פעילויות השבוע

In this talk, we will start going over Takesaki‘s annals paper that proves that every separable C*-algebra A can be represented as continuous ”noncommutative“ functions with values in B(H) (H separable) on the space of representations of A on H. Furthermore, the universal enveloping von Neumann algebra of A is identified with all the bounded ”noncommutative“ functions on the same space

The question of determining if a given algebraic variety is rational is a notoriously difficult problem in algebraic geometry, and attempts to solve rationality problems have often produced powerful new techniques. A well-known open rationality problem is the determination of a criterion for when a cubic hypersurface of five-dimensional projective space is rational. After discussing the history of this problem, I will introduce the two conjectural rationality criteria that have been put forth and then discuss a package of tools I have developed with my collaborators to bring these two conjectures together. Our theory of Relative Bridgeland Stability has a number of other beautiful consequences such as a new proof of the integral Hodge Conjecture for Cubic Fourfolds and the construction of full-dimensional families of projective Hyper Kahler manifolds. Time permitting I’ll discuss a few of the many applications of the theory of relative stability conditions to problems other than cubic fourfolds.

בהרצאה נדון על נקודת מבט שהתפתחה מאוד בעשורים האחרונים בחקר תורת החבורות שבה מתבוננים על חבורה כאובייקט גיאומטרי. נתאר שתי דרכים לבנות שפה (או נקודות גבול) לחבורה אינסופית, ונדבר על קשרים בין ההסתכלות הזו לבין תכונות אלגבריות של החבורה.

The conjecture of Loxton and var der Poorten is a criterion for a formal power series to be the expansion at 0 of a rational function, and is related to a famous theorem of Cobham in the theory of finite automata. It was proved by Adamczewski and Bell in 2013. Recently, Schafke and Singer found a novel approach that lead also to a simple conceptual proof of Cobham‘s theorem. We shall explain these results and the cohomological machinery behind them, and discuss what is missing from the picture to establish an elliptic analogue.

The usual weighted average of points $(z_0, ..., z_q)$ in the real vector space $R^n$, with weights $(w_0, ..., w_q)$, is translation invariant. Hence it can be seen as an average of points in a torsor Z over the Lie group $G = R^n$ (A $G$-torsor is a $G$-manifold with a simply transitive action.)

In this talk I will explain how this averaging process can be generalized to a torsor Z over a unipotent Lie group $G$. (In differential geometry, a unipotent group is a simply connected nilpotent Lie group. $R^n$ is an abelian unipotent group.)

I will explain how to construct the unipotent weighted average, and discuss its properties (functoriality, symmetry and simpliciality). If time permits, I will talk about torsors over a base manifold, and families of sections parametrized by simplices. I will indicate how I came about this idea, while working on a problem in deformation quantization.

Such an averaging process exists only for unipotent groups. For instance, it does not exist for a torus $G$ (an abelian Lie group that‘s not simply connected). In algebraic geometry the unipotent averaging has arithmetic significance, but this is not visible in differential geometry.

Notes for the talk can be founds here: https://www.math.bgu.ac.il/~amyekut/lectures/average-diff-geom/abstract.html