Time Zone Notice. All talk times listed below are in US Central Time (America/Chicago). To view a talk in your local time zone, click the “click here to view in your time zone” link under that talk.
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Talk 01 — Zhou Feng (Technion - Israel Institute of Technology, Israel) — [computing…]Local time: click here to view in your time zoneTitle: Dimension of overlapping self-conformal measures on the plane
Abstract
A self-conformal measure $\mu$ on $\mathbb{C}$ arises as the stationary measure of a random walk driven by a holomorphic iterated function system $\Phi$. Understanding the dimensional properties of $\mu$ is a central problem in fractal geometry, especially in the presence of overlaps and nonlinearity. When $\Phi$ preserves $\mathbb{R}$, Rapaport recently made a breakthrough by showing that, if $\Phi$ satisfies a mild Diophantine condition and does not preserve any point, then the dimension of $\mu$ attains its natural upper bound. In the complex case, however, it turns out that additional obstructions may occur. In particular, a nonlinear version of the saturation phenomenon observed by Hochman in the self-similar setting may appear. Nevertheless, we are able to show that the expected dimension equality still holds under the Diophantine condition together with essentially necessary algebraic assumptions. This is joint work with Ariel Rapaport.
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Talk 02 — Tianara Borges (UPenn) — [computing…]Local time: click here to view in your time zoneTitle: Multiple pins in distance graphs
Abstract
Falconer-type problems study Hausdorff-dimension thresholds that force nontrivial geometric structure. A basic example is the pinned distance set: for a compact set $E \subset \mathbb{R}^d$ and a point $y \in E$, define
$\Delta_y(E)=\{|x-y| : x\in E\}.$
Falconer’s conjecture predicts that if $\dim(E) > d/2$, then $\Delta_y(E)$ has positive Lebesgue measure for many $y \in E$.
More generally, one can study distance sets associated with chains, trees, triangles, necklaces, and other finite graphs. For a graph with $n$ vertices and $m$ edges, each $n$-tuple of points in $E$ determines a vector in $\mathbb R^m$ recording the lengths of all edges.
A natural problem is to determine Hausdorff-dimension thresholds on $E$ that ensure this $m$-dimensional configuration set has positive Lebesgue measure.
This talk, based on joint work with Ben Foster, Yumeng Ou, Eyvindur Palsson, and Francisco Romero Acosta, develops a framework for multiply pinned distance sets.
We investigate how the geometry changes when pins are placed at several (nonadjacent) vertices of a graph. In particular, we ask: under what Hausdorff-dimension thresholds on $E$ can one still guarantee positive $m$-dimensional Lebesgue measure for the resulting multiply pinned distance set? How do these thresholds depend on the number of pins and on their placement within the graph?
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Talk 03 — Dima Zakharov (Massachusetts Institute of Technology) — [computing…]Local time: click here to view in your time zoneTitle: Furstenberg and Kakeya problems for sticky sets of tubes.
Abstract
We prove a sticky version of the Furstenberg set estimate in three dimensions, conjectured by Wang and Wu. This conjecture is a common generalization of the Kakeya conjecture in $\mathbb R^3$, proven recently by Wang and Zahl, and of the Furstenberg estimate in the plane, proven by Ren and Wang. As a byproduct, we obtain a new proof of the sticky Kakeya estimate in $\mathbb R^3$ and hope to extend our techniques further.
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Talk 04 — Emily Casey (University of Minnesota) — [computing…]Local time: click here to view in your time zoneTitle: Spherical functions and boundary regularity
Abstract
Since the resolution of Carleson's $\varepsilon^2$-conjecture in the plane by Jaye, Tolsa, and Villa, and in higher dimensions by Fleschler, Tolsa, and Villa, it is known that the tangent points of certain domains are characterized via qualitative control on a ``spherical spectral'' function. In this talk, we will discuss the classes of domains which are characterized via quantitative conditions on this same ``spherical spectral'' function. This talk will include discussions of joint work with Xavier Tolsa and Michele Villa, and with Max Engelstein and Tatiana Toro.
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Talk 05 — Sha Wu (Hunan University, China) — [computing…]Local time: click here to view in your time zoneTitle: Line spectra for the arc-length measure on two segments
Abstract
We study the spectrality of arc-length measures supported on the union of two line segments in the plane. We show that any such spectral measure must admit a line spectrum. Moreover, when the two segments are non-parallel, such spectral measure admits only line spectra. Thus, in this case every spectrum is one dimensional. In addition we show that this property fails for unions of three or more segments in the plane. We construct some arc-length spectral measures supported on the union of at least three line segments such that none of its spectra is contained in a line. Finally, we work in the general framework of arc-length measures supported on finite unions of curves in $\mathbb{R}^d$. We show that the size of any orthogonal set for such a measure inside a ball of radius $R$ grows at most linearly in $R$. We also give an alternative proof of this bound, and in fact obtain a more general result of growth rate of orthogonal sets for Ahlfors--David regular measures in $\mathbb{R}^d$ (not restricted to the one-dimensional setting). Joint work with Mihail Kolountzakis, Ruxi Shi.
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Talk 06 — Haipeng Chen (Shenzhen Technology University, China) — [computing…]Local time: click here to view in your time zoneTitle: Lower Assouad-type dimensions and spectra: Theory and recent progress
Abstract
Fractal dimensions and dimension spectra are fundamental to fractal geometry. Recently, the study of dimension spectra has emerged as an active topic, yielding a wide range of significant results. This talk begins with a review of standard dimension definitions. We then focus on the lower Assouad dimension (also known as the uniformity dimension) and its variations, collectively known as lower Assouad-type dimensions. We discuss these concepts, including the lower Assouad spectrum, in the context of characterizing the local separation of fractal sets. In particular, we present recent results regarding attainable forms and dimension interpolations.