## Departmental PhD Thesis Exam – Damir Kinzebulatov

Everyone welcome.

Refreshments will be served in the Math Lounge before the exam.

Monday, October 24, 2011, 9:30 a.m., in BA 6183, 40 St. George St.

PhD Candidate: Damir Kinzebulatov

Thesis Title: Geometric analysis on solutions of some differential inequalities and within restricted classes of holomorphic functions

http://www.math.toronto.edu/dkinz/thesis.pdf

Thesis Abstract:

The thesis pursues two directions of research.

Parts 1 and 2 are devoted to study of solutions of certain differential inequalities.

Namely, in Part 1 we show that a germ of an analytic set (real or complex) admits a Gagliardo-Nirenberg type inequality with a certain exponent $s \geq 1$. At a regular point $s=1$, and the inequality becomes classical. As our examples show, $s$ can be strictly greater than 1 even for an isolated singularity. The ultimate goal would be not only to capture the geometric meaning of invariant s extending the results of Bos and Milman for subanalytic domains to include analytic sets, but also to find a relation between the equation defining the germ of an analytic set, and the value of this invariant.

In Part 2 we prove the property of unique continuation (also known as quasianalyticity) for solutions $S$ of differential inequality $|\Delta u| \leq |Vu|$ for a large class of potentials $V$. This result can be applied to the problem of absence of positive eigenvalues for self-adjoint Schroedinger operator $-\Delta+V$ defined in the sense of form sum. The original motivation for us and stil an ultimate goal for this project is to understand the local geometry of the solutions in $S$ by means of their desingularization for which it suffices to show stability’ of the unique continuation property with respect to just two simple transformations (involved in the desingularization algorithm). The results of Part 2 are contained in a joint paper with Leonid Shartser.

In Part 3 and 4 we derive the basic elements of complex function theory within some subalgebras of holomorphic functions (including extension from submanifolds, corona type theorem, properties of divisors, approximation property). Our key instruments and results are the analogues of Cartan theorems A and B for the coherent sheaves’ on the maximal ideal spaces of these subalgebras, and of Oka-Cartan theorem on coherence of the sheaves of ideals of the corresponding complex analytic subsets. (The ultimate goal is to transfer some results of Oka-Cartan theory to coherent sheaves on the maximal ideal space of the subalgebra $H^\infty(X)$ of bounded holomorphic functions on $X$, with the hope that this may shed some light on corona problem for $H^\infty(X)$, where $X$ is a pseudoconvex domain.)

More precisely, in Part 3 we consider the algebras of holomorphic functions on regular coverings of complex manifolds whose restrictions to each fiber belong to a translation-invariant Banach subalgebra of bounded functions endowed with $\sup$-norm.

The model examples of such subalgebras are Bohr’s holomorphic almost periodic functions on tube domains, and all fibrewise bounded holomorphic functions on regular coverings of complex manifolds (arising e.g. in study of holomorphic $L^2$-functions, Caratheodory hyperbolicity, corona type problems, Hartogs type theorems etc).

In Part 4 the primary object of study is the subalgebra of bounded holomorphic functions on the unit disk whose moduli can have boundary discontinuities of only the first kind.

The results of Parts 3 and 4 are contained in joint papers with Alexander Brudnyi.