Steven Clontzmathematician, professor, puzzler
http://clontz.org/
Fri, 15 Mar 2019 15:57:34 +0000Fri, 15 Mar 2019 15:57:34 +0000Jekyll v3.7.42019 Spring Topology and Dynamics Conference<p>Slides from my talk are available at
<a href="https://www.overleaf.com/read/wmrgkzfzbbvf">https://www.overleaf.com/read/wmrgkzfzbbvf</a>.</p>
<p>If you’re looking for a link to the video of
my talk last year on the <a href="https://topology.pi-base.org">pi-Base</a>,
that’s here:
<a href="https://youtu.be/iuNZWllVDKg?t=3910">https://youtu.be/iuNZWllVDKg?t=3910</a>.</p>
Wed, 20 Feb 2019 13:38:00 +0000
http://clontz.org/blog/2019/02/20/stdc/
http://clontz.org/blog/2019/02/20/stdc/SBG in MyLabMath<p>In response to a <a href="https://twitter.com/katemath/status/1026848341177638914">discussion</a> on Twitter
with Kate Owens and Drew Lewis,
I wanted to document my experience using the Study Plan
in Person’s Study Plan feature within its MyLabMath
(MyMathLab? MyLab Math?) product.</p>
<p>For various reasons outside the scope of this blog post,
I chose to make the <a href="https://prof.clontz.org/classes/2018/06/ma125/">course</a>
(click for the syllabus) grade 85% based on the Study Plan,
and 15% on the Final Exam.
With 85% based on the Study Plan, I of course needed to ensure two
things:</p>
<ol>
<li>Completion of the Study Plan should reflect true mastery
of the course objectives.</li>
<li>I should be able to reasonably guarantee that academic honesty
had been respected by my students.</li>
</ol>
<p>For #1, I began by reviewing my options. In the Study Plan,
instructors can see a master list of all “objectives” by visiting
<code class="highlighter-rouge">Study Plan > Manage > Coverage and Scoring</code>. Each objective may be
selected as <code class="highlighter-rouge">Scored and recommended</code> (counts towards the Study Plan grade)
and/or <code class="highlighter-rouge">Available for practice</code> (appears in the student Study Plan).
In my implementation, I always used both options together (and I’m not sure
why anyone wouldn’t at least include all scored objectives in practice),
so I won’t distinguish them further here.</p>
<p>To design my course, I went through and first checked/unchecked the sections
of the textbook that my <a href="https://www.southalabama.edu/colleges/artsandsci/mathstat/syllabi/ma125.html">department course syllabus</a>
designated. Then within each section, I checked/unchecked objectives
according to the following criteria:</p>
<ul>
<li>I covered all the “learning objectives” given on my department’s course
syllabus.</li>
<li>Even if it wasn’t explicitly given as a “learning objective”, I included
objectives that either supported future objectives or I simply thought
were important.</li>
<li>I tried to limit myself to four or less objectives per section, with
five as my absolute maximum.</li>
<li>There existed multiple reasonable exercises for students to complete
when practicing and being assessed in MyLab Math.</li>
</ul>
<p>Per the last point: each objective can be expanded into its supporting
exercises in MyLab Math. Often, many of the exercises were either too easy
or too perfidious to be worthwhile as either practice or assessment.
(I suppose there’s an argument to only include these in practice; I chose
not to.) So I selected the exercises that seemed reasonable tractable and
relevant for assessment; there were a few interesting objectives I
abandoned due to a lack of quality exercises.</p>
<p>Here is where I diverged from the intended use case of the Study Plan.
