פעילויות השבוע

An index theory for elliptic operators on a closed manifold was developed by Atiyah and Singer. For a family of such operators parametrized by points of a compact space X, they computed the K^0(X)-valued analytical index in purely topological terms. An analog of this theory for self-adjoint elliptic operators on closed manifolds was developed by Atiyah, Patodi, and Singer; the analytical index of a family in this case takes values in the K^1 group of a base space.

If a manifold has non-empty boundary, then boundary conditions come into play, and situation becomes much more complicated. The integer-valued index of a single boundary value problem was computed by Atiyah and Bott. This result was recently generalized to K^0(X)-valued family index by Melo, Schrohe, and Schick. The self-adjoint case, however, remained open.

In the talk I shall present a family index theorem for self-adjoint elliptic operators on a surface with boundary. I compute the K^1(X)-valued analytical index in terms of the topological data of the family over the boundary. The talk is based on my preprint arXiv:1809.04353.

There are many formulas that express interesting properties of a finite group G in terms of sums over its characters. For estimating these sums, one of the most salient quantities to understand is the character ratio:

Trace(ρ(g)) / dim(ρ),

for an irreducible representation ρ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of the mentioned type for analyzing certain random walks on G.

Recently (https://www.youtube.com/watchv=EfVCWWWNxvg&feature=youtu.be), we discovered that for classical groups G over finite fields there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant rank.

Rank suggests a new organization of representations based on the very few “Small” ones. This stands in contrast to Harish-Chandra’s “philosophy of cusp forms” (P-of-CF), which is (since the 60‘s) the main organization principle, and is based on the (huge collection) of “Large” representations.

This talk will discuss the notion of rank for the group GL(n) over finite fields, demonstrate how it controls the character ratio, and explain how one can apply the results to verify mixing time and rate for random walks.

This is joint work with Roger Howe (Yale and Texas A&M). The numerics for this work was carried by Steve Goldstein (Madison).

בשנת 1963 ארדש ורני הוכיחו את משפט שבמבט ראשון נראה בלתי סביר:

משפט: נבנה גרף על הטבעיים באופן הבא: לכל זוג טבעיים נקבע אם יש צלע ביניהם על-ידי הטלת מטבע (נאמר, הוגן). אזי בהסתברות 1, כל שני גרפים שנבנה כך יהיו איזומורפיים.

הגרף המתקבל באופן זה (בהסתברות 1), נקרא ”הגרף המקרי“, והוא האנאלוג (גם במובנים טכניים מדויקים) בתורת הגרפים לרציונליים כטיפוס סדר. בהרצאה נוכיח את המשפט של ארדש ורני ונראה איך לבנות את הגרף המקרי (בכמה דרכים שונות), נסקור מעט מתכונותיו: כל גרף בן מניה הוא תת-גרף של הגרף המקרי, בכל חלוקה של קודקדי הגרף לשתי קבוצות — הגרף המושרה על אחת מהן לפחות הוא הגרף המקרי עצמו והוא אחד מבין שלושה גרפים בלבד שלהם תכונה זו (מהם השנים הנותרים?).

לבסוף, נראה איך להשתמש בלוגיקה של הגרף המקרי על מנת להוכיח את הטענה הבאה:

תהי $P$ תכונה מסדר ראשון של גרפים (אנו נסביר בדיוק למה הכוונה). נסמן $P(n)$ את ההסתברות שלגרף על $n$ קודקדים יש התכונה $P$. אזי הגבול, כאשר $n$ שואף לאינסוף, של $P(n)$ הוא $0$ או $1$.

https://www.math.bgu.ac.il/~amyekut/lectures/flat-comp-revis/abstract.html

A cut-and-project set is constructed by restricting a lattice $L$ in $(d+m)$-space to a domain bounded in the last m coordinates, and projecting these points to the the space spanned by its d-dimensional orthogonal complement. These point sets constitute an important example of so-called quasicrystals.

During the talk, we shall present and give some classification results of the moduli spaces of cut-and-project sets, which were introduced by Marklof-Strömbergsson. These are obtained by considering the orbit closure of the special linear group in $d$-space acting on the lattice $L$ inside the space of unimodular lattices of rank $d+m$. Theorems of Ratner imply that these are meaningful objects.

We then describe quantitative counting result for patches in generic cut-and-project sets. Patches are local configuration of point sets whose multitude reflects aperiodicity.

The count follows some old argument of Schmidt using moment bounds. These bounds are obtained by integrability properties of the Siegel transform, which in turn follow from reduction theory and a symmetrisation argument of Rogers. This argument is of independent interest, giving an alternative account to recent work of Kelmer-Yu (which is based on the theory of Eisenstein series) on counting points in generic symplectic lattices.

This is a joint endeavour with Yotam Smilansky and Barak Weiss.