## 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.