פעילויות השבוע

Open Gromov-Witten (OGW) invariants should count pseudoholomorphic maps from curves with boundary to a symplectic manifold, with Lagrangian boundary conditions and various constraints on boundary and interior marked points. The presence of boundary of real codimension 1 poses an obstacle to invariance. In a joint work with J. Solomon (2016-2017), we defined genus zero OGW invariants under cohomological conditions. The construction is rather abstract. Nonetheless, in a recent work, also joint with J. Solomon, we find that the generating function of OGW has many properties that enable explicit calculations. Most notably, it satisfies a system of PDE called open WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equation. For the projective space, this PDE generates recursion relations that allow the computation of all invariants. Furthermore, the open WDVV can be reinterpreted as an associativity of a suitable version of a quantum product.

No prior knowledge of any of the above notions will be assumed.

In this talk we will discuss two central problems in multivariable operator theory: (1) joint invariant subspaces of the tuple of shift operators on the Hardy space over the unit polydisc, and (2) a commutant lifting theorem of the tuple of shift operator on certain analytic reproducing kernel Hilbert spaces over the unit ball. If time permits, we will also talk about Nevanlinna-Pick interpolation theorem and a relevant factorization result for multipliers on reproducing kernel Hilbert spaces over the unit ball.

Benjamini-Schramm convergence is a probabilistic notion useful in studying the asymptotic behavior of sequences of metric spaces. The goal of this talk is to discuss this notion and some of its applications from various perspectives, e.g. for groups, graphs, hyperbolic manifolds and locally symmetric spaces, emphasizing the distinction between the hyperbolic rank-one case and the rigid high-rank case. Understanding the ”sofic“ part of the Benjamini-Schramm space, i.e. all limit points of ”finitary“ objects, will play an important role. From the group-theoretic perspective, I will talk about sofic groups, i.e. groups which admit a probabilistic finitary approximation, as well as a companion notion of permutation stability. Several results and open problems will be discussed.

לפירות שמקיימים את המשוואה בתמונה יש לפחות 80 ספרות. נראה איך ניתן למצוא אותם - באמצעות פונקציות מרוכבות על מרחב כל השריגים, וגיאומטריה של עקומים מעוקבים.