פעילויות השבוע

(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.

Whitney studied the embeddings of (C^\infty) manifolds into R^N. A simple initial idea is: start from a map M-> R^N, and deform it generically. Hopefully one gets an embedding, at least an immersion. This fails totally because of the ”stable maps“. They are non-immersions, but are preserved in small deformations. The theory of stable maps was constructed in 50‘s-60‘s by Thom, Mather and others. The participating groups are infinite-dimensional, and the engine of the theory was vector fields integration. This chained all the results to the real/complex-analytic case. I will discuss the classical case, then report on the new results, extending the theory to the arbitrary field (of any characteristic).

A fascinating question in the geometry of numbers and diophantine approximation pertains to the maximal covering radius of a lattice with respect to a fixed function. An important covering radius is the multiplicative covering radius, since it is invariant under the diagonal group and relates to the Littlewood‘s conjecture. Minkowski conjectured that the multiplicative covering radius of a unimodular lattice in $R^d$ is bounded by above by $1/2^d$ and that this upper bound is unique to the diagonal orbit of the standard lattice. Minkowski‘s conjecture is known to be true for $d\leq 10$, yet there isn‘t a general proof for higher dimensions.

In this talk, I will discuss the function field (positive characteristic) analogue of Minkowski‘s conjecture, which we stated and proved for every dimension. The proofs and the results are surprisingly different from the real case and have implications in geometry of numbers and dynamics. This talk is based on joint work with Uri Shapira.