Pearson, for some unintelligible reason, has designed the Study Plan as
though only the student is interested in knowing their mastery of course
material. So while there is the ability for instructors to see Study
Plan progress and use it as a part of the grade on MyLabMath, there are
<del>two</del> three serious anti-patterns in Pearson’s design (that I’ve reported to them,
not that I anticipate any changes):</p>
<ul>
<li><em>UPDATE 2018-08-08:</em> <strong>Study Plan mastery is set by student account, not by
the course.</strong></li>
</ul>
<p>This isn’t bad in of itself, but the only way to fairly assess based on
work done during your course is to enable an option in the Study Plan to
scrub all previous progress in the Study Plan from previous semesters. I wish
there was a way to have a course-level Study Plan that somehow shadows the
student-level Study Plan. There’s possibly a major security issue here if a student
has access to a second MyLabMath course that can improve the Study Plan;
that depends on how Pearson has implemented things behind the scenes and
I cannot tell.</p>
<ul>
<li>The default/natural way to configure the Study Plan for students to
demonstrate mastery is through a “Quiz Me” feature. After students
click the “Practice” button and correctly solve a few relevant exercises,
the “Quiz Me” button becomes available, which generates a random quiz
based upon that sole objective.
<strong>There is no way to password-protect this feature.</strong></li>
</ul>
<p>Of course, the above issue directly contradicts Thing #2 listed above. Fortunately,
this can be disabled. Instead, students can be required to instead answer a relevant exercise
from an assigned assessment via <code class="highlighter-rouge">Study Plan > Manage > Mastery</code>. This brings up the
second anti-pattern.</p>
<ul>
<li>Pearson expects instructors to design their assessments first, and then generate
Study Plans based upon those assessments.
<strong>This is in conflict with <a href="https://cft.vanderbilt.edu/guides-sub-pages/understanding-by-design/">backwards design</a>†, and there is no automatic
way to generate assessments from a carefully curated Study Plan.</strong></li>
</ul>
<p>However, despite the shrugging I got from Pearson when I asked, I did find a way
to hand-create assessments from an existing Study Plan without too much pain.</p>
<ul>
<li>I created a quiz for each section (or two sections if they summed to four/five objectives)
using <code class="highlighter-rouge">Assignments > Manage > Create Assignment</code>.</li>
<li>I named the quiz for that section(s), and did not create a companion Study Plan
since I already had done that work at the beginning as a part of my backwards design.</li>
<li>Under <code class="highlighter-rouge">Add/Remove Content</code>, I chose the Chapter and Objective, and then under Availability
I chose “Questions that are in the Study Plan”.</li>
<li>Then for each Objective that would list questions (which were in my Study Plan),
I hit the “check all” box and clicked “Pool” to create a quiz exercise that would be
chosen randomly from that pool.</li>
<li>I set a password for the quiz and let it be available for the entire semester.
<em>UPDATE 2018-08-08:</em> Quizzes could be attempted infinitely many times.</li>
</ul>
<p>Since a password was set, students were required to visit our on-campus Math Technology Lab
to complete quizzes. For this summer MTWRF class, I actually did not cover new material
on Fridays to let students visit the lab and work on quizzes. They could also reclaim
their Friday afternoons to do other things, and work on quizzes at another time during
the week that the lab was open as well. (I made an error here: I should have required students
to either spend a sufficient amount of time or made a sufficient amount of progress
in the lab working on quizzes in MyLabMath, since I believe
that data is available, or else they should have been forced to come to class on Friday.)</p>
<p><em>UPDATE 2018-08-08:</em> In the gradebook, there were only two entries: 85% for the Study Plan,
and 15% for the Final Exam. Note in particular that Quizzes were worth 0% directly, but
they updated the Study Plan, so each correct response was essentially
worth about 1% of the overall grade for the course.</p>
<p>Overall I think this went well. A major downside was the inability to assess twice in
MyLabMath: while I don’t usually consider a standard “mastered” until it has been
demonstrated as mastered twice on separate weeks, there’s no way to enforce this in
MyLabMath directly (as far as I know). So this was part of the justification for the final,
as students knew they would need to hit the high points of the course one more time
at the semester’s end. However, because the final was so weakly weighted, I’m unconvinced
it was very useful. (Of course, there are other reasons to give a final besides
pedagogical soundness…) My final was given on paper, separate from MyLabMath, and
did not affect the Study Plan at all.</p>
<p><strong>TL;DR:</strong> Using the MyLabMath Study Plan for SBG is better than using MyLabMath
for pretty much anything else. They could make the product a lot better, but I haven’t
been given much evidence that Pearson cares. Given a repeat of the situation I was in
(a short-notice prep for a summer freshman math course), I’d probably do everything
again the same way, with the above noted exceptions.</p>
<hr />
<p><small>† What an unfortunate name: I get that we want to start by thinking about
what our students should master by the end of the course, but “backwards” certainly
comes off as… well, backwards.</small></p>
Tue, 07 Aug 2018 21:48:00 +0000
http://clontz.org/blog/2018/08/07/mylabmath-sbg/
http://clontz.org/blog/2018/08/07/mylabmath-sbg/Mathfest<p>I’ll be co-organizing two workshops and giving one talk
at MAA Mathfest 2018 in Denver. More info is available
at <a href="/mathfest">Clontz.org/mathfest</a>.</p>
Sun, 29 Jul 2018 14:13:00 +0000
http://clontz.org/blog/2018/07/29/mathfest/
http://clontz.org/blog/2018/07/29/mathfest/52nd Spring Topology and Dynamical Systems Conference<p>Here are a few useful resources related to my involvement at the
<strong>52nd Spring Topology and Dynamical Systems Conference</strong>
at Auburn University, March 14-17.</p>
<h2 id="conference-livestream">Conference Livestream</h2>
<p>My students Cody Martin and Sam Formichella are assisting me in livestreaming
several plenary talks and set-theoretic topology talks for the conference.
