403/503- Homework 1

Recall that we defined Nim addition and Nim multiplication on the natural numbers by setting:

1. is the result of writing in binary and adding without carrying.
2. is the unique binary operation on that is commutative, associative, distributive over and satisfies:

• and
• whenever . Here, for all , call these numbers “Fermat’s 2-powers”.

The first problem is that it is not quite clear that is even well-defined (i.e., are there any functions at all that behave as required of ? Is there really only one such function?). Begin by checking this, by showing that the rules above give us that is the result of writing 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 can indeed be written as such a sum in a unique way, and that this indeed shows that is completely characterized by our description.)

Show that if we set , then and each are fields (over , with addition and multiplication given by .

In lecture I erroneously mentioned that is algebraically closed. This is not the case. For example, show that the equation has no solutions in when we interpret “ is a solution” to mean that .

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)

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 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 existentially definable in ? (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 grid so that no three of them are collinear?
• Can we characterize Euclidean Ramsey sets?
• The Riemann hypothesis.

507- Problem list (III)

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 is the -th Ramsey number, does exist?
• Hindman’s problem: Is it the case that for every finite coloring of the positive integers, there are and such that , , , and are all of the same color?
• Does the polynomial Hirsch conjecture hold?
• Does ? (See also this post (in Spanish) by Javier Moreno.)
• Mahler’s conjecture on convex bodies.
• Nathanson’s conjecture: Is it true that for “almost all” finite sets of integers ?
• The (bounded) Burnside’s problem: For which is the free group finite?
• Is the frequency of 1s in the Kolakoski sequence asymptotically equal to ? (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.

Part IV.

403/503- Advanced linear algebra

Math 403/503: Advanced linear algebra.

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.

I will use this website to post additional information, and encourage you to use the comments feature. If you leave a comment, please use your full name, which will simplify my life filtering spam out.