פעילויות השבוע

Cut-and-project point sets are constructed by identifying a strip of a fixed n-dimensional lattice (the ”cut“), and projecting the lattice points in that strip to a d-dimensional subspace (the ”project”). Such sets have a rich history in the study of mathematical models of quasicrystals, and include well-known examples such as the Fibonacci chain and vertex sets of Penrose tilings. Dynamical results concerning the translation action on the hull of a cut-and-project set are known to shed light on certain properties of the point set itself, but what happens when instead of restricting to translations we consider all volume preserving linear actions?

A homogenous space of cut-and-project sets is defined by fixing a cut-and-project construction and varying the n-dimensional lattice according to an SL(d,R) action. In the talk, which is based on joint work with René Rühr and Barak Weiss, I will discuss this construction and introduce the class of Ratner-Marklof-Strömbergsson measures, which are probability measures supported on cut-and-project spaces that are invariant and ergodic for the group action. A classification of these measures is described in terms of data of algebraic groups, and is used to prove analogues of results about a Siegel summation formula and identities and bounds involving higher moments. These in turn imply results about asymptotics, with error estimates, of point-counting and patch-counting statistics for typical cut-and-project sets.

The classical isoperimetric inequality in Euclidean space $\mathbb{R}^n$ states that among all sets (”bubbles“) of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the $n$-sphere $\mathbb{S}^n$ and on $n$-dimensional Gaussian space $\mathbb{G}^n$ (i.e. $\mathbb{R}^n$ endowed with the standard Gaussian measure). Furthermore, one may consider the ``multi-bubble“ isoperimetric problem, in which one prescribes the volume of $p \geq 2$ bubbles (possibly disconnected) and minimizes their total surface area – as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to $p=1$; the case $p=2$ is called the double-bubble problem, and so on.

In 2000, Hutchings, Morgan, Ritor'e and Ros resolved the double-bubble conjecture in Euclidean space $\mathbb{R}^3$ (and this was subsequently resolved in $\mathbb{R}^n$ as well) – the boundary of a minimizing double-bubble is given by three spherical caps meeting at $120^\circ$-degree angles. A more general conjecture of J.~Sullivan from the 1990‘s asserts that when $p \leq n+1$, the optimal multi-bubble in $\mathbb{R}^n$ (as well as in $\mathbb{S}^n$) is obtained by taking the Voronoi cells of $p+1$ equidistant points in $\mathbb{S}^{n}$ and applying appropriate stereographic projections to $\mathbb{R}^n$ (and backwards).

In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for $p \leq n$ bubbles in Gaussian space $\mathbb{G}^n$ – the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) $p+1$ equidistant points. In the talk, we describe our approach in that work, as well as recent progress on the multi-bubble problem on $\mathbb{R}^n$ and $\mathbb{S}^n$. In particular, we show that minimizing bubbles in $\mathbb{R}^n$ and $\mathbb{S}^n$ are always spherical when $p \leq n$, and we resolve the latter conjectures when in addition $p \leq 5$ (e.g. the triple-bubble conjectures when $n\geq 3$ and the quadruple-bubble conjectures when $n\geq 4$).

Given a rational Hecke eigenform $f$ of weight $2$, Eichler-Shimura theory gives a construction of an elliptic curve over ${\mathbb Q}$ whose associated modular form is $f$. Mazur, van Straten, and others have asked whether there is an analogous construction for Hecke eigenforms $f$ of weight $k>2$ that produces a variety for which the Galois representation on its etale ${\mathrm H}^{k-1}$ (modulo classes of cycles if $k$ is odd) is that of $f$. In weight $3$ this is understood by work of Elkies and Sch"utt, but in higher weight it remains mysterious, despite many examples in weight $4$. In this talk I will present a new construction based on families of K3 surfaces of Picard number $19$ that recovers many existing examples in weight $4$ and produces almost $20$ new ones.

תורת הקשרים חוקרת היבטים מתמטיים ותכונות טופולוגיות של קשרים, ומהווה אובייקט חשוב ובסיסי בחקר של יריעות במימדים נמוכים, ובעלת שימושים חשובים בפיזיקה, בכימיה וביולוגיה מהצד השני, מספרים ראשוניים הם אבני הבסיס של המספרים השלמים, ומהווים אובייקט מרכזי בתורת המספרים

בשנות ה-60 של המאה הקודמת, בארי מזור זיהה קשר מעניין ועמוק בין קשרים ומספרים ראשוניים

בהרצאה אסביר על קשרים ועל מספרים ראשוניים, ואנסה לתת מעט אינטואיציה לאנלוגיה, לשימושים שלה, ולקשר המעניין שהיא נותנת בין פיזיקה ותורת המספרים