## Daniel Donado – Metric spaces on omega_1 under determinacy

August 24, 2014

Daniel was my first masters student, completing his M.S. in June 2010. I co-advised him together with my friend and colleague Ramiro de la Vega, at the Universidad de los Andes, in Bogota. The following picture is from his Facebook profile.

Daniel

Daniel’s thesis, Metric spaces on $\omega_1$ under the axiom of determinacy (in Spanish), is part of a vastly unexplored field: general topology in the absence of choice.

Work on this area has been mostly about highlighting pathologies, illustrating how vastly different results can be when we keep standard definitions but work in setting where the axiom of choice fails badly. Even in the setting of the real numbers with the standard topology, things may not work as expected: Every set of reals may be Borel (in fact $F_{\sigma\sigma}$), there may be Borel infinite Dedekind-finite sets, etc. In his PhD thesis at UC Berkeley, Apollo Hogan showed that, instead, we can carry out a systematic and detailed study of general topology if instead of dealing with arbitrary models with absurd failures of choice, we concentrate on settings where the absence of choice is compensated by a rich combinatorial structure. Specifically, Apollo (who was a student at Berkeley at the same time I was there) considered topology under the axiom of determinacy. Daniel’s thesis is a survey of some of the results discovered by Apollo.

Daniel begins by reviewing some of the basic consequences of $\mathsf{AD}$, the axiom of determinacy. To state $\mathsf{AD}$, we need to consider certain ideal games between two players, that I will just call $I$ and $II$. In all these games the format is the same: players $I$ and $II$ alternate with $I$ playing first. In each turn, the corresponding player picks a natural number, repetitions being allowed, and both players having knowledge of all the moves both have made previously. They play for infinitely many rounds. At the end, a sequence of natural numbers $\langle n_0,n_1,n_2,\dots\rangle$ has been produced, with $n_0,n_2,n_4,\dots$ being the numbers picked by player $I$, and $n_1,n_3,\dots$ being the ones picked by $II$. We have one of these games for each set $A\subseteq\mathbb N^{\mathbb N}$, where $\mathbb N^{\mathbb N}$ is the set of all infinite sequences of natural numbers. In the game associated to such an $A$, player $I$ wins iff the sequence $\langle n_0,n_1,n_2,\dots\rangle$ is in $A$. Otherwise, player $II$ is the winner.

A strategy $\sigma$ for player $I$ is a function that tells player $I$ what to play each time. Formally, this is just a function from the set of finite sequences of numbers to $\mathbb N$. A strategy for $II$ is defined similarly. We say that a strategy for $I$ is winning if and only if player $I$ wins the game whenever they play following the strategy. That is, in any such game we have  $n_0=\sigma(\langle\rangle), n_2=\sigma(\langle n_1\rangle), n_4=\sigma(\langle n_1,n_3\rangle)$, etc, and at the end we have that $\langle n_0,n_1,n_2,\dots\rangle\in A$. Winning strategies for player $II$ are defined similarly. We say that $A$ is determined iff one of the players has a winning strategy.

It is easy to give examples of determined sets $A$. Using the axiom of choice, we can give examples of undetermined sets, but deep theorems in descriptive set theory indicate that no undetermined set can be particularly simple. For instance, it is a celebrated theorem of Martin that all Borel sets are determined. Here, $\mathbb N^{\mathbb N}$ is made into a topological space by taking the product topology of countably many copies of the discrete set $\mathbb N$.

The axiom of determinacy is the statement that all $A\subseteq \mathbb N^{\mathbb N}$ are determined. In particular, this statement contradicts the axiom of choice. See here for slides of a talk I gave a few years ago containing a quick introduction to the subject.

The short remainder of this post (after the fold) is by necessity more technical.

## 287 Communication in the mathematical sciences – Syllabus

August 24, 2014

Instructor: Andrés E. Caicedo
Fall 2014

Time: MW 1:30-2:45 pm.
Place: Mathematics building, Room 139.

This class satisfies the CID requirement in mathematics.

Contact Information

• Office: 239-A Mathematics building.
• Phone number: (208)-426-1116. (Not very efficient.)
• Office Hours: W 12:00-1:15 pm. (Or by appointment.)
• Email: caicedo@math.boisestate.edu

Textbook

MR1443666
Mount Holyoke College
Laboratories in mathematical experimentation. A bridge to higher mathematics.
Textbooks in Mathematical Sciences. Springer-Verlag, New York, 1997. xx+278 pp.
ISBN: 0-387-94922-4.

