## Final Report Summary - OAGUB (Operator-algebraic geometry in the unit ball)

The title of this project is Operator Algebraic Geometry in the Unit Ball - OAGUB for short. Let me begin by explaining what this title means.

Classical algebraic geometry deals with the study of the sets of numerical solutions to systems of polynomial equations. For example, one can consider the polynomial p(x,y,z) = xy + yz + zx in the three variables x,y and z, and ask what are the triplets of numbers (a,b,c) that satisfy p(a,b,c) = 0. "Operator algebraic geometry" is the study of sets of solutions to systems of polynomial equations, but now we search not only for numerical solutions, but also for more general solutions - operator solutions (for example, one can consider also matrix solutions to a certain equation). For example, we can consider the polynomial p(x,y,z) = xy + yz + zx in the three variables x,y,z, and ask what are the 3-tuples of operators (X,Y,Z) that satisfy p(X,Y,Z) = XY + YZ + ZX = 0. Here, by a “3-tuple of operators”, we simply mean three operators X, Y and Z which act on the same Hilbert space.

Broadly speaking, the project OAGUB deals with the question: "What can we say about a tuple of operators that satisfies a given system of polynomial equations?" My ultimate goal was to study such questions, as a route to understanding the connection between operator theory, complex function theory and complex-algebraic geometry, in a broad sense.

At the outset of this project, I chose to focus on d-tuples T = (T_1, ... T_d) of operators that commute with one another, i.e., satisfy T_i T_j = T_j T_i for all i,j=1,...,d. In addition, I studied tuples that are “row contractions”, that is, also satisfy the norm condition ||T|| < 1 (this norm condition is a constraint on the "size" of the solutions, and is what accounts for the use of the terminology "unit ball" appearing in the title). In this setting what happens is that every "classical solution" (that is, every numerical solution) to the system of equations gives rise to an operator solution, but the set of operator solutions is almost always much bigger than the set of classical solutions, and the operator solutions behave in a very different way (for example, their "size" under application of a series of bounded polynomials can be unbounded - this cannot happen for classical solutions).

One kind of problem that is related to this goal, is the study of the universal operator algebras generated by row contractions of commuting operators which satisfy certain polynomial (or more generally, analytic) relations. The goal here is to classify the operator algebras up to certain kinds of isomorphisms in terms of the underlying algebraic (or analytic) varieties that are determined by the polynomial (analytic) relations.

Another one of the concrete targets of my research is a conjecture due to William Arveson (called "Arveson's essential normality conjecture") that says, roughly, that the operator solutions - in spite of there being "much more of them" than the classical solutions - can essentially be understood by looking at classical solutions. Here the word "essentially" has a very precise mathematical meaning, which is that the operator solutions behave like numerical solutions modulo the compact operators.

During the first two years of the project I made significant progress on understanding universal algebras generated by commuting tuples of operators satisfying some relations. I also obtained several results bringing us closer to a solution of Arveson's essential normality conjecture described above, though we did not solve it fully. On the other hand, I encountered some profound difficulties, which made some of my original approaches appear futile.

I was naturally led to consider a higher level approach, and in the second two-year period, I chose to concentrate on operator-algebraic approaches to the problem. This, in turn, led me to view the multivariable operator theory from a fully noncommutative point of view. I realized that the emergent theory of noncommutative functions serves a framework in which to study all problems of interest, and I therefore gradually switched my focus to the study of multivariable operator theory on noncommutative varieties in the noncommutative unit ball.

The European Commission's objectives for funding this project are dual. There is first is the obvious objective which lies on the surface of things, and that is to promote scientific and mathematical research, in order to enrich humanity's treasure of knowledge. The second objective for funding this project is to facilitate my reabsorption as a researcher in Israel (and in the European Union Scientific community) after spending two years as a postdoctoral fellow in North America.

The second objective of this project has also been met during the four years that this project has been carried out. I successfully integrated at the Technion, and am up for promotion and tenure this year. The grant allowed me to fund students, and to host visitors from abroad with which I collaborated. The financial support given by this grant helped me concentrate on my research and grow as a mathematician.

During the second period of this project I was involved with the training of one PhD student and two postdoctoral students. This was very important not only to my own career, but also for the development of the careers of these young mathematicians at the start of their career.