A schedule and live video are available on my
<a href="/math/stdc-2018">STDC2018 Livestream</a> page.</p>
<h2 id="team-based-learning-and-standards-based-grading">Team-Based Learning and Standards-Based Grading</h2>
<p>My slides are <a href="https://docs.google.com/presentation/d/1ulHbe6SlAoUjq5hQH70W5rt0P96mFeESJZ7UG8pOurs/edit?usp=sharing">available at this link</a>.</p>
<h2 id="pi-base-a-usable-map-of-the-forest">pi-Base: A Usable Map of the Forest</h2>
<p>My slides are <a href="https://docs.google.com/presentation/d/1RnQYkNL6Mt4ye99BV7XHLsu-h5-s5qLfNFdl22edo60/edit?usp=sharing">available at this link</a>.</p>
Fri, 09 Mar 2018 14:13:00 +0000
http://clontz.org/blog/2018/03/09/spring-topology/
http://clontz.org/blog/2018/03/09/spring-topology/pi-Base: A usable map of the forest<p>Here are my notes for today’s talk on <a href="http://pi-base.org">pi-Base</a> for
the University of South Alabama’s ACM and Math Clubs.</p>
<h2 id="clontzorg">Clontz.org</h2>
<ul>
<li>Notes/links from this talk are available on my <a href="http://clontz.org">website</a>.</li>
</ul>
<h2 id="relationship-of-math-and-comp-sci">Relationship of Math and Comp Sci</h2>
<ul>
<li>Math folks: knowing computer science/engineering can help you get a job.
<ul>
<li>Don’t have to take classes; just pick up a fun sideproject!</li>
</ul>
</li>
<li>CS folks: knowing math can help you get a job.
<ul>
<li>Having a math background makes people think you’re (a) smart,
(b) a problem-solver (in general).</li>
</ul>
</li>
<li>I hope to see future collaborations between ACM and Math Club, and I
encourage members of both groups to be involved with the other and
leadership to work together to support their memberships.</li>
</ul>
<h2 id="what-is-topology">What is Topology?</h2>
<ul>
<li>Topology is the a of mathematical structure that generalizes geometry
and calculus.
<ul>
<li>Classic example: <a href="https://en.wikipedia.org/wiki/Topology#/media/File:Mug_and_Torus_morph.gif">donut and coffee cup</a></li>
<li>My favorite example: <a href="http://mappmath.org/puzzles/">topology of fonts</a></li>
<li>Big data application: <a href="https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf">persistent homology</a></li>
</ul>
</li>
<li>One goal of topology is to identify the properties of topological spaces
that characterize them.
<ul>
<li>Compactness: \([0,1]\) is compact, but
\(\mathbb{R}\) is not.</li>
<li>Hausdorff: \([0,1]\) is Hausdorff, but an indiscrete space is not.</li>
<li>Normal: \([0,1]\) must be normal, because it is compact & Hausdorff.</li>
</ul>
</li>
</ul>
<h2 id="what-is-pi-base">What is pi-Base?</h2>
<ul>
<li>Topologists have studied thousands of topological spaces and thousands of
topological properties over the past century.
