## 403/503- Homework 1

January 27, 2011

Recall that we defined Nim addition $\oplus$ and Nim multiplication $\otimes$ on the natural numbers ${\mathbb N}=\{0,1,\dots\}$ by setting:

1. $n\oplus m$ is the result of writing $n,m$ in binary and adding without carrying.
2. $\otimes$ is the unique binary operation on ${\mathbb N}$ that is commutative, associative, distributive over $\oplus$ and satisfies:
• $\displaystyle F_n\otimes F_n=\frac32 F_n$ and
• $\displaystyle F_n\otimes m=F_nm$ whenever $m< F_n$. Here, $F_n=2^{2^n}$ for all $n\in{\mathbb N}$, call these numbers “Fermat’s 2-powers”.

The first problem is that it is not quite clear that $\otimes$ is even well-defined (i.e., are there any functions at all that behave as required of $\otimes$? Is there really only one such function?). Begin by checking this, by showing that the rules above give us that $n\otimes m$ is the result of writing $n,m$ as sums of (multiplications by powers of 2) of Fermat 2-powers, and expanding using associativity, distributivity, and the two rules about how to multiply with Fermat 2-powers. (Verify that any $n$ can indeed be written as such a sum in a unique way, and that this indeed shows that $\otimes$ is completely characterized by our description.)

Show that if we set ${\mathbb F}_{2^{2^n}}=\{0,1,\dots,F_n-1\}$, then ${\mathbb N}$ and each ${\mathbb F}_{2^{2^n}}$ are fields (over ${\mathbb Z}_2$, with addition and multiplication given by $\otimes,\oplus$.

In lecture I erroneously mentioned that ${\mathbb N}$ is algebraically closed. This is not the case. For example, show that the equation $x^3+x+1=0$ has no solutions in ${\mathbb N}$ when we interpret “ $n$ is a solution” to mean that $(n\otimes n\otimes n)\oplus n\oplus 1=0$.

Nim addition and multiplication were introduced by John Conway. A bit of online searching will give you references for this exercise, but please abstain from looking for them. I will provide references and some additional details once the homework has been turned in.

This set is due February 11 at the beginning of lecture.

## 507- Problem list (IV)

January 21, 2011

For Part III, see here.

(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list. If you have corrections/updates, please email me. Sorry for the delay with posting this.)

• Is there a dense subset of ${\mathbb R}^2$ with all pairwise distances rational?
• Is every polygonal region illuminable?
• Does the odd greedy expansion for Egyptian fractions terminate?
• Erdős conjecture on arithmetic progressions.
• Is ${\mathbb Z}$ existentially definable in ${\mathbb Q}$? (And similar extensions of Hilbert’s tenth problem. See also this question on MathOverflow.)
• Is any/none algebraic irrational real-time computable?
• Do perfect boxes exist?
• The lonely runner conjecture. (See also these posts by R. Lipton: 1, 2.)
• Hadwiger conjecture on convex bodies.
• What is the maximum number of points that can be placed in an $n\times n$ grid so that no three of them are collinear?
• Can we characterize Euclidean Ramsey sets?
• The Riemann hypothesis.

## Very nice news from Google

January 21, 2011

See here:

Maths is very important to Google. It’s the basis of everything we do: from the algorithms that deliver answers to your search queries, to the way in which your Gmails are grouped in conversations, to the technology advances which are enabling us to develop driverless cars.

[…]

And so yesterday, together with the Advisory Board of the International Mathematical Olympiad, we were proud to announce that we are making a gift of one million euros to the organisation to help cover the costs of the next five global events (2011-15).

## 507- Problem list (III)

January 20, 2011

For Part II, see here.

(Many thanks to Robert Balmer, Nick Davidson, and Amy Griffin for help with this list.)

