פעילויות השבוע

The dimension of the space of multilinear products of higher commutators is equal to the number of derangements, $[e^{-1}n!]$. Our search for a combinatorial explanation for this fact led us to study representations of left regular bands, whose resolution is obtained through analysis of cubical partial representations. There are applications in combinatorics, probability, and nonassociative algebra.

במרכז של תורת הקשרים עומדת השאלה: בהנתן שרוך קשור בקשר כלשהו, אשר קצוותיו תפורים זה לזה, האם ניתן ”לפתוח“ את הקשר בלי לחתוך את השרוך. תחום זה משתמש בטכניקות ממגוון תחומים אחרים במתמטיקה, החל מטופולוגיה ועד לתורת ההצגות.

משפט פארי מילנור הוא משפט המאפשר, בתנאים מסויימים, לתת תשובה לשאלה הנ“ל בעזרת אנליזה. בהרצאה אספר על המשפט ועל שתי הוכחות יפות שניתנו באופן בלתי תלוי על ידי פארי ומילנור.

Given a group G, an Eilenberg-MacLane space X = K(G,1) provides a topological model of both G and Aut(G). The latter is understood via Whitehead’s theorem as the group of pointed homotopy equivalences of X up to homotopy. When X has rich structure, such as the case of a closed surface group, this point of view leads to a rich understanding of Aut(G). Motivated by dynamics and mathematical physics, Biswas, Nag, and Sullivan initiated the study of the universal hyperbolic solenoid, the inverse limit of all finite covers of a closed surface of genus at least two. Following their work, Odden proved that the mapping class group of the universal hyperbolic solenoid is isomorphic to the abstract commensurator of a closed surface group. In this talk I will present a general topological analog of Odden’s theorem, realising Comm(G) as a group of homotopy equivalences of a space for any group of type F. I will then use this realisation to classify the locally finite subgroups of the abstract commensurator of a finite-rank free group. This is joint work with Daniel Studenmund.

This talk will be a survey of the mathematics of the gap labelling problem for quasicrystals, but will assume no knowledge of physics.

In one standard approximation, the possible energy levels of an electron moving in a crystal form a collection of bands. These energy levels constitute the spectrum of a suitable Schr$\"{o}$dinger operator, and the gaps between the bands are gaps in the spectrum.

Quasicrystals are not periodic, but exhibit long range order. The structure of the spectrum of the Schr$\"{o}$dinger operator for quasicrystals is addressed by the ``Gap Labelling Conjecture‘‘. This conjecture was made in 1989, and some results are known.

An infinite quasicrystal has an associated action of ${\mathbb{Z}}^d$ on the Cantor set $X$, and thus a transformation group C*-algebra $A$. The physics is supposed to give an invariant measure on $X$, and hence a tracial state on $A$. The gaps in the spectrum of Schr"{o}dinger operator correspond to the values of this tracial state on projections in $A$, and the Gap Labelling Theorem states that these values all already occur as values of the measure on compact open subsets of $X$.

In this talk, I will give a more careful description of the situation, including sketches of how the objects above are constructed and how they are related to each other. Then I will say something about the results that have been proved, and outline what goes into their proofs.