Activities This Week

An observation by Jens Marklof shows that the primitive rational points of a fixed denominator along the periodic unipotent orbit of volume equal to the square of the denominator equidistribute inside a proper submanifold of the modular surface. This concentration as well as the equidistribution are intimately related to classical questions of number theoretic origin. We discuss the distribution of the primitive rational points along periodic orbits of intermediate size. In this case, we can show joint equidistribution with polynomial rate in the modular surface and in the torus. We also deduce simultaneous equidistribution of primitive rational points in the modular surface and of modular hyperbolas in the two-torus under certain congruence conditions. This is joint work with M. Einsiedler and N. Shah.

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.