• The Erdös-Turán conjecture on additive bases of order 2.
• If $R(n)$ is the $n$-th Ramsey number, does $\lim_{n\to\infty}R(n)^{1/n}$ exist?
• Hindman’s problem: Is it the case that for every ﬁnite coloring of the positive integers, there are $x$ and $y$ such that $x$, $y$, $x + y$, and $xy$ are all of the same color?
• Does the polynomial Hirsch conjecture hold?
• Does $P=NP$? (See also this post (in Spanish) by Javier Moreno.)
• Mahler’s conjecture on convex bodies.
• Nathanson’s conjecture: Is it true that ${}|A+A|\le|A-A|$ for “almost all” finite sets of integers $A$?
• The (bounded) Burnside’s problem: For which $m,n$ is the free group $B(m,n)$ finite?
• Is the frequency of 1s in the Kolakoski sequence asymptotically equal to $1/2$? (And related problems.)
• A question on Narayana numbers: Find a combinatorial interpretation of identity 6.C7(d) in Stanley’s “Catalan addendum” to Enumerative combinatorics.

January 20, 2011

Instructor: Andrés E. Caicedo.
Contact Information: See here.
Time: MWF 11:40 am – 12:30 pm.
Place: Mathematics/Geosciences building, Room 124.
Office Hours: MF 10:40-11:30 am.
Text: Axler, Sheldon. Linear algebra done right. Springer, 2nd edition (1997).

Contents: Math 403/503 is intended to be a second course in linear algebra, where an abstract approach emphasizing the role of linear transformations is preferred to a more computational approach based on properties of matrices. From Course Description in the Department’s site:

Concepts of linear algebra from a theoretical perspective. Topics include vector spaces and linear maps, dual vector spaces and quotient spaces, eigenvalues and eigenvectors, diagonalization, inner product spaces, adjoint transformations, orthogonal and unitary transformations, Jordan normal form.

Grading: Based on homework. No late homework is allowed. Collaboration is encouraged, although you must turn in your own version of the solutions, and give credit to books/websites/… you consulted and people you talked/emailed/… to.

## Set theory and its applications

January 15, 2011

Set theory and its applicationsL. Babinkostova, A. E. Caicedo, S. GeschkeM. Scheepers, eds. Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011. ISBN-10: 0-8218-4812-7 ISBN-13: 978-0-8218-4812-8

The Boise Extravaganza in Set Theory (BEST) started in 1992 as a small, locally funded conference dedicated to Set Theory and its Applications. A number of years after its inception BEST started being funded by the National Science Foundation. Without this funding it would not have been possible to maintain the conference. The conference remained relatively small with many opportunities for its participants to meet informally. We like to think that during these years BEST has made it possible for the numerous set theorists who have participated in it to absorb, besides the new developments featured in the conference talks, also part of the folklore and traditions of the ﬁeld of set theory and its relatives. An explicit effort was made to bring together role models from various career stages in set theory as well as the new generation to support some notion of continuity in the ﬁeld.
This volume has a similar purpose. In it the reader will ﬁnd valuable papers ranging from surveys that put in print here set theoretic knowledge that has been around for several decades as unpublished lore, to hybrid survey-research papers, to pure research papers. Readers can be assured of the authority of each paper since each has been carefully refereed. The reader will also ﬁnd that the subjects treated in these papers range over several of the historically strongly represented areas of set theory and its relatives. Rather than expounding the virtues of each paper individually here, we invite the reader to learn from the authors.
Bringing to publication such a collection of papers is not possible without the generous dedication of authors and referees and the services of a publisher. We would like to thank all authors and referees for their selﬂess contributions to this volume. And we particularly would like to thank the publisher, Contemporary Mathematics, and Christine Thivierge, for the guidance they provided during this process.

January 6, 2011

Today at lunch we got a trifecta:

The effort have the potential to pay off handsomely today

That’s just ok. But then:

People try thing, because they just don’t want it enough.

And finally:

Photographic memory! Remember to put in film, or its digital!?