Date |
Speaker |
Title & Abstract |
Host |
| Oct. 29 |
Greg Blekherman
(Virginia Tech) |
Title:
Nonnegative Polynomials and Sums of Squares: Real Algebra meets Convex Geometry
Abstract. A multivariate real polynomial is non-negative if its value is at least zero for all points in $\mathbb{R}^n$. Obvious examples of non-negative polynomials are squares and sums of squares. What is the relationship between non-negative polynomials and sums of squares? I will review the history of this question, beginning with Hilbert's groundbreaking paper and Hilbert's 17th problem. I will discuss why this question is still relevant today, for computational reasons, among others. I will then discuss my own research which looks at this problem from the point of view of convex geometry. I will show how to prove that there exist non-negative polynomials that are not sums of squares via "naive" dimension counting. I will discuss the quantitative relationship between non-negative polynomials and sums of squares and also show that there exist convex polynomials that are not sums of squares. |
Bill Helton & Jiawang Nie |
| Nov. 19 |
Prof. Herbert Heyer
(Univ. Tuebingen, Germany) |
Title:
Hypergroup stationarity of random fields
Abstract. Traditionally weak stationarity of a random field $\{X(t) : t\in \TTT\}$ over an index space $\TTT$ is defined with respect to a translation operation in $\TTT$. But this classical notion of stationarity does not extend to related random fields, as for example to the field of averages of $\{X(t): t\in \TTT\}$. In order to equip this latter field with a stationarity property one introduces a generalized translation in $\TTT$ which arises from a generalized convolution structure in the space $M^b(\TTT)$ of bounded measures on $\TTT$. There are two fundamental constructions providing such (hypergroup) convolution structures on the index spaces $\ZZZ_+$ and $\RRR_+$, in terms of polynomial sequences and families of special functions, respectively. In the present talk emphasis will be put on polynomially stationary random fields $\{X(n): n\in\ZZZ_+\}$ which were studied for the first time by R.~Lasser and M.~Leitner about 20 years ago. In the meantime the theory has developed interesting applications such as regularization, moving averages and prediction. For square-integrable radial random fields over graphs, J.P.~Arnaud has coined a notion of stationarity which yields spectral and Karhunen type representations. These fields are related to polynomially stationary random fields over $\ZZZ_+$, where the underlying polynomial sequence generates the Cartier-Dunau convolution structure in $M^b(\ZZZ_+)$. An analogous approach related to special function stationarity of random fields over $\RRR_+$ seems promising, but requires further progress. |
Patrick Fitzsimmons |
| Nov. 26 | No colloquium. Thanksgiving Holiday. | ||
Date |
Speaker |
Title & Abstract |
Host |
| Jan. 7 |
Prof. Rahul Pandharipande
(Princeton) |
Mark Gross | |
| Feb. 4 |
Prof. Michael Overton
(Courant, NYU) |
Philip Gill, Bill Helton, & Jiawang Nie | |
| Feb. 11 |
Prof. Richard Palais
(UC Irvine) |
Lei Ni & Nolan Wallach | |
| Feb. 25 |
Prof. Xiaofeng Shao
(UIUC) |
Dimitris Politis | |
| March 11 |
Prof. Ezra Getzler
(Northwestern Univ.) |
Ben Weinkove |
Date |
Speaker |
Title & Abstract |
Host |
| April 15 |
Prof.
Steve Zelditch
(Northwestern Univ.) |
Lei Ni |