Contents (Thanks to Samuel Coskey for suggestions)

The course decription at the Department’s site describes this course as follows:

Integrates mathematics content with the opportunity to develop proof writing and communication skills important in the mathematical sciences. Content is drawn from discrete and foundational math and elementary analysis. Introduction to and engagement with written and verbal communication practices characteristic to mathematical sciences. Introduction to and use of technologies that support communication in the mathematical sciences.

Following up on the theme introduced in Math 187, this course will help you transition from computational to proof-based mathematics. Skill at computations is of course still essential. However, in deeper mathematical study, we ask broader questions, and require that the answers be justified by proofs. In this course we will practice proving theorems, and along the way we will participate in the whole mathematical process.

A key realization is that, in actual mathematical research, we are not told what statement to prove, but must instead ask good questions and investigate them methodically. Even if there is a statement we are interested in, we typically do not know whether it is indeed true. In this class we will get practical experience in discovery, conjecture, and exposition of mathematical truth. We will learn to gather data to inform our conjectures, usually with the aid of a computer program. In the process we will learn to distinguish between evidence and proof, and to use both in support of our statements.

In most classes, you are expected to work individually and you are assessed in a timed environment. In a CID course such as this one, we focus instead on the activities that actually take place in the discipline: collaboration with our peers, writing research papers, attending and giving talks, and so on.  Accordingly, we will take a look at some of the key technologies that mathematicians use to carry out and share their work. There are many mathematical programming languages, but we will use Sage. We will typeset papers in LaTeX, and presentations in Beamer.

LaTeX

Please visit Sharelatex, where you can start practicing right away and work in groups. LaTeX has been the primary tool for the dissemination of mathematics (and many other sciences, take a look at the ArXiv to get an idea of how widely used the program is), and it has been so for almost 35 years, even though it has changed very little in that time. It is important to master the LaTeX system, since the language it provides for expressing mathematics will certainly be the standard for many years to come. MathJax and other technologies are expected to eventually replace LaTeX as the standard, but for the time being, knowing it is essential. for instance, Scott Aaronson lists as the first of his Ten Signs a Claimed Mathematical Breakthrough is Wrong that the authors do not use (La)TeX.

LaTeX is available as free software, and abundant documentation exists. A few useful references are The (not so) short introduction to LaTeX, the NASA guide to LaTeX commands, and The comprehensive LaTeX symbol list. I recommend that you also bookmark and visit frequently the Q&A site on Stack Exchange.

Beamer

Beamer is a LaTeX documentclass designed especially for creating powerpoint-style presentations. The output is a pdf file that you can click through page-by-page while you speak. In the code, you simply enclose each slide within \begin{frame}{My Title} and \end{frame}. Inside, simply use the LaTeX commands that you are used to. The ultimate Beamer reference is the full user guide, but this may contain too much information, and to get started the less intimidating short intro (and its source) may be more useful.

Sage and python

Sage is a computer algebra system and programming language. In the long term, knowing it may be just as important and useful if not more than knowing LaTeX. While you are welcome to use any language you prefer, I recommend Sage because it is free to use online, and has been developed as open source software from the beginning. The recently developed Sage Cloud makes its use even more convenient.

When getting started, check out the guided tour or the introductory video series. There is also a nice web site for the nuts and bolts of Python at learnpython. You can start using Sage very quickly by logging into sagenb.org using your google mail account.

All of your time in-class will be devoted to our research projects. Come to class every day prepared to listen, think, and discuss mathematics with your group. Buy a notebook just for Math 287 and keep careful track of your thoughts and data. Assign yourself homework each class so you can bring new ideas to the next meeting. Since attendance and participation are crucial to your success, absences will lower your grade rapidly. [Documented absences are always excused, and you may have one or two free unexplained absences.]

At the end of each unit you will turn in a written article called a “lab report” that summarizes and explains the results of your activities. The majority of your grade will be based on these reports, each of which receives a letter grade.

You will also be asked to give several (typically very short) presentations on your work, individually or as a group. In total, these will be worth about the same as a lab report.

Further details will be provided once we get started.

Additional information will be posted in this blog, and students are encouraged to use the comments feature. Please use full names, which will simplify my life filtering spam out.

## 414/514 Advanced Analysis (Analysis I) – Syllabus

August 20, 2014

Instructor: Andrés E. Caicedo
Fall 2014

Time: MWF 10:30-11:45 am.
Place: Mathematics building, Room 139.

