SPRING 2011
| Jan 28 | 4:00 pm | J. Maurice Rojas, Texas A&M University New multiplier sequences via discriminant amoebae In their classic 1914 paper, Polya and Schur introduced and characterized two types of multiplier sequences (linear operators acting diagonally on the monomial basis of $\mathbb{R}[x]$) sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered by Gelfand, Kapranov, and Zelevinsky, we introduce a new class of sign-independently real-rooted polynomials and describe diagonal operators preserving this new class. A pleasant circumstance in our description is that this class of operators essentially coincides with the class of all log-concave sequences. This is joint work with Mikael Passare and Boris Shapiro, and is about to appear in an issue of the Moscow Mathematical Journal in memory of Vladimir Igorevich Arnold. |
| Feb 8-10 | John McCarthy, Washington University, gives the Frontiers lectures. |
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| Feb 18 | 4:00 pm | Kai Wang, Fudan University Essential normality of the cyclic submodule generated by any polynomial Kunyu Guo and I showed that the closure $[I]$ in the Drury-Arveson space of a homogeneous principal ideal $I$ in $\mathbb{C}[z_1,\dots ,z_n]$ is essentially normal. The proof exploited combinatorial relations between the weights of the shift operators on the space. One can show that this result extends to the Bergman space for the ball $B_n$ and closely related spaces, but thus far no one has been able to extend the result to other ideals in pursuit of a conjecture of Arveson. I will discuss joint work with Ron Douglas in which we use techniques from harmonic analysis involving the radial derivative and the complex tangential derivative to extend the result to the closure of any principal polynomial ideal in the Bergman space. In particular, the commutators and cross-commutators of the restrictions of the multiplication operators are shown to be in the Schatten p-class for p>n. Moreover, the maximal ideal space $X_I$ of the resulting C*-algebra for the quotient module is shown to be contained in $Z(I)\cap B_n$, where $Z(I)$ is the zero variety for $I$, and to contain all points in $B_n$ that are limit points of $Z(I)\cap B_n$. As a consequence, $X_I = Z(I)\cap B_n$ sin the case that $I$ is quasi-homogeneous. |
| Feb 25 | 4:00 pm | Peter Pivovarov, Texas A&M University A probabilistic take on isoperimetric-type inequalities Let $K\subset \mathbb{R}^n$ be a convex body of volume one. Let $X_1 , \dots , X_N$ be independent random vectors distributed uniformly in $K$ and form their convex hull $K_N$. A result of Groemer's states that the expected volume of $K_N$ is smallest when $K$ is the Euclidean ball $B$ of volume one. Similar results hold for other convex sets associated with $K$, e.g., random zonotopes, centroid bodies and their $L_p$ analogs. I will discuss a streamlined approach to inequalities of this type, a key feature being the use of laws of large numbers. This is joint work with Grigoris Paouris. |
| Mar 4 | 4:00 pm |
Joint with the Probability Seminar in Milner 216 Alexandra Rodkina, University of the West Indies Discretized Itô formula and stability of stochastic difference equations [pdf abstract] |
| Mar 11 | 4:00 pm | Florent Baudier, Texas A&M University The Lipschitz extension problem after Nigel J. Kalton As you probably know Nigel Kalton passed away in August 2010. Shortly before his death he was about to submit a series of four amazing papers. Kalton solved a couple of long standing and famous open problems in non-linear Banach space geometry. The purpose of this talk is to describe those problems with a particular emphasis on the Lipschitz extension problem. Last January in Cambridge, Assaf Naor, in an outstanding talk, explained how to extract a quantitative version of Kalton's non extension result. If time permits we will sketch the proof. |
| Mar 24-27 | Group Actions on Measure Spaces workshop. | |
| Apr 1 | 4:00 pm |
Joint with the Banach Space Seminar Ellen Veomett, California State University, East Bay Spaces of small metric cotype Rademacher type and cotype of Banach spaces have played an important role in the geometry of Banach spaces, providing information about when certain kinds of embeddings exist. Recently, Mendel and Naor defined a notion of metric cotype which extends the Rademacher cotype of a Banach space to all metric spaces. In this talk, I will briefly explain my initial interest in type and cotype, which arose in the context of approximations of convex bodies. I will then discuss some recent results from work with K. Wildrick on metric cotype. We show that, while there are no Banach spaces of cotype smaller than $2$, metric spaces which are bi-Lipschitz equivalent to ultrametric spaces have infimal cotype $1$. Moreover, if $s_X$ is the supremal value of $s$ for which a metric space $X$ is an $s$-snowflake space (as defined by Tyson and Wu), then $X$ has infimal cotype $1$ if $s_X > 1$, and cotype at least $2$ if $s_X = 1$. Thus, no metric space has infimal cotype strictly between $1$ and $2$. Finally, I will briefly discuss the scaling function $m(n)$ which appears in the metric cotype inequality. This scaling function can be chosen to grow like $n^{1/q}$ for a $K$-convex Banach space of cotype $q$, but must grow faster if one is to show that an $s$-snowflake has infimal cotype $1$ (assuming $s>1$). This brings up a question of whether a restriction on the scaling function $m(n)$ ought to be included in the definition of metric cotype. |
