Colloquium

: In 1921, Landau wrote a letter to Hardy including a proof of the inequality \begin{align*} \sum_{n=0}^{\infty}|\phi(n)-\phi(n+1)|^{p}\geq \left(\frac{p-1}{p}\right)^{p}\sum_{n=1}^{\infty}\frac{|\phi(n)|^{p}}{n^{p}} \end{align*} which holds for all finitely supported $\phi:\N_0\to\R$ such that $\phi(0)=0$ (here $1<p<\infty$ is a fixed number). This inequality was stated before…
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Periods of automorphic forms provides an important tool to classify automorphic representations and arise time and time again in problems with arithmetic origin. Motivated by that, I shall describe some of our investigation of local counterparts of these periods that…
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The free uniform spanning forest (FUSF) of an infinite connected graph G is obtained as the weak limit of uniformly chosen spanning trees of finite subgraphs of G. It is easy to see that the FUSF is supported on spanning…
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In infinite ergodic theory, distributional limits replace the absolutely normalized pointwise ergodic theorem.We’ll review the subject and then see that every random variable on the positive reals occurs as the distributional limit of some infinite ergodic transformation. We’ll consider consequences…
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I will first recall the notions of Schwartz functions and tempered distributions – on the line and then on general real affine algebraic manifolds. Then I will pose a problem on the existence of distributions that are semi-invariant under a…
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We intend to present an update regarding several open problems in Nonlinear Analysis, which have been of recent research interest. These problems concern, for instance, the generic method, infinite products of operators, as well as the asymptotic properties of holomorphic…
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A group G has the Liouville property with respect to some generating set S if the only bounded harmonic functions on the Cayley graph of (G,S) are the constant functions. On such Cayley graphs it is interesting to ask how slowly can…
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A complete system of functions in $L_2({\mathbb R})$, can it be “localized” around a sparse set?

I will give an overview of recent results regarding the behavior of the Poisson bracket under perturbations of the functions relative to different topologies. It turns out that it exhibits both rigidity and flexibility. For instance, if the Poisson bracket…
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Given a map of discrete groups f:G –> H when is this map induced from a normal subgroup inclusion of topological groups X \subseteq Y by taking path components of X and Y?For example, the trivial map on the integers…
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