Contact Information

• Office: 239-A Mathematics building.
• Phone number: (208)-426-1116. (Not very efficient.)
• Office Hours: W 12:00-1:15 pm. (Or by appointment.)
• Email: caicedo@math.boisestate.edu

Text
We will use three textbooks and complement with papers and handouts for topics not covered there.

• MR1886084 (2003e:00005).
Pugh, Charles Chapman
Real mathematical analysis.
Undergraduate Texts in Mathematics. Springer-Verlag, New York, 2002. xii+437 pp.
ISBN: 0-387-95297-7.
• MR0655599 (83j:26001).
van Rooij, A. C. M.; Schikhof, W. H.
A second course on real functions.
Cambridge University Press, Cambridge-New York, 1982. xiii+200 pp.
ISBN: 0-521-23944-3; 0-521-28361-2.
Gelbaum, Bernard R.; Olmsted, John M. H.
Counterexamples in analysis.
Corrected reprint of the second (1965) edition. Dover Publications, Inc., Mineola, NY, 2003. xxiv+195 pp.
ISBN: 0-486-42875-3.

The book by van Rooij and Schikhof will be our primary reference, supplemented naturally by the Counterexamples book.  The book assumes some knowledge beyond what is covered in our undergraduate course Math 314: Foundations of Analysis, and does not cover the theory in dimension $n>1$; for these topics, we will follow Pugh’s text.

Contents
Math 414/514 covers Analysis on Euclidean spaces (${\mathbb R}^n$) with emphasis on the theory in dimension one. The approach is theoretical, as opposed to the more computational approach of calculus, and a certain degree of mathematical maturity is required. The course is cross-listed, and accordingly the level will be aimed at beginning graduate students.

From the Course Description on the Department’s site:

Introduction to fundamental elements of analysis on Euclidean spaces including the basic differential and integral calculus. Topics include: infinite series, sequences and series of function, uniform convergence, theory of integration, implicit function theorem and applications.

Here is a short list of topics we expect to cover. This list may change based on students’ interest:

1. Set theoretic preliminaries.
• Cantor’s approach to infinite cardinalities. Countable vs. uncountable sets. Sets of size continuum. The Bernstein-Cantor-Schröder theorem.
• The axiom of choice. Zorn’s lemma. Countable and dependent choice.
• Transfinite recursion. The first uncountable ordinal $\omega_1$.
2. Axiomatization and construction of the set of reals.
• The least upper bound property; uniqueness of $\mathbb R$ up to isomorphism.
• Dedekind cuts, and complete orders.
• Metric spaces, and Cauchy completions. Banach contraction mapping theorem.
3. Topology on $\mathbb R$.
• Open and closed sets. Compact sets and Cantor sets. Baire space.
• Borel sets. Analytic sets.
• Notions of smallness.
• Meagerness and the Baire category theorem. The Baire-Cantor stationary principle.
• Sets of Jordan content zero and of measure zero.
• Introduction to the theory of strong measure zero sets.
4. Continuity.
• Sets of discontinuity of functions.
• Monotonicity. Functions of bounded variation.
5. Differentiability.
• The problem of characterizing derivatives. Baire class one functions. The intermediate value property. Sets of continuity of derivatives.
• The mean value theorem. L’Hôpital’s rule.
• The dynamics of Newton’s method.
• The Baire hierarchy of functions.
• Continuous nowhere differentiable functions.
6. Power series.
• Real analytic functions. Taylor series.
• $C^\infty$ functions. Zahorsky’s characterization of the sets of points where a $C^\infty$ function fails to be analytic.
7. Integration.
• Riemann integration. Lebesgue’s characterization of Riemann integrability.
• Weierstraß approximation theorem.
• Lebesgue integration. The fundamental theorem of calculus.
• The Henstock-Kurzweil integral. Denjoy’s approach to reconstructing primitives.
8. Introduction to multivariable calculus.
• (Frechet) derivatives.
• The inverse and implicit function theorems.

Based on homework. No late homework is allowed. Collaboration is encouraged, although students must turn in their own version of the solutions, and give credit to books/websites/… that were consulted and people with whom the problems where discussed.

There will be no exams. However, an important component of being proficient in mathematics is a certain amount of mental agility in recalling notions and basic arguments. I plan to assess these by requesting oral presentations of solutions to some of the homework problems throughout the term. If I find the students lacking here, it will be necessary to have an exam or two. The final exam is currently scheduled for Wednesday, December 17, 2014, 12:00-2:00 pm.

Additional information will be posted in this blog, and students are encouraged to use the comments feature. Please use full names, which will simplify my life filtering spam out.

