{"id":21468,"date":"2026-04-13T09:55:32","date_gmt":"2026-04-13T13:55:32","guid":{"rendered":"https:\/\/www.crmath.ca\/page-calendrier\/abstracts-hyperbolic\/"},"modified":"2026-04-27T16:20:50","modified_gmt":"2026-04-27T20:20:50","slug":"abstracts-koosis","status":"publish","type":"page-calendrier","link":"https:\/\/www.crmath.ca\/en\/page-calendrier\/abstracts-koosis\/","title":{"rendered":"abstracts-koosis"},"content":{"rendered":"

Eugenia Malinnikova (Stanford University)<\/h4>\n

Critical sets of harmonic functions<\/p>\n

\nAbstract<\/summary>\n
Han, Hardt, and Lin in the 1990s proved that the size of the singular set of a harmonic function is bounded in terms of its doubling index, which is closely related to the frequency function. In 2017, Naber and Valtorta showed that if a harmonic function in the unit ball in R^d has frequency bounded by N then the (d\u22122)-dimensional Hausdorff measure of its critical set in the half ball is bounded by exp(CN^2). It is conjectured that the optimal bound should instead be polynomial in N. This is known in dimension two, but the higher-dimensional case remains surprisingly subtle. In this talk, we will review the initial ideas of Han, Hardt, and Lin, and the approach of Naber and Valtorta, we will then sketch some ideas which allow to improve the estimate in dimension three to a bound that is much closer to polynomial. The talk is based on ongoing joint work with Ben Foster and Josef Greilhuber.<\/div>\n<\/details>\n

Fedor Nazarov (Kent State University)<\/h4>\n

Every weak type $L^1$ bound for the maximal function has an underlying covering lemma<\/p>\n

\nAbstract<\/summary>\n
Let $X$ be a separable metric space, $\\mu$ a Borel measure on $X$, and $B$ some family of open sets of finite positive measure. Define the maximal function of a non-negative integrable function $f$ by
\n$$
\nMf(x)=\\sup_\\mu(R)^\\int_R f\\,d\\mu\\,.
\n$$
\nThe weak type $L^1$ bound for $M$, i.e., the inequality
\n$$
\n\\mu(\\)\\le Ct^\\int_X f\\,d\\mu\\,,
\n$$
\nin various settings is usually derived from some covering selection property, the most general version of which seems to be as follows:There exist constants $c,C\\in(0,+\\infty)$ such that for every finite family $B_0\\subset B$, there is a subfamily $B_1\\subset B_0$ satisfying $\\mu(\\cup_R)\\ge c\\mu(\\cup_R)$ and $\\sum_\\chi_R\\le C$ $\\mu$-almost everywhere.<\/p>\n

We shall show that there cannot be any other reason for a weak type $L^1$ bound of the above type, namely, that if the weak type bound holds for the maximal function $M$ associated with the family $B$, then $B$ necessarily has this covering selection property. Time permitting, we'll discuss analogues of this theorem for the similar bounds for $M$ involving $L\\log L$ and other Orlich type expressions on the right-hand side.<\/p>\n

This is a joint work with Paul Hagelstein and Blanca Radillo-Murguia.<\/p>\n<\/div>\n<\/details>\n

Alexei Poltoratski (University of Wisconsin)<\/h4>\n

From spectral gaps to causal depth<\/p>\n

\nAbstract<\/summary>\n
I will talk about the classical gap and type problems in Fourier analysis which focus on functions and measures with gaps in the support of their Fourier transform. These historic problems, studied by many famous mathematicians including Paul Koosis, can be viewed in the context of the uncertainty principle in harmonic analysis. They have multiple connections and applications in various areas of analysis and spectral theory. The second part of the title refers to recently found connections between the type problem and the theory of holographic horizons in high-energy physics.<\/div>\n<\/details>\n

Malabika Pramanik (Universit\u00e9 de la Colombie-Britannique)<\/h4>\n

Directional Maximal Operators and the Geometry of Finite-Order
\nLacunarity<\/p>\n