| Apr 8 | 4:00 pm | Constanze Liaw, Texas A&M University Regularizations of singular integral operators In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal, so the integral formally defining the operator or its bilinear form is not well defined (the integrand is not in $L^1$) even for nice functions. However, since the kernel only has singularities on the diagonal, the bilinear form is well defined say for bounded compactly supported functions with separated supports. One of the standard ways to interpret the boundedness of a singular integral operators is to consider regularized kernels, where the cut-off function is zero in a neighborhood of the origin, so the corresponding regularized operators with kernel are well defined (at least on a dense set). Then one can ask about uniform boundedness of the regularized operators. For the standard regularizations one usually considers truncated operators. The main result of the paper is that for a wide class of singular integral operators (including the classical Calderon-Zygmund operators in non-homogeneous two weight settings), the $L^p$ boundedness of the bilinear form on the compactly supported functions with separated supports (the so-called restricted $L^p$ boundedness) implies the uniform $L^p$ boundedness of regularized operators for any reasonable choice of a smooth cut-off of the kernel. If the kernel satisfies some additional assumptions (which are satisfied for classical singular integral operators like Hilbert Transform, Cauchy Transform, Ahlfors-Beurling Transform, Generalized Riesz Transforms), then the restricted $L^p$ boundedness also implies the uniform $L^p$ boundedness of the classical truncated operators. |
| Apr 15 | 4:00 pm | Plamen Simeonov, University of Houston-Downtown q-blossoming and h-blossoming and applications We introduce two new variants of the standard blossom: the q-blossom and the h-blossom by altering the diagonal property of the standard blossom in two different ways. We use these blossoms to develop recursive evaluation algorithms for q-Bezier and for h-Bezier curves. We show that each of these blossoms satisfies a dual functional property for the corresponding Bezier curves. We obtain several new identities including q- and h-versions of the classical Marsden identity. Using two of the recursive evaluation algorithms, recursive subdivision procedures for q- and h-Bezier curves are constructed. Starting from the original control polygon of a q- (h-)Bezier curve, the subdivision procedure generates a sequence of control polygons that converges rapidly to the original q- (h-)Bezier curve. We show that the homogeneous version of the q- (h-)blossom can be used to compute q- (h-)derivatives of polynomials. Joint work with Ron Goldman (Rice University) and Vasilis Zafiris (UH-Downtown). |
| Apr 29 | 4:00 pm | Ali Kavruk, University of Houston Tensor products of operator systems In this talk we will introduce tensor products on the category of operator systems and examine several examples including the minimal, the maximal, the maximal commuting and some asymmetric tensor products. We refine the notion of nuclearity to this setting by studying the operator systems that preserve various pairs of tensor products. This allows us to describe several properties such as exactness, the weak expectation property (WEP) and the local lifting property (LLP) in terms of "nuclearity". As an application of this theory we discuss some new equivalences of the Kirchberg Conjecture and the Smith-Ward Problem. This presentation is based on joint work with V.I. Paulsen, I.G. Todorov and M. Tomforde and my current research. |
| May 13 | 4:00 pm |
Joint with the Banach Space Seminar Dan Freeman, University of Texas at Austin Weak Grothendieck compactness principles The Grothendieck compactness principle states that every norm compact subset of a Banach space is contained in the closed convex hull of a norm null sequence. We will discuss two different ways in which this principle can be modified when the norm topology of a Banach space is replaced by its weak topology. First, we will prove that every weakly compact subset of a Banach space is contained in the closed convex hull of a weakly null sequence if and only if the Banach space has the Schur property. Second, we will give a characterization of what Banach spaces with symmetric bases have the property that a subset is weakly compact if and only if it is contained in the rearrangement invariant convex hull of a weakly null sequence. This is joint work with P.N. Dowling, C.J. Lennard, E. Odell, B. Randrianantoanina, and B. Turett. |