Category Archives: Expository

Mathematics on mathematics

Expository, Thoughts on mathematics, Uncategorized

 

This post is the outline of a talk (or perhaps the talk is an outline of this post) that I will give on February 28 on our “open day” to prospective students in our department. This is supposed to be a story, it is intended to give a flavor, and neither the history nor the math are 100% precise, because it is a 15 minute talk! The big challenge is to take some rough ideas from this post, throw away the rest, and make that into an interesting quarter of an hour lecture. Comments are very welcome.

1. Introduction

What is mathematics? I am not going to answer that. You have all met mathematics in your life, in school, but also in other places as well, because math is everywhere. So you have some kind of idea what mathematics is about. However, I suspect that the most profound aspects of math have been hidden from you. I am here today to try to give you a taste of this mathematics which you have not yet seen. It is only fair to let you know that — for better and for worse — the mathematician’s mathematics, the mathematics that you will study if you do an undergraduate degree in math, is of a dramatically different nature from the math you learn in high-school or the math-is-everywhere kind of math which you meet in various popular accounts.

You have met various different kinds of mathematics: combinatorics, geometry, algebra, integral and differential calculus (aka HEDVA — which literally means “joy” in Hebrew). It seems as if mathematics splits into various branches, where in each branch there are different tasks that one should do. The objects of study of geometry are triangles, circles, trapezoids, etc; one has to prove that a certain triangle has this or that property, or one has to compute some angle or length or area. The objects of study in algebra are certain symbolic expressions or equations; one has to find the root of an equation, or to simplify an expression. In HEDVA the objects of study are functions; one has to compute the minimum of a function, or its anti-derivative, and so on.

The theme of this talk is that the objects of study in mathematics do not have to be only triangles or functions or equations, but they can also be geometry or analysis or algebra. Mathematics can also be used to study mathematics itself. This is profound. But perhaps more surprisingly, this has practical consequences.

Of course, I have no time to tell you precisely how this works. For this, I recommend that you come here and study mathematics. Continue reading Mathematics on mathematics →

The remarkable Hilbert space H^2 (part III – three open problems)

d shift space, Expository, Open problem, Operator theory, Research, Talks, ,

This is the last in the series of three posts on the d–shift space, which accompany/replace the colloquium talk I was supposed to give. The first two parts are available here and here. In this post I will discuss three open problems that I have been thinking about, which are formulated within the setting of $latex H^2_d$.

Continue reading The remarkable Hilbert space H^2 (part III – three open problems) →

Matthew Kennedy awarded CMS 2012 Doctoral Prize

Awards, Expository, Operator theory

One of the nicest things that happen on math blogs is that people write expository posts on other people’s work. Tim Gowers and Terry Tao have set a fine example in their expositions of the works of Fields Medalists or Abel Prize laureates. These are among the most interesting and important posts out there, I think.

Recently, at the Canadian Mathematical Society’s Winter Meeting, Matthew Kennedy was awarded the CMS 2012 Doctoral Prize (at the meeting several other prizes were awarded by the CMS as you can see on the meeting’s homepage). See here for the media release, and here for a description of the prize and a list of past recipients.

Matt gave a plenary lecture at the CMS meeting surveying (some of) his work. Here are the slides, which are certainly worth looking at (thanks to Matt for allowing me to post them). Note that on the last slide there are photos of two gentlemen with no captions; these are Ken Davidson (on the left, Matt’s PhD supervisor) and Heydar Radjavi (Matt’s undergraduate research supervisor).

In this post I will describe a couple of Matt’s first really big results, one of which didn’t make it into his talk. These are the existence of wandering vectors for (certain) free semigroup algebras, and the reflexitivity of free semigroup algebras. These result appeared in the tour-de-force paper “Wandering vectors and the reflexivity of free semigroup algebras”; here are links to arxiv, mathscinet, and the official version in Crelle.  Continue reading Matthew Kennedy awarded CMS 2012 Doctoral Prize →

The remarkable Hilbert space H^2 (Part II – multivariable operator theory and model theory)

d shift space, Expository, Operator theory, Talks,

This post is the second post in the series of posts on the d–shift space, a.k.a. the Drury–Arveson space, a.k.a. $latex H^2_d$ (see this previous post about the space $latex H^2$).

Continue reading The remarkable Hilbert space H^2 (Part II – multivariable operator theory and model theory) →

The remarkable Hilbert space H^2 (Part I – definition and interpolation theory)

d shift space, Expository, Operator theory, Talks,

This series of posts is based on the colloquium talk that I was supposed to give on November 20, at our department. As fate had it, that week studies were cancelled.