<ul>
<li><a href="https://en.wikipedia.org/wiki/Counterexamples_in_Topology">Counterexamples in Topology</a> surveyed about 160 of
these spaces and about 60 of these properties in 1970.</li>
<li>Paraphrasing the famous topologist Mary Ellen Rudin in her
review of Counterexamples,
<a href="http://www.jstor.org/stable/2318037?origin=crossref&seq=2#page_scan_tab_contents">topology is a dense forest of counterexamples, and a usable map of the forest is a fine thing</a>.</li>
<li><a href="http://pi-base.org">pi-Base</a> began as a sideproject of my colleague
<a href="http://jdabbs.com/">James Dabbs</a> to digitize this information into a user-friendly
web app.</li>
<li>Later pi-Base was opened up to community contributions.</li>
</ul>
</li>
<li>Features:
<ul>
<li><a href="https://topology.jdabbs.com/properties/P000013">Properties</a></li>
<li><a href="https://topology.jdabbs.com/spaces/S000026">Spaces</a></li>
<li><a href="https://topology.jdabbs.com/spaces/S000026/properties/P000013">Theorems</a></li>
<li>Canonical naming as in <a href="http://oeis.org/A000045">OEIS</a>.</li>
</ul>
</li>
</ul>
<h2 id="where-is-pi-base-going">Where is pi-Base going?</h2>
<ul>
<li>Currently I have a small grant to convert pi-Base into a modern tool for
mathematical researchers.
<ul>
<li>Major flaw in current version: lack of citations and peer-review.</li>
<li>ACM officer Cody Martin worked for me last summer to add citations
from pi-Base to Counterexamples.</li>
<li>Once this audit is complete, all contributions will require references
to a peer-reviewed manuscript to be marked as verified.</li>
</ul>
</li>
<li>pi-Base will become a treasure trove for undergraduate research.
<ul>
<li>There are still many unknowns in topology.</li>
<li>pi-Base can automatically catalog space/property pairs that are missing.</li>
<li>Three possibilities:
<ul>
<li>The question has been answered in literature not in pi-Base.</li>
<li>The question hasn’t been answered because it’s hard.</li>
<li>The question hasn’t been answered because no one’s tried: perfect
for undergrads!</li>
</ul>
</li>
</ul>
</li>
</ul>
Tue, 12 Sep 2017 10:16:00 +0000
http://clontz.org/blog/2017/09/12/acm-pibase-talk/
http://clontz.org/blog/2017/09/12/acm-pibase-talk/Alternate Standard Deviation Formula<p>While teaching MA 110 (Finite Mathematics), I casually introduced
standard deviation using the formula provided by our textbook,
where \(\sum x\) represents the sum of a dataset and
\(\overline{x}=\frac{\sum x}{n}\) is its mean:</p>
<p>\[
s
=
\sqrt{
\frac{
\sum (x^2) - n(\overline{x})^2
}{
n-1
}
}
\]</p>
<p>But, this isn’t my favorite formula for standard deviation, because it’s
not really clear what it’s trying to measure from its formula. The formula
I was more accustomed to makes it clear that standard deviation measures
the differences between datapoints and the mean, and then sums up the
squares of those differences. That is to say, it gives a measure of how
different the datapoints are from the average.</p>
<p>\[
s
=
\sqrt{
\frac{
\sum (x-\overline{x})^2
}{
n-1
}
}
\]</p>
<p>But while the textbook uses that first formula, some of our online homework
uses the second formula when working out example solutions. So, for the
record, here is a proof that both formulas do in fact measure the same thing.</p>
<p>Note that we need only prove that the numerators of each fraction are equal.
