פעילויות השבוע

Severi varieties parameterize reduced irreducible curves of given geometric genus in a given linear system on an algebraic surface. The first irreducibility result for Severi varieties was established in 1986, and it is due to Harris, who considered the classical case of planar curves in characteristic zero. Few more irreducibility results have been obtained since then, but none of the known approaches is applicable in positive characteristic. In my talk, I will discuss the history and the state of the art in the irreducibility problem, and will also announce new results obtained in a joint work with Karl Christ and Xiang He.

The Gaussian Entire Function (GEF) is the random Taylor series, whose coefficients are independent centered complex Gaussians such that the n-th coefficient has variance 1/n!. The zero set of the GEF is a random point process in the plane, which is invariant with respect to isometries. The topic of this talk is the zero distribution of the GEF conditioned on the event that no zero lies in a given (large) region.

If the hole is a disk of radius r, Ghosh and Nishry observed a striking feature. As r tends to infinity, the density of particles vanishes not only on the given hole, but also on an annulus beyond the (rescaled) hole — a forbidden region emerges. Here, we study this problem for general simply connected holes, and find a curious connection to quadrature domains and a seemingly novel type of free boundary problem.

This reports on joint work in progress with Alon Nishry.

C*-algebras have been intensely studied in recent years, especially through the lens of classification via K-theoretic invariants. Prominent advances include results for Cuntz-Krieger algebras of directed graphs. One such result of Cuntz and Krieger shows that the K-theory groups of such algebras essentially coincide with Bowen-Franks groups of the subshift of finite type associated to the graph.

On the other hand, classifying non-self-adjoint operator algebras is an effort initiated by Arveson in his late 60s paper on algebras arising from one-sided measure preserving dynamics. This was later taken up by Davidson and Katsoulis in the topological scenario, where they classified non-self-adjoint operator algebras arising from multidimensional one-sided dynamical systems on compact Hausdroff spaces.

In this talk we will connect, through examples, these traditionally unrelated classification schemes. We survey some pertinent results from the literature and uncover a striking hierarchy of classification for irreversible and reversible operator algebras.

One of the classical enumerative problems in algebraic geometry is that of counting of complex or real rational curves through a collection of points in a toric variety.

We explain this counting procedure as a construction of certain cycles on moduli of rigid tropical curves. Cycles on these moduli turn out to be closely related to Lie algebras.

In particular, counting of both complex and real curves is related to the quantum torus Lie algebra. More complicated counting invariants (the so-called Gromov-Witten descendants) are similarly related to the super-Lie structure on the quantum torus.

[No preliminary knowledge of tropical geometry or the quantum torus algebra is expected.]

משוואות דיפרנציאליות רגילות ומשוואות דיפרנציאליות חלקיות לרוב נלמדות בגישות שונות זו מזו. בהרצאה אציג קשר מיוחד ביניהן, כך שמשוואה חלקית תיראה כמו מערכת של אינסוף משוואות דיפרנציאליות רגילות. אנו נראה זאת על ידי פתרון אלגנטי של מערכת משוואות מהצורה $Mx''+Kx=0$ עבור $K$ ו-$M$ מטריצות מוגדרות חיובית וסימטריות, ונסביר אילו עקרונות בדרך פתרון זו יכולים לסייע בפתרון משוואת הגלים. בדרך זו, ”יצוץ“ הצורך להשתמש בפירוק פורייה של פונקציה כדי לפתור משוואות חלקיות.