Georgia Institute of Technology School of Mathematics | Georgia Institute of Technology | Atlanta, GA

We introduce the averages $K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) f(x+n)$, where $d(n)$ denotes the divisor function and $D(N) = \sum _{n=1} ^N d(n) $. We shall see that these averages satisfy a uniform, scale free, $\ell^p$-improving estimate for $p \in (1,2)$, that is

$$ \Bigl( \frac{1}{N} \sum |K_Nf|^{p'} \Bigl)^{1/p'}  \leq C  \Bigl(\frac{1}{N} \sum |f|^p \Bigl)^{1/p} $$

as long as $f$ is supported on the interval $[0,N]$.

We will also see that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse bounds for $p \in (1,2)$, which implies that $K^*$ is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.

The seminar will be held on Zoom, and can be accessed by the link

https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09

We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning.  By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved.  Connections to geometric analysis, stochastic processes, and deep learning are emphasized.  The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points.  Applications to image processing and bioinformatics will be shown, demonstrating state-of-the-art empirical performance.  Extensions to active learning, generative modeling, and computational geometry will be discussed.

The classical Livshits theorem characterizes coboundaries over a transitive Anosov flow as precisely those functions which integrate to zero over all periodic orbits of the flow. I will present a variant of this theorem which uses a weaker assumption and gives a weaker conclusion that the function is an ``abelian coboundary.” Such weaker version corresponds to studying the cohomological equation on infinite abelian covers e.g., for geodesic flows on abelian covers of hyperbolic surfaces. I will also discuss a connection to the marked length spectrum rigidity of Riemannian metrics. Joint work with Federico Rodriguez Hertz.

Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, patterns can be assigned to a spectrum. On one extreme, patterns are completely described by algebraic systems where each individual pattern is obtained by repeatedly applying simple operations on primitive elements. On the other extreme, patterns are perceived as visual or frequency regularities without any prior knowledge of the underlying mechanisms. In this presentation, we aim at demonstrating some mathematical techniques for representing patterns traversing the aforementioned spectrum, which leads to qualitative analysis of the patterns’ properties and quantitative prediction of the modeled behaviors from various perspectives. We investigate lattice patterns from material science, shape patterns from computer graphics, submanifold patterns encountered in point cloud processing, color perception patterns applied in underwater image processing, dynamic patterns from spatial-temporal data, and low-rank patterns exploited in medical image reconstruction. For different patterns and based on their dependence on structured or unstructured data, we introduce suitable mathematical representations using techniques ranging from group theory to deep neural networks.

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