So…</p>
<p>\[
\sum (x-\overline{x})^2
\]</p>
<p>Applying \((a-b)^2=a^2-2ab+b^2\), we get:</p>
<p>\[
= \sum (x^2-2x\overline{x}+(\overline{x})^2)
\]</p>
<p>We’re allowed to add up each term separately.</p>
<p>\[
= \sum (x^2)-2\sum (x\overline{x})+\sum((\overline{x})^2)
\]</p>
<p>Now, since the last term is just adding the same number
\((\overline{x})^2\) once for each of the \(n\) members of the dataset,
we may simplify it as follows.</p>
<p>\[
= \sum (x^2)-2\sum (x\overline{x})+n(\overline{x})^2
\]</p>
<p>We may also factor out the constant \(\overline{x}\) from the middle sum.</p>
<p>\[
= \sum (x^2)-2\overline{x}\sum x+n(\overline{x})^2
\]</p>
<p>It follows from \(\overline{x}=\frac{\sum x}{n}\) that
\(n\overline{x}=\sum x\), giving the following.</p>
<p>\[
= \sum (x^2)-2\overline{x}(n\overline{x})+n(\overline{x})^2
\]</p>
<p>The solution is revealed by rearranging and combining like terms.</p>
<p>\[
= \sum (x^2)-2n(\overline{x})^2+n(\overline{x})^2
\]</p>
<p>\[
= \sum (x^2)-n(\overline{x})^2 \hspace{1em}\Box
\]</p>
Fri, 21 Apr 2017 11:52:00 +0000
http://clontz.org/blog/2017/04/21/alternate-standard-deviation/
http://clontz.org/blog/2017/04/21/alternate-standard-deviation/Job Hunting Workshop Notes<p>First things first, here are the relevant links.</p>
<ul>
<li><a href="http://overleaf.com">Overleaf.com</a> and <a href="http://sharelatex.com">ShareLaTeX.com</a>
<ul>
<li>Online LaTeX editors</li>
<li>Typeset professional documents and save/share them in the cloud</li>
</ul>
</li>
<li><a href="http://researchgate.net">ResearchGate.net</a>
<ul>
<li>Professional network for academics</li>
<li>Connect with colleagues at other institutions (useful
for potential name recommendation, but cannot replace
networking in real life)</li>
</ul>
</li>
<li><a href="http://mathjobs.org">MathJobs.org</a>
<ul>
<li>One of the main job posting websites for academic jobs
in mathematics</li>
</ul>
</li>
</ul>
<p>Some more detailed notes follow.</p>
<hr />
<h2 id="typesetting-professional-documents-with-overleafsharelatex">Typesetting Professional Documents with Overleaf/ShareLaTeX</h2>
<p>Something you can start doing now, no matter how far off your job hunt is,
is learn to typeset professional documents. In addition to writing research
papers, it’s good practice to use professional-looking templates based on the
LaTeX typesetting system. Learning LaTeX isn’t hard, especially if you use
cloud LaTeX services like <a href="http://overleaf.com">Overleaf</a> and <a href="http://sharelatex.com">ShareLaTeX</a>.</p>
<p>Both websites allow you to select pre-made templates for CVs, cover letters,
and so on. You can then open up the LaTeX code within your web browser,
make your edits to customize its contents to yourself, and then save/share the
result online or download a PDF. First impressions are important, and using
LaTeX to typeset documents is the industry standard, so it’s wise to use it when
presenting yourself to prospective future collegues.</p>
<h2 id="networking-in-real-life-and-online">Networking in real life and online</h2>
<p>Your main goal as an applicant is to get your foot in the door. In a sea of
qualified applicants, you only have a few opporunities to catch a search committee’s
attention. One important way is to have a colleague at the school you’re applying to.
