Activities This Week

Arakelov geometry offers a framework to develop an arithmetic counterpart of the usual intersection theory. For varieties defined over the ring of integers of a number field, and inspired by the geometric case, one can define a suitable notion of arithmetic Chow groups and of an arithmetic intersection product. In a joint work with Roberto Gualdi (University of Regensburg), we prove an arithmetic analogue of the classical Shioda-Tate formula, relating the dimension of the first Arakelov-Chow vector space of an arithmetic variety to some of its geometric invariants. In doing so, we also characterize numerically trivial arithmetic divisors, confirming part of a conjecture by Gillet and Soulé.

Horospherical group actions on homogeneous spaces are famously known to be extremely rigid. In finite volume homogeneous spaces, it is a special case of Ratner’s theorems that all horospherical orbit closures are homogeneous. Rigidity further extends in rank-one to infinite volume but geometrically finite spaces. The geometrically infinite setting is far less understood.

We consider $\mathbb{Z}$-covers of compact hyperbolic surfaces and show that they support quite exotic horocycle orbit closures. Surprisingly, the topology of such orbit closures delicately depends on the choice of a hyperbolic metric on the covered compact surface. In particular, our constructions provide the first examples of geometrically infinite spaces where a complete description of horocycle orbit closures is known. Based on an ongoing joint work with James Farre and Yair Minsky.

הרעיון של “כיוונים שונים ללכת בהם לאינסוף” במרחבים שונים, גרפים או חבורות נחקר בדרכים רבות. הדבר הוליד מושגים שונים של “שפה באינסוף” שניתן להגדיר בכל ההקשרים הללו.

בהרצאה זו נכיר את אחת הדוגמאות האלה, שנקראת Horoboundary, ומקורה בעבודתו של גרומוב על חבורות ומרחבים היפרבוליים. נראה דרכים שונות להגדיר את השפה הזו, בעזרת פונקציות על המרחב ובעזרת גיאודזים במרחב, ונתחיל לשאול שאלות שונות שרובן עוד ללא מענה. בהתאם לזמן, נכיר דוגמאות שבהן גיאודזים לא מספרים את כל הסיפור, ונסכם בשאלות הפתוחות ובשימושים האפשריים של הרעיונות האלה לכל מיני שאלות אחרות.

(Part 2 of the talk from last week.)

In this talk, I will discuss a noncommutative (nc) analogue of the Gleason problem and its application in the “NC Cowen-Douglas” class. The Gleason problem was first studied by Andrew Gleason in studying the maximal ideals of a commutative Banach algebra. In particular, he showed that if the maximal ideal consisting of functions in the Banach algebra $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ vanishing at the origin is finitely generated then it has to be generated by the coordinate functions where $\mathcal{A} ( \mathbb{B} ( 0, 1 ) )$ is the Banach algebra of holomorphic functions on the open unit ball $\mathbb{B} ( 0, 1 )$ at $0$ in $\mathbb{C}^n$ which can be continuously extended up to the boundary. The question – whether the maximal ideals in algebras of holomorphic functions are generated by the coordinate functions – has been named the Gleason problem. It turns out that the existence of a local solution of the Gleason problem in a reproducing kernel Hilbert space provides a sufficient condition for the membership of the tuple of adjoint of multiplication operators by coordinate functions in the Cowen-Douglas class.

After briefly discussing these classical aspects of the Gleason problem, I will introduce its nc counterpart for uniformly analytic nc functions and show that such a problem in the nc category is always locally uniquely solvable unlike the classical case. As an application one obtains a characterization of nc reproducing Hilbert spaces of uniformly analytic nc functions on a nc domain in $\mathbb{C}^d_{ \text{nc} }$ so that the adjoint of the $d$ - tuple of left multiplication operators by the nc coordinate functions are in the nc Cowen-Douglas class. Along the way, I will recall necessary materials from nc function theory.

This is a part of my ongoing work jointly with Professor Vinnikov on the nc Cowen-Douglas class.