| Date | Speaker | Title | Abstract |
| 09/06/2006 | Prof. Robin Thomas | Some of my favorite coloring problems | I will discuss several problems in the area of graph coloring, especially coloring graphs on surfaces and problems that could be attacked using algebraic methods. Some are undoubtedly hard, but I will focus on those that are of current interest and where I think decent progress can be made. |
| 09/13/2006 | Prof. Doron Lubinsky | Orthogonal Polynomials and their Relatives | We discuss some features of orthogonal polynomials and their cousins (such as biorthogonal and Muntz orthogonal polynomials). Some applications will also be briefly outlined. |
| 09/20/2006 | Prof. Tom Trotter | Proving and Disproving | The famous "1/3 - 2/3" conjecture of Kislytsin asserts that if P is a poset which is not a chain, then there is an incomparable pair (x,y) for which the fraction of linear extensions in which x is less than y is at least 1/3 and at most 2/3 of the total number of linear extensions. This conjecture was first posed in 1966 and remains open today. In this talk, we sketch proofs of three interesting partial results on this conjecture: 1. The conjecture holds for any poset which has width 2 (size of largest antichain is two; alternatively, union of two chains). 2. The conjecture holds for any poset which is a semiorder (unit length intervals with [a, a+1] < [ b, b+1] when a+1 < b in R). 3. There is a width 2 semiorder for which the conjecture fails. |
| 09/27/2006 | Prof. Michael Loss | Sharp constants in conformaly invariant inequalities. | I will talk about a integral inequality that goes back to Hardy, Littlewood and Sobolev. The sharp constant for this inequality was calculated by Lieb in 1983 overcoming substantial difficulties. In this talk I'll give a simple geometric prove due to Carlen and myself that yields also the sharp constant. |
| 10/04/2006 | Prof. Ernie Croot | Arithmetic Progressions in Subsets of Finite Abelian Groups. | Suppose that G is a finite abelian group. Denote by r_3(G) the size of the largest subset of G having no three-term arithmetic progressions, which are triples n,n+d,n+2d. What can one say about r_3(G) ? In this talk I will survey some of the known and unknown problems related to this basic question. |
| 10/11/2006 | Prof. William Green | The Weierstrass Uniform Approximation Theorem and its History. | In 1885 Weierstrass proved that every continuous function on a closed bounded interval is uniformly approximable there by polynomials. Over the nest few decades, many other mathematicians gave additional illuminating proofs of this useful result. In the late 1930's, Marshall Stone discovered an important extension of Weierstrass's result to continuous functions on compact Hausdorff spaces, and by 1914 C. Muntz had already published a generalization that took the result in yet a different direction. We give a brief introduction to these results and to the kinds of arguments that have been used to establish them. |
| 10/18/2006 | William McClain | Basic Fourier Analysis in Additive Combinatorics | I'll show some problem solving techniques in Additive Combinatorics that are derived from Fourier Analysis. I'll offer different perspectives on interpreting the problem solving techniques and present them in the most elementary way possible. First we'll see a basic proof of Roth's theorem addressing 3-term Arithmetic Progressions in subsets of the integers. Then we'll see the complications and adaptations of the problem solving techniques used to tackle problems concerning 4-term arithmetic progressions in subsets of the integers. |
| 10/25/2006 | Prof. Michael Lacey | Small Ball Inequalities | This talk will combine some of our favorite topics: combinatorics, orthogonality, analysis, probabilistic reasoning. We will work with Haar functions in dimensions 2 and 3, Such functions are supported on dyadic rectangles. The objects of interest are sums of such functions where the volume of the rectangles is held fixed---the `hyperbolic' assumption. And we seek a lower bound on the sup norm of the such sums. It is easy to establish such a lower bound on average, and the conjecture is that the average case estimate by a factor that is the square root log of the volume. This is true in dimension 2--a theorem of Talagrand--we will recall a short mysterious proof due to Temlyakov. The case of three dimensions is much harder, as will be explained. There are partial results due to Jozef Beck and the speaker and Dmitry Bilyk. These questions arise in the setting of probability, approximation theory, and number theory. |
| 11/01/2006 | Prof. John Etnyre | TBA | TBA |
| 11/08/2006 | Prof. Howie Weiss | Density Dependent Population Models: dynamical systems and real-life modeling. | TBA |
| 11/15/2006 | Prof. Yingfey Yi | TBA | TBA |
| 11/22/2006 | Thanksgiving | ||