That means that as soon as you have research to share, you need to start going to
conferences and meeting people in your field.</p>
<p>There’s a very small chance that you’ll
know someone on the search committee. Howeer, many search committees will ask their
colleagues for input. Those collegues will probably (at most) look through the list of
applicant names, their schools, and their fields. So, you want your name to be familiar
to that colleague, at least enough that they will take the extra step to read your cover
letter.</p>
<p>There’s no substitute for meeting colleagues in real life. However, another tool you
can use to follow up on a real life connection is <a href="http://researchgate.net">ResearchGate</a>. While
search committees should read every cover letter they receive, your colleague may not
be serving on that committee. In that case, they may only have a list of names,
schools, and fields of research; thus, you want to increase your chances of name
recognition.</p>
<h2 id="finding-jobs-and-submitting-applications">Finding jobs and submitting applications</h2>
<p>While there are other sites for academic mathematics jobs, the only one I used during
my search was <a href="http://mathjobs.org">MathJobs</a>. You should begin looking on this site 13 months
before you want to start; for example, if you’re looking for a job to start in August
2018, you should begin looking in July 2017.</p>
<p>Here’s my recommended workflow. Start by making an account. Once you’ve logged in,
go through all the job postings by clicking All on the View Jobs page. You’ll
probably notice that 80% of the postings don’t apply to you, depending on your
field of research and geographical preferences. Mark those jobs with an X so they
will be hidden by default when logged in. Other jobs you will know that you want
to apply. Mark those with a checkmark.</p>
<p>Now once you’ve filtered all the postings, you can click the yellow jagged cloud
on future logins to see what’s been posted since the last time you visited. I
recommend doing this once a week so you don’t get overwhelmed.</p>
<p>There are other places for advice on how to write CVs, cover letters, etc.,
beyond my general advice above on using professional LaTeX typesetting.
One particular thing I’d like to mention: make sure you make the case in your
cover letter for why you want that job particularly. A connection with
a faculty member or the local community is a great way to catch the attention
of search committees you have a particular interest in.</p>
<hr />
<p>Best of luck!</p>
Fri, 31 Mar 2017 18:16:00 +0000
http://clontz.org/blog/2017/03/31/job-hunting-workshop/
http://clontz.org/blog/2017/03/31/job-hunting-workshop/Katamari on the Rocks<p>So here’s an arrangement by myself (Steven Clontz, alias Steben on the Overclocked ReMix forums) of the Katamari Damacy theme, Katamari on the Rocks, for marching band. The percussion was arranged by Christopher Nelson (alias Rainman DX on the Overclocked ReMix forums). This was written somewhere around 2006, when I was a student at Auburn University (hence why I stuck a bit of “War Eagle” in the arrangement).</p>
<p>Over the last decade I’d get requests for the arrangement every couple years or so, but somewhere down the road I had misplaced the hard drive that contained them. Fortunately, I recently had the presence of mind to track down Chris, who had kept up with his copies better than I.</p>
<p>The arrangement’s Finale source files and a burned MP3 of the resulting MIDI have been stored in the following GitHub repo for posterity.</p>
<p><a href="https://github.com/StevenClontz/katamari-on-the-rocks-marching-band">https://github.com/StevenClontz/katamari-on-the-rocks-marching-band</a></p>
<p>Here’s a direct link to the MP3 if you want to give it a listen.</p>
<p><a href="https://github.com/StevenClontz/katamari-on-the-rocks-marching-band/blob/master/katamari-on-the-rocks.mp3?raw=true">https://github.com/StevenClontz/katamari-on-the-rocks-marching-band/blob/master/katamari-on-the-rocks.mp3?raw=true</a></p>
Sat, 18 Mar 2017 16:28:00 +0000
http://clontz.org/blog/2017/03/18/katamari-on-the-rocks/
http://clontz.org/blog/2017/03/18/katamari-on-the-rocks/k-Markov Strategies in Selection Games<p>Slides for my talk may be found at <a href="https://docs.google.com/presentation/d/1arkW_nLv8ph8fjn-_zqgCcSJAfdGgxIB2d7mDxPPe_Q/edit?usp=sharing">Google Slides</a>.</p>
<p>I’ve also put my notes on a more in-depth talk on my website
<a href="/blog/2017/02/05/auburn-chalk-talk/">here</a>.</p>
Sat, 04 Mar 2017 14:30:00 +0000
http://clontz.org/blog/2017/03/04/stdc2017/
http://clontz.org/blog/2017/03/04/stdc2017/Markov Strategies in Selection Games<p>Here is an outline of the talk I gave at the AU set-theoretic topology
seminar on 2017 Feb 06.</p>
<ul>
<li>Definition of the <a href="https://en.wikipedia.org/wiki/Banach%E2%80%93Mazur_game">Banach-Mazur Game</a>:
<ul>
<li>Introduced by Mazur as Problem 43 of the <a href="https://en.wikipedia.org/wiki/Scottish_Book">Scottish Book</a>
<a href="http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf">[pdf]</a>, a notebook kept by mathematicians frequenting
the Scottish Cafe in Lwów, Poland in the 1930s and 1940s.</li>
<li>Let \(S\subseteq\mathbb R\). Two players alternate choosing
decreasing intervals
\(N_0\supseteq E_0\supseteq N_1\supseteq E_1\supseteq\dots\).