\nAbstract<\/summary>\n
Directional maximal operators in the plane are governed by the
\ngeometric structure of the underlying set of directions. A central problem
\nis to characterize those slope sets for which the associated operator is
\nbounded on nontrivial Lebesgue spaces. Classical results indicate that
\nfinite-order lacunarity plays a decisive role, but earlier formulations
\nleft a gap between combinatorial definitions and the geometric mechanisms
\nunderlying Kakeya-type phenomena.In this talk, I will describe recent joint work with Ed Kroc and Juyoung
\nLee that develops a refined notion of admissible finite-order lacunarity.
\nThe key idea is to encode direction sets by rooted, labelled trees and to
\ninterpret lacunarity through structural invariants such as splitting
\nnumber. This framework provides a unified way to relate three complementary
\nperspectives: combinatorial sparsity of the slope set, geometric incidence
\nproperties of associated rectangle families, and analytic boundedness of
\ndirectional maximal operators.A central ingredient is a pruning procedure applied to the slope tree,
\nwhich produces a subtree of slopes exhibiting controlled Euclidean
\nseparation. This enables a probabilistic construction of Kakeya-type
\nconfigurations adapted to the pruned geometry. As a consequence, we obtain
\na dichotomy: admissible finite-order lacunary sets give rise to bounded
\noperators on all nontrivial Lebesgue spaces, while sublacunary sets
\ngenerate Kakeya-type phenomena and lead to unboundedness on all such
\nspaces.<\/p>\n

The talk will emphasize the structural ideas behind this approach, as well
\nas some ongoing directions and open questions.<\/p>\n<\/div>\n<\/details>\n

Thomas Ransford (Universit\u00e9 Laval)<\/h4>\n

Negative powers of Hilbert-space contractions<\/p>\n

\nR\u00e9sum\u00e9<\/summary>\n
I shall discuss the following conjecture due to Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a sequence $u_n\\to\\infty$ with the following property: if $T$ is any Hilbert-space contraction whose spectrum is contained in $E$, then either $T$ is unitary or $\\limsup_\\|T^\\|\/u_n=\\infty$. This conjecture is intimately related to the properties of inner functions.<\/p>\n<\/div>\n<\/details>\n

Sasha Volberg (Michigan State University)<\/h4>\n

The Boolean surface area of polynomial threshold functions<\/p>\n

\nAbstract<\/summary>\n
Abstract_Volberg_Alexander<\/a><\/div>\n<\/details>\n

Reem Yassawi (Queen Mary University of London)<\/h4>\n

U-adic integers for some Pisot U-numerations<\/p>\n

\nAbstract<\/summary>\n
Abstract_Yassawi_Reem<\/a><\/div>\n<\/details>\n<\/div><\/div><\/div><\/div><\/div>\n","protected":false},"author":16,"template":"","class_list":["post-21468","page-calendrier","type-page-calendrier","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/www.crmath.ca\/en\/wp-json\/wp\/v2\/page-calendrier\/21468","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.crmath.ca\/en\/wp-json\/wp\/v2\/page-calendrier"}],"about":[{"href":"https:\/\/www.crmath.ca\/en\/wp-json\/wp\/v2\/types\/page-calendrier"}],"author":[{"embeddable":true,"href":"https:\/\/www.crmath.ca\/en\/wp-json\/wp\/v2\/users\/16"}],"version-history":[{"count":12,"href":"https:\/\/www.crmath.ca\/en\/wp-json\/wp\/v2\/page-calendrier\/21468\/revisions"}],"predecessor-version":[{"id":21686,"href":"https:\/\/www.crmath.ca\/en\/wp-json\/wp\/v2\/page-calendrier\/21468\/revisions\/21686"}],"wp:attachment":[{"href":"https:\/\/www.crmath.ca\/en\/wp-json\/wp\/v2\/media?parent=21468"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}