## 403 – HW 4 – Algebraic graph theory

April 28, 2014

This set is due Thursday, May 15, at 11am.

1. (Norman Biggs, Algebraic Graph Theory, Proposition 2.3) Let $\Gamma$ be a (finite, simple) graph with $n$ vertices, and let $A$ be its adjacency matrix. Suppose that the characteristic polynomial of $A$ is ${\rm char}_A(x)=x^n+c_1x^{n-1}+c_2x^{n-2}+\dots+c_n$. Verify that:

1. $c_1=0$.
2. $-c_2$ is the number of edges in $\Gamma$.
3. $-c_3$ is twice the number of triangles in $\Gamma$.

2.  (Chris Godsil, Gordon F. Royle, Algebraic Graph Theory, Theorem 8.5.1) Let $G$ be a finite simple graph on $n$ vertices that is $k$-regular, that is, such that each vertex has degree $k$. Suppose $G$ has $n$ vertices and $A$ is its adjacency matrix.

1. Verify that $k$ is an eigenvalue of $A$.
2. Let $\lambda_2,\dots,\lambda_n$ be the other eigenvalues of $A$. Let $\bar G$ denote the complementary graph to $G$: This graph has the same vertices, but two vertices are joined by an edge iff they are not joined by an edge in $G$. Let $B$ be the adjacency matrix of $\bar G$. Show that there is a basis of $\mathbb R^n$ consisting of vectors that are common eigenvectors of $B$ and $A$, and that the eigenvalues of $B$ are $n-k-1,-1-\lambda_2,\dots,-1-\lambda_n$.

## 403 – HW 3 – Computing eigenvalues

April 16, 2014

This set is due Thursday, May 8, at the beginning of lecture. (There will be another homework set, due the scheduled day of the final exam, Thursday May 15, at 11am, so I recommend you try to complete this set earlier than the scheduled deadline.)

You can work on your own, or in groups of up to three members. In case you cannot find anybody to work with, and do not know how to program, let me know as soon as possible, and we will find an alternative. As usual, you can still collaborate with others not in your group, but please make sure to give appropriate credit and indicate clearly who you worked with, what references you consulted, etc.

1. Give an example of a matrix for which the power method fails. (Include a proof that this is indeed the case.)

2. Write a program that, given a square matrix $A$ (diagonalizable and) with real entries, computes approximations to its eigenvalues using the $QR$-algorithm. Ideally, the user can decide the dimensions of the matrix and, more importantly, the (tolerance) error within which the approximations will be found. Apply your method to a $10\times 10$ symmetric matrix, and check the number of iterations the process requires, as a function of the tolerance error.

Please turn in: The code (best if you email it to me), a write up explaining what your code does, the matrix you applied the method to, and the result. To help verify that your algorithm is proceeding correctly, at each step of the iteration have your program indicate clearly what the matrices $Q$ and $R$ are, and what the new (output) matrix is.

Please make the algorithm as explicit as possible. Meaning: Do not use shortcuts already built into the software; most CASs already have functions that perform the Gram-Schmidt process to a given set of vectors, or functions that give the $QR$ decomposition of a matrix. Instead, I want you to program these subroutines as well.

The programming language you use is up to you. Maple, Mathlab, Sage are standard choices, but if you prefer a different language, it should be fine. Let me know, just in case.

3. Do the same, but now for Francis’s algorithm. Apply it to the same $10\times 10$ matrix. (Here there are more matrices and some vectors the algorithm may want to display along the way. For instance, whenever a matrix is put into upper Heissenberg form, indicate what the reflectors used along the way are.)

## 314 – C+C=[0,2]

April 1, 2014

Recall that the Cantor set $C$ is defined as the intersection $\bigcap_n C_n$ where

$C_0=[0,1]$

and $C_{n+1}$ is obtained by removing from each closed interval that makes up $C_n$ its open middle third, so

$C_1=[0,1/3]\cup[2/3,1]$,

$C_2=[0,1/9]\cup[2/9,1/3]\cup[2/3,7/9]\cup[8/9,1]$,

etc. Each $C_n$ is the union of $2^n$ closed intervals, each of length $1/3^n$.

Let’s prove that $C+C=\{x+y\mid x,y\in C\}$ is the interval ${}[0,2]$. (Cf. Abbott, Understanding analysis, Exercise 3.3.6.)

1.