The first player wins if \(\bigcap_{n<\omega}N_n\) contains a point of
\(S\); the first player wins otherwise.</li>
<li>Banach (1935) showed that the second player has a winning strategy for
this game if and only if \(S\) is <em>meager</em>, a countable union
of nowhere-dense sets. The prize was a bottle of wine, awarded by Mazur.</li>
<li>Our variant of interest: two players choose<br />
decreasing non-empty open subsets
\(E_0\supseteq N_0\supseteq E_1\supseteq N_1\supseteq\dots\)
of a topological space \(X\), and let the <em>second</em> player win if and
only if \(\bigcap_{n<\omega}N_n\not=\emptyset\).
Call this game \(BM(X)\).</li>
</ul>
</li>
<li>Limited information strategies
<ul>
<li>Let \(M\) be a set of possible moves for a game.</li>
<li>A (perfect information) strategy is a function
\(\sigma:M^{<\omega}\to M\) that determines the next move for a player
based upon all the previous moves of her opponent.</li>
<li>A \(k\)-tactical strategy is a function
\(\sigma:M^{\leq k}\to M\) that determines the next move for a player
based upon the most recent \(k\) previous moves of her opponent.</li>
<li>A \(k\)-Markov strategy is a function
\(\sigma:M^{\leq k}\times\omega\to M\)
that determines the next move for a player
based upon the most recent \(k\) moves of her opponent and the current
round number.</li>
<li>A coding strategy is a function
\(\sigma:M^{\leq 2}\to M\)
that determines the next move for a player
based upon the most recent moves of both players.</li>
<li>A strategy is <em>winning</em> if a player that follows the strategy is
guaranteed to win the game regardless of the moves of the opponent.</li>
<li>A winning (perfect information) strategy for the second player in
\(BM(X)\) may always be improved to a winning coding strategy.
(Debs 1985; Galvin,Telgarsky 1986)</li>
<li>A winning Markov (that is, \(1\)-Markov)
strategy for the second player in
\(BM(X)\) may always be improved to a winning tactical
(that is, \(1\)-tactical, also known as stationary) strategy.
(Galvin,Telgarsky 1986)</li>
<li>Debs (1985) found examples of spaces for which the second player
in \(BM(X)\) has winning
(\(2\)-tactical; see Bartoszynski,Scheepers,Just 1993)
strategies, but no winning tactical strategies.</li>
<li><a href="http://www.telgarsky.com/1987-RMJM-Telgarsky-Topological-Games.pdf">Telgarksy conjectures (1987)</a>
that there exist spaces \(X_k\) for \(k<\omega\)
such that the second player in
\(BM(X_k)\) has a winning \((k+1)\)-tactical
strategy, but no winning \(k\)-tactical strategy.</li>
</ul>
</li>
<li>Countable/Finite Games
<ul>
<li>Scheepers (1992) published the first of many papers under the
title of <em>Meager-Nowhere Dense Games: \(n\)-Tactics</em>.</li>
<li>Let \(MG(X)\) be a game where the first player
chooses meager subsets \(M_{n+1}\supseteq M_n\) during round \(n+1\),
followed by the second player choosing a nowhere dense subset
\(N_{n+1}\). The second player wins if
\(\bigcup_{n<\omega}N_n\supseteq\bigcup_{n<\omega}M_n\).</li>
<li>The special case where \(X=\kappa\) has the co-finite topology
(and therefore the first player chooses countable sets and the second
player chooses finite sets) was studied by Scheepers in
<em>Concerning \(n\)-tactics in the countable-finite game</em>.