The project website can be viewed at the address: the website includes links to personal and contact information.

Classical algebraic geometry deals with the study of the sets of numerical solutions to systems of polynomial equations. For example, one can consider the polynomial p(x,y,z) = xy + yz + zx in the three variables x,y and z, and ask what are the triplets of numbers (a,b,c) that satisfy p(a,b,c) = 0. "Operator algebraic geometry" is the study of sets of solutions to systems of polynomial equations, but now we search not only for numerical solutions, but also for more general solutions - operator solutions (for example, one can consider also matrix solutions to a certain equation). For example, we can consider the polynomial p(x,y,z) = xy + yz + zx in the three variables x,y,z, and ask what are the 3-tuples of operators (X,Y,Z) that satisfy p(X,Y,Z) = XY + YZ + ZX = 0. Here, by a “3-tuple of operators”, we simply mean three operators X, Y and Z which act on the same Hilbert space.

Broadly speaking, the project OAGUB deals with the question: "What can we say about a tuple of operators that satisfies a given system of polynomial equations?" My ultimate goal was to study such questions, as a route to understanding the connection between operator theory, complex function theory and complex-algebraic geometry, in a broad sense.

At the outset of this project, I chose to focus on d-tuples T = (T_1, ... T_d) of operators that commute with one another, i.e., satisfy T_i T_j = T_j T_i for all i,j=1,...,d. In addition, I studied tuples that are “row contractions”, that is, also satisfy the norm condition ||T|| < 1 (this norm condition is a constraint on the "size" of the solutions, and is what accounts for the use of the terminology "unit ball" appearing in the title). In this setting what happens is that every "classical solution" (that is, every numerical solution) to the system of equations gives rise to an operator solution, but the set of operator solutions is almost always much bigger than the set of classical solutions, and the operator solutions behave in a very different way (for example, their "size" under application of a series of bounded polynomials can be unbounded - this cannot happen for classical solutions).

One kind of problem that is related to this goal, is the study of the universal operator algebras generated by row contractions of commuting operators which satisfy certain polynomial (or more generally, analytic) relations. The goal here is to classify the operator algebras up to certain kinds of isomorphisms in terms of the underlying algebraic (or analytic) varieties that are determined by the polynomial (analytic) relations.

Another one of the concrete targets of my research is a conjecture due to William Arveson (called "Arveson's essential normality conjecture") that says, roughly, that the operator solutions - in spite of there being "much more of them" than the classical solutions - can essentially be understood by looking at classical solutions. Here the word "essentially" has a very precise mathematical meaning, which is that the operator solutions behave like numerical solutions modulo the compact operators.

During the first two years of the project I made significant progress on understanding universal algebras generated by commuting tuples of operators satisfying some relations. I also obtained several results bringing us closer to a solution of Arveson's essential normality conjecture described above, though we did not solve it fully. On the other hand, I encountered some profound difficulties, which made some of my original approaches appear futile.

I was naturally led to consider a higher level approach, and in the second two-year period, I chose to concentrate on operator-algebraic approaches to the problem. This, in turn, led me to view the multivariable operator theory from a fully noncommutative point of view. I realized that the emergent theory of noncommutative functions serves a framework in which to study all problems of interest, and I therefore gradually switched my focus to the study of multivariable operator theory on noncommutative varieties in the noncommutative unit ball.

The European Commission's objectives for funding this project are dual. There is first is the obvious objective which lies on the surface of things, and that is to promote scientific and mathematical research, in order to enrich humanity's treasure of knowledge. The second objective for funding this project is to facilitate my reabsorption as a researcher in Israel (and in the European Union Scientific community) after spending two years as a postdoctoral fellow in North America.

The second objective of this project has also been met during the four years that this project has been carried out. I successfully integrated at the Technion, and am up for promotion and tenure this year. The grant allowed me to fund students, and to host visitors from abroad with which I collaborated. The financial support given by this grant helped me concentrate on my research and grow as a mathematician.

During the second period of this project I was involved with the training of one PhD student and two postdoctoral students. This was very important not only to my own career, but also for the development of the careers of these young mathematicians at the start of their career.

The project website can be viewed at the address: the website includes links to personal and contact information.