In 1996, C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any non-zero square integrable function $g$ and subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT Conjecture, and is still largely unresolved.
This talk will detail two recent papers concerning Rogers-Shephard inequalities and Zhang inequalities for various classes of measures, the first of which is a reverse form of the Brunn-Minkowsk inequality, and the second of which can be seen to be a reverse affine isoperimetric inequality; the feature of both inequalities is that they each provide a classification of the n-dimensional simplex in the volume case. The covariogram of a measure plays an essential role in the proofs of each of these inequalities.
Let f be a real-valued Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution.
What is the probability that f remains above a certain fixed line for a long period of time?
We give simple spectral(and almost tight) conditions under which this probability is asymptotically exponential, that is, that the limit of log P(f>a on [0,T])/ T, as T approaches infinity, exists.
In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space.
In this talk, we present an operator theoretic analogue of the F. and M. Riesz Theorem. We first recast the classical theorem in operator theoretic terms. We then establish an analogous result in the context of representations of the Cuntz algebra, highlighting notable differences from the classical setting. At the end, we will discuss some extensions of these ideas. This is joint work with R. Clouâtre and R. Martin.
Zoom Link:
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
This talk is based on a chapter that I wrote for a book in honor of John Benedetto's 80th birthday. Years ago, John wrote a text "Real Variable and Integration", published in 1976. This was not the text that I first learned real analysis from, but it became an important reference for me. A later revision and expansion by John and Wojtek Czaja appeared in 2009. Eventually, I wrote my own real analysis text, aimed at students taking their first course in measure theory. My goal was that each proof was to be both rigorous and enlightening.
Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$.