Call this game \(Sch^{\cup,\supsetneq}(\kappa)\).</li>
<li>Scheepers showed that any \((k+3)\)-tactical strategy for the second
player in \(Sch^{\cup,\supsetneq}(\kappa)\)
may be improved to a \(k\)-tactical strategy.</li>
<li>Modification: \(Sch^{\cap}(\kappa)\): the first player chooses
any countable set \(C_n\), and the second player chooses any finite set
\(F_n\). The second player wins if
\(\bigcup_{n<\omega}F_n\supseteq\bigcap_{n<\omega}C_n\).</li>
<li>The axiom \(\mathcal A’(\kappa)\) implies that the second player
has a winning \(2\)-tactical strategy in
\(Sch^{\cup,\supsetneq}(\kappa)\) (Scheepers) and
a winning \(2\)-Markov strategy in
\(Sch^{\cap}(\kappa)\) (Clontz).</li>
<li>\(A’(\aleph_n)\) holds in ZFC for \(n<\omega\) (Clontz,Dow to appear);
\(A’(\kappa)\) may be forced to hold for arbitrarily large
\(\kappa\leq\mathfrak c\) (Scheepers).</li>
</ul>
</li>
<li>Selection Games
<ul>
<li>Let \(G_{fin}(\mathcal A,\mathcal B)\)
be the game where the first player chooses
\(A_n\in \mathcal A\) and the second player chooses a finite subset
\(B_n\in[A_n]^{<\omega}\). The second player wins if
\(\bigcup_{n<\omega}B_n\in\mathcal B\).</li>
<li>For each cardinal \(\kappa\), let \(L(\kappa)=\kappa\cup\{\infty\}\)
be the space with \(\kappa\) discrete and \(\infty\) having
co-countable neighborhoods.</li>
<li>Let the Menger game be
\(Men(X)=G_{fin}(\mathcal O,\mathcal O)\) where \(\mathcal O\)
is the set of open covers of \(X\).</li>
<li>In regards to \(k\)-Markov strategies for the second player,
\(Men(L(\kappa))\) and \(Sch^{\cap}(\kappa)\) are equivalent
(Clontz).</li>
</ul>
</li>
<li>Markov strategies in Selection Games
<ul>
<li>Any \((k+2)\)-Markov strategy for the second player in
\(G_{fin}(\mathcal A,\mathcal B)\) may be improved to a
\(2\)-Markov strategy.
<ul>
<li>Source: <a href="https://www.researchgate.net/publication/282155672_Applications_of_limited_information_strategies_in_Menger%27s_game">Applications of limited information strategies in Menger’s game (to appear, CUMC)</a></li>
</ul>
</li>
<li>Assume \(|\bigcup\mathcal A|\leq\aleph_0\) and \(\mathcal B\)
is closed under supersets. Then any winning (perfect-information)
strategy for the second player in \(G_{fin}(\mathcal A,\mathcal B)\)
may be improved to a Markov strategy.
<ul>
<li>Source: <a href="https://www.researchgate.net/publication/309202868_Relating_games_of_Menger_countable_fan_tightness_and_selective_separability">Relating games of Menger, countable fan tightness, and selective separability (preprint)</a></li>
</ul>
</li>
<li>Corollaries:
<ul>
<li>Let \(\mathcal D\) give the dense subsets of a space \(X\).
The second player having a winning strategy in
\(G_{fin}(\mathcal D,\mathcal D)\) characterizes
strategic selective separability, \(SS^+\). Thus all countable
\(SS^+\) spaces are Markov selectively separable, \(SS^{+mark}\).
(Barman,Dow 2012).</li>
<li>The second player having a winning strategy in
\(G_{fin}(\mathcal O,\mathcal O)\) characterizes the
strategic Menger property.
All second-countable strategic Menger spaces are Markov Menger.
Markov Menger spaces are exactly the
\(\sigma\)-relatively-compact spaces
(equivalent to \(\sigma\)-compact for regular spaces). (Clontz)</li>
</ul>
</li>
<li>Question (Gruenhage): Does there exist an \(SS^+\) space that is not
\(SS^{+mark}\)?</li>
</ul>
</li>
</ul>
Sun, 05 Feb 2017 19:54:00 +0000
http://clontz.org/blog/2017/02/05/auburn-chalk-talk/
http://clontz.org/blog/2017/02/05/auburn-chalk-talk/