פעילויות השבוע

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

Coarse structures are used to study the large scale geometry of a space. For example, although the integers and the real line are different topologically, they look the same from ”far away“, in the sense that any geometric configuration in the real line can be approximated by one in the integers, up to some uniformly bounded error. A coarse group is a group object in the category of coarse spaces, for example, this means the group operation is only ”coarsely associative,“ etc. In joint work with Federico Vigolo we study coarse groups. This talk will be an advertisement for our work, as we walk through examples that illustrate some of our main results, and connections to other subjects.