
intermediate growth; that is, he found groups whose growth functions were
neither bounded above by polynomials, nor bounded below by exponential
functions. Grigorchuk's examples were subgroups of the automorphism group of
the infinite rooted binary tree.

In Nekrashevych's 2005 monograph, Self-similar groups, he defined the
{\sl iterated monodromy group} (IMG) of a post-critically finite complex
polynomial. (A polynomial $f$ is {\sl post-critically finite} if the set of
forward iterates $f(f(f(f...(f(c))))$ is finite, for each critical point $c$.)
The iterated monodromy group of a quadratic post-critically finite polynomial
acts on the infinite rooted binary tree, like Grigorchuk's groups.

Bux and Perez showed that the iterated monodromy group of $z^{2} + i$ has
intermediate growth. This gave the first positive evidence for a conjecture
which Bux and Perez attribute to Nekrashevych:

If $p(z) = z^{2} + c$ is a post-critically finite quadratic polynomial and $0$
is not a post-critical point of $p$, then IMG$(p)$ has intermediate growth.

In this talk (or series of talks), I will confirm this conjecture for two more
post-critically finite polynomials. I will make the talks as self-contained as
possible. The results come from undergraduate research in SUMSRI that I supervised
during the summer.



Thursday, October 6, 2011      3:10 pm - Rm 118 BAC

Thursday, September 8, 2011      3:10 pm - Rm 118 BAC
Thursday, September 1, 2011      3:10 pm - Rm 118 BAC
Friday, August 26, 2011      3:10 pm - Rm 118 BAC

Title: How much do discrete subspaces of a space characterize its topology?
Abstract: We will give a cycle of talks on topological properties determined to some
extent by discrete subspaces of a given space. The first property we are going to talk
about is discrete generability which is a convergence property with a flavor of a global one.
A space $X$ is discretely generated if for any $A\subset X$ and any point $x\in \overline A$
there exists a discrete subset $D\subset A$ such that $x\in \overline D$. It is evident that
discretely generated spaces form a class that contains both Frechet-Urysohn spaces and
scattered spaces so discrete generability is a local property with components of a global one.
We will present a survey of both recent and old results on discrete generability; the material
of the respective lectures is supposed to be sufficient to give a reasonably complete view of
the area.



Thursday, September 8, 2011      3:10 pm - Rm 118 BAC

Thursday, September 1, 2011      3:10 pm - Rm 118 BAC

Friday, August 26, 2011      3:10 pm - Rm 118 BAC

Abstract: We will give a cycle of talks on topological properties determined to some
A space $X$ is discretely generated if for any $A\subset X$ and any point $x\in \overline A$
there exists a discrete subset $D\subset A$ such that $x\in \overline D$. It is evident that