Noncommutative Analysis

Category: Expository

The disc trick (and some other cute moves)

This post is about a chain of little tricks that I discovered with collaborators and used in several papers. It is just a collection of simple moves that lets you deduce the existence of a zero preserving map of a certain class between two gauge invariant spaces, given the existence of a map from that class (things will be very clear soon, I hope). These tricks were later used by some other people, who applied it in different settings.

I am writing this post as notes for my upcoming Pizza & Beer seminar talk. The section at the end of the notes contains references and links to papers where this was used.

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The complex matrix cube problem – new results from summer projects

In this post I will summarize the results obtained by my group in the “Summer Projects Week” that took place two weeks ago at the Technion. As in last time (see here for a summary of last year’s project) the title of the project I suggested was “Numerical Explorations of Open Problems from Operator Theory”. This time, I was lucky to have Malte Gerhold and Satish Pandey, my postdocs, volunteer to help me with the mentoring. The students who chose our project were Matan Gibson and Ofer Israelov, and they did fantastic work.

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The perfect Nullstellensatz just got more perfect

After giving a talk about the perfect Nullstellensatz (the commutative free Nullstellensatz) at the Technion Math department’s pizza and beer seminar, I had a revelation: I think it holds over other fields as well, not just over the complex numbers! (And in particular, contrary to what I thought before, it holds over the reals. It seems to hold over other fields as well). 

To explain, I will need some notation. 

Let k be a field. We write A = k[z_, \ldots, z_d] – the algebra of all polynomials in d (commuting) variables over the field k

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The complex matrix cube problem (in “Summer Projects in Mathematics at the Technion”)

Next week I will participate as a mentor in the Technion’s Summer Projects in Mathematics. The project I offered is called “Numerical explorations of open problems from operator theory”, and it suggests three open problems in operator theory where theoretical progress seems to be stuck, and for which I believe that some computer experiments can help us get a feeling of what is going on. I also hope that thinking seriously about designing experiments can help us to understand some general facets of the theory.

I have been in contact with the students in the last few weeks and we decided to concentrate on “the matrix cube problem”. On Sunday, when the week begins, I will need to present the background to the project to all participants of this week, and I have seven minutes (!!) for this. As everybody knows, the shorter the presentation, the harder the task is, and the more preparation and thought it requires. So I will take use this blog to practice my little talk.

Introduction to the matrix cube problem

This project is in the theory of operator spaces. My purpose is to give you some kind of flavour of what the theory is about, and what we will do this week to contribute to our understanding of this theory.

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The perfect Nullstellensatz

Question: to what extent can we recover a polynomial from its zeros?

Our goal in this post is to give several answers to this question and its generalisations. In order to obtain elegant answers, we work over the complex field \mathbb{C} (e.g., there are many polynomials, such as x^{2n} + 1, that have no real zeros; the fact that they don’t have real zeros tells us something about these polynomials, but there is no way to “recover” these polynomials from their non-existing zeros). We will write \mathbb{C}[z] for the algebra of polynomials in one complex variable with complex coefficients, and consider a polynomial as a function of the complex variable z \in \mathbb{C}. We will also write \mathbb{C}[z_1, \ldots, z_d] for the algebra of polynomials in d (commuting) variables, and think of polynomials in \mathbb{C}[z_1, \ldots, z_d] – at least initially – as a functions of the variable z = (z_1, \ldots, z_d) \in \mathbb{C}^d

[Update June 24, 2019: contrary to what I thought, the main theorem presented below holds over arbitrary fields, not just over the complex numbers, very much by the same proof. See this post.]

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