Activities This Week

Lobachevsky started to work on computing volumes of hyperbolic polytopes long before the first model of the hyperbolic space was found. He discovered an extraordinary formula for the volume of an orthoscheme via a special function called dilogarithm.

We will discuss a generalization of the formula of Lobachevsky to higher dimensions. For reasons I do not fully understand, a mild modification of this formula leads to the proof of a conjecture of Goncharov about the depth of multiple polylogarithms. Moreover, the same construction leads to a functional equation for polylogarithms generalizing known equations of Abel, Kummer, and Goncharov.

Guided by these observations, I will define cluster polylogarithms on a cluster variety.

Consider a dynamical system (X,T) consisting of a compact set X in the Euclidean space and a transformation T on X. Takens-type time-delay embedding theorems state that for a generic real-valued observable h on X, one can reconstruct uniquely the initial state x of the system from a sequence of values of h(x), h(Tx), …, h(T^{k-1} x), provided that k is large enough. In the deterministic setting, the number of measurements k has to be at least twice the dimension of the state space X. This was proved in several categories and can be seen as dynamical versions of the classical (non-dynamical) embedding theorems. We provide a probabilistic counterpart of this theory, in which one is interested in reconstructing almost every state x, subject to a given probability measure. We prove that in this setting it suffices to take k greater than the Hausdorff dimension of the considered measure, hence reducing the number of measurements at least twice. Using this, we prove a related conjecture of Shroer, Sauer, Ott and Yorke in the ergodic case. We also construct an example showing that the conjecture does not hold in its original formulation. This is based on joint works with Krzysztof Barański and Yonatan Gutman.

The tracial Rokhlin property for actions of finite groups is now well known, along with weakenings and versions for other classes of discrete groups. The Rokhlin property for actions of infinite infinite compact groups has also been studied. We define and investigate the tracial Rokhlin property for actions of second countable compact groups on simple unital C*-algebras. The naive generalization of the verrsion for finite groups does not appear to be good enough. We have a property which, first, allows one to prove the expected theorems, second, is “almost” implied by the version for finite groups when the group is finite, and, third, admits examples.

This is joint work with Javad Mohammadkarimi.

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.