Analysis Seminar, Dept. of Mathematics and Applied Math. -- Σεμινάριο Ανάλυσης

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Analysis Seminar in Crete (2026-27)

Abstract: For $N$ lattices in $\RR^d$ with volume $1$ and pairwise trivial intersections, every nonzero common tiling function has support diameter $\Omega(N^{1/d})$, while for lattice families whose fundamental domains have uniformly bounded diameters, the standard convolution construction gives an $O(N)$ upper bound, leaving a gap that has remained open since the work of Kolountzakis and Wolff. We close this gap by constructing, for every $d\geq 2$ and all $N\ge 2$, lattice families satisfying the same volume and intersection conditions that admit a nonnegative common tiling function with support diameter $O(N^{1/d})$, thereby also answering a question posed by Kolountzakis and Papageorgiou. We also obtain the optimal $O(\sqrt N)$ upper bound by constructing, for any prescribed family of plane lattices whose volumes lie in a fixed bounded set independent of $N$, a pairwise trivially intersecting family with the same respective volumes and with bases arbitrarily close to suitable bases of the prescribed lattices.

Abstract: We deal with the existence of bounded and measurable common fundamental domains for two lattices in $\mathbb{R}^d$ as well as three fully commensurable lattices, i.e. lattices whose sum is again a lattice. The concept of a fundamental domain of a lattice $L\subseteq \mathbb{R}^d$ is similar to a set that tiles $\mathbb{R}^d$ by translations of $L$. In fact every fundamental domain of a lattice, tiles $\mathbb{R}^d$. While the converse is not generally true, sets that tile the space by lattice translations behave as fundamental domains for most practical applications. We show that for any two lattices of equal volume, there always exists a bounded and measurable common fundamental domain. This generalizes the result by S. Grepstad and M. Kolountzakis which states that for any two lattices of equal volume there exists a bounded and measurable set that tiles $\RR^d$ with both lattices. As for two lattices of unequal volume, $L, M\subseteq \RR^d$, answering a question raised by Han and Wang, we show that if ${\rm vol} (L)<{\rm vol} (M)$ then there exists a bounded and measurable set that tiles $\RR^d$ by translations of $L$ and packs $\RR^d$ by translations of $M$. Finally, we completely characterize when three subgroups of equal order in a finite abelian group share a common transversal. As a consequence, we determine precisely when three full-rank lattices of equal volume with a discrete sum in $\RR^d$ admit a common fundamental domain, thereby answering an open question from 1997 for the case $n=3$.

Thu, 08 Oct 2026, 11:15, Room: A303
Speaker: Stefanos Koustas (University of Athens)

Abstract: The Mizohata-Takeuchi conjecture was recently disproved by Cairo and Zhang (2025), shifting the focus away from its classical and well-known setting. In this talk, we establish a new local Mizohata-Takeuchi estimate for Hörmander-type operators with non-degenerate phases, showing an explicit dependence on the phase's signature. Our result is driven by refined decoupling estimates—originating from the work of Guth, Iosevich, Ou, Wang adapted to this broader class of operators and generalizes previous work of Carbery, Iliopoulou, Wang (2023), which treats the maximal signature case.

Wed, 21 Oct 2026, 12:15, Room: A303
Speaker: Vassilis Nestoridis (University of Athens)

Abstract: Let Ω be a domain that is the interior of a simple closed curve ∂Ω in the complex plane. In the 1980s, N. Danikas and V. Nestoridis proved that, when Ω = 𝔻 is the open unit disk, for every function f ∈ A(𝔻), that is, f is continuous on the closed unit disk and holomorphic in 𝔻, and for every z ∈ 𝔻, the value f(z) equals the average of f over an arc I ⊂ ∂𝔻 with respect to the one-dimensional Lebesgue measure dθ. In today’s talk, we extend this result to a general Jordan domain Ω and functions f ∈ A(Ω), replacing the measure dθ with a harmonic measure μ for Ω. Shortly after the original result of Danikas and Nestoridis, it was shown that there are measures μ on ∂𝔻 for which the result does not hold. The conjecture that harmonic measures are the only measures on ∂Ω for which the result holds remains open. The original result also holds for functions in the larger Hardy space H1(𝔻). It is not known whether this extends to every harmonic measure on 𝔻. It is also of interest to investigate how the Hardy space H1(Ω) should be defined for a general domain Ω so that the original result holds for functions f ∈ H1(Ω). Must the boundary ∂Ω have finite length?

Abstract: The joint intersectivity conjecture of Bergelson, Leibman, and Lesigne gives a simple and optimal arithmetic criterion for the multidimensional polynomial Szemeredi theorem to remain valid without the usual zero constant term assumption. The corresponding result when all transformations are powers of a single transformation was proved by Bergelson, Leibman, and Lesigne in 2008 using the Host-Kra theory, but no comparable structural theory is available for several commuting transformations. I will discuss two recent advances: in joint work with Borys Kuca, we settled the conjecture for arbitrary pairs of jointly intersective polynomials, including the difficult dependent case, and more recently, in joint work with Davide Sclosa, we proved it for jointly intersective families of arbitrary length under the assumption of pairwise independence.