The usual proof consists in showing inductively that $C_n+C_n=[0,2]$ for all $n$. This is easy: Note first that

$\displaystyle C_{n+1}=\frac13 C_n+\left(\frac13C_n+\frac23\right)$,

where

$\displaystyle \frac13C_n=\left\{\frac x3\mid x\in C_n\right\}$

and

$\displaystyle\frac13 C_n+\frac23=\left\{\frac{x+2}3\mid x\in C_n\right\}$.

This equality is verified by induction. Using this, we can use induction again to verify that, indeed, $C_n+C_n=[0,2]$ for all $n$.

We clearly have that $C+C\subseteq \bigcap_n C_n+C_n=[0,2]$. To prove the converse, for each $z\in[0,2]$ and each $n$, pick $x_n,y_n\in C_n$ such that $x_n+y_n=z_n$. The sequence of $x_n$ is bounded, so it has a convergent subsequence $x_{n_k}$. The corresponding subsequence $y_{n_k}$ has itself a convergent subsequence $y_{n_{k_m}}$. One argues that their limit values $x,y$ belong to $C$, because they belong to each $C_n$, since these sets are nested and closed. Finally, it follows immediately that $x+y=z$ as well.

2.

A very elegant different argument is obtained by using an alternative characterization of $C$: Note that each $x\in[0,1]$ can be written in base three as

$\displaystyle x=0.x_1x_2x_3\dots=\sum_{n=1}^\infty\frac {x_n}{3^n}$

where each $x_i$ is $0$, $1$, or $2$. By induction, one easily verifies that $x\in C_n$ iff it admits such an expansion with $x_n\ne1$. It follows that $x\in C$ iff it admits an expansion where no $x_i$ is $1$.

Given $z\in[0,2]$, we have $z/2\in[0,1]$, so we can write $z/2=a+b$ where the ternary expansion of $a$ has only $0$s and $2$s (so $a\in C$), and the expansion of $b$ has only $0$s and $1$s: If

$z/2=0.t_1t_2\dots$,

we can set $a=0.a_1a_2\dots$ where $a_i=0$ unless $t_i=2$, in which case $a_i=2$ as well, and similarly $b=0.b_1b_2\dots$ where $b_i=0$ unless $t_i=1$, in which case $b_i=1$ as well.

We then have that $z=2(a+b)=(a+2b)+a$, and both $a+2b$ and $a$ are in $C$.

This construction has the further advantage of making clear that the typical $z$ admits continuum many ($=|\mathbb R|$) representations as sum of two members of $C$: If we can split $b=c+d$ (where the expansions of $c,d$ only have $0$s and $1$s), we can set

$z=(a+2c)+(a+2d)$.

This gives us as many representations as subsets of $\{n\in\mathbb N\mid b_n=1\}$.

3.

The related problem of describing $C\cdot C$ appears to be much more complicated. See here and here.

## BEST 2014, First Announcement

March 26, 2014

BOISE EXTRAVAGANZA IN SET THEORY (BEST)
http://diamond.boisestate.edu/~best/
June 18 – 20, 2014
University of California, Riverside

The 21-st meeting of BEST will be hosted at University of California, Riverside, as a symposium of the 95th annual meeting of the American Association for the Advancement of Science – Pacific Division (AAAS-PD). Contributed and invited talks at BEST will be held on Wednesday, Thursday and Friday.

In addition to four invited speakers, the conference program has reserved speaking slots for students, post docs and pre-tenure tenure track faculty. NSF supported funding to assist up to eight student speakers, four post-doc speakers and two pre-tenure tenure track faculty speakers is available. For details on applying to the BEST program committee for these, please visit the conference website. In addition, the AAAS-PD provides up to \$150 in travel funding for students. Please see the BEST conference website for more details on these also. There are a number of deadlines associated with applications for a travel grant.

The four invited speakers at BEST 2014 are:

Dr. Joel Hamkins, CUNY Graduate Center
Dr. Dima Sinapova, University of Illinois at Chicago
Dr. Nam Trang, Carnegie Mellon University
Dr. Andrew Marks, Caltech University

Special features of BEST 2014 include:

The BEST 2014 conference marks the end of three weeks of intensive set theory meetings in California. In addition to the BEST symposium there are several other symposia and workshops of interest offered at the AAAS-PD annual meeting. On Thursday, June 19, the AAAS-PD hosts a banquet at which awards of excellence are given to student speakers selected by a panel of judges. On Friday, June 20, Dr. Joel Hamkins will deliver a plenary lecture from 12:15 to 1:15 about our field to the general audience of the AAAS-PD annual meeting.