Several people in our department thought that it would be a nice idea if alongside the usual colloquium talks given by invited speakers which highlight their recent achievements, we would also have some talks by department members that will be more of an exposition to the fields they work in. So my talk was supposed to be an exposition to the setting in which much of the research I do goes on.

The topic of the “talk”  is the Hilbert space $latex H^2_d$. There will be three parts to this series:

  1. Definition and interpolation theory. 
  2. Multivariate operator theory and model theory
  3. Current research problems

Continue reading The remarkable Hilbert space H^2 (Part I – definition and interpolation theory) →

William Arveson

Arveson, Expository, Functional analysis, Operator theory,

William B. Arveson was born in 1934 and died last year on November 15, 2011. He was my mathematical hero; his written mathematics has influenced me more than anybody else’s. Of course, he has been much more than just my hero, his work has had deep and wide influence on the entire operator theory and operator algebras communities. Let me quickly give an example that everyone can appreciate: Arveson proved what may be considered as the “Hahn-Banach Theorem” appropriate for operator algebras. He did much more than that, and I will expand below on some of his early contributions, but I want to say something before that on what he was to me.

When I was a PhD student I worked in noncommutative dynamics. Briefly, this is the study of actions of (one-parameter) semigroups of *-endomorphisms on von Neumann algebras (in short E-semigroups). The definitive book on this subject is Arveson’s monograph “Noncommutative Dynamics and E-Semigroups”. So, naturally, I would carry this book around with me, and I would read it forwards and backwards. The wonderful thing about this book was that it made me feel as if all my dreams have come true! I mean my dreams about mathematics: as a graduate student you dream of working on something grand, something important, something beautiful, something elegant, brilliant and deep. You want your problem to be a focal point where different ideas, different fields, different techniques, in short, all things, meet.

When reading Arveson there was no doubt in my heart that, e.g., the problem classifying E-semigroups of type I was a grand problem. And I was blown away by the fact that the solution was so beautiful. He introduced product systems with such elegance and completeness that one would think that this subject has been studied for the last 50 years. These product systems were measurable bundles of operator spaces – which turn out to be Hilbert spaces! – that have a group like structure with respect to tensor multiplication. And they turn out to be complete invariants of E-semigroups on $latex B(H)$. The theory set down used ideas and techniques from Hilbert space theory, operator space theory, C*-algebras, group representation theory, measure theory, functional equations, and many new ideas – what more could you ask for? Well, you could ask that the new theory also contribute to the solution of the original problem.

It turned out that the introduction of product systems immensely advanced the understanding of E-semigroups, and in particular it led to the full classification of type I ones.

So Arveson became my hero because he has made my dreams come true. And more than once: when reading another book by him, or one of his great papers, I always had a very strong feeling: this is what I want to do. And when I felt that I gave a certain problem all I thought I had in me, and decided to move on to a new problem, it happened that he was waiting for me there too.

I wish to bring here below a little piece that I wrote after he passed away, which explains from my point of view what was one of his greatest ideas.

For a (by far) more authoritative and complete review of Arveson’s contributions, see the two recent surveys by Davidson (link) and Izumi (link).

Continue reading William Arveson →

Functional Analysis – Introduction. Part I

Advanced Analysis, 20125401, Expository, Functional analysis, Operator theory,

I begin by making clear a certain point. Functional analysis is an enormous branch of mathematics, so big that it does not seem appropriate to call it “a branch”, it sometimes looks more like another tree. When I will talk below about functional analysis, I will mean “textbook functional analysis” and not “research functional analysis”. By this I mean that I will only refer to the core of the theory which is several decades old and which is more-or-less agreed to be the essential and basic part of the subject.

The goal of this post is to serve as an introduction to the course “Advanced Analysis, 201.2.5401”, which is a basic graduate course on (textbook) functional analysis. In the lectures I will only have time to give a limited description of the roots of the subject and the motivation will have to be brief. Here I will aim to describe what was the climate in which this tree grew, where are its roots and what are its fruits.

To prepare this introduction I am relying on the following sources. First and foremost, my love of the subject and my point of view on it were strongly shaped by my teachers, and in particular by Boris Paneah (my Master’s thesis advisor) and Baruch Solel (my PhD. thesis advisor). Second, I learned a lot on the subject from the book “Mathematical Thought from Ancient to Modern Times” by M. Kline and from the notes sections of Rudin’s and Reed-Simon’s books “Functional Analysis”.

And a warning to the kids: this is a blog, not a book, and if you really want to learn something go read the books (the books I mentioned have precise references).

Continue reading Functional Analysis – Introduction. Part I →

Functional analysis – a preface to the introduction

Advanced Analysis, 20125401, Expository, Thoughts on mathematics,

I am planning to write a post that will be an introduction to the course “Advanced Analysis”, which I shall be teaching in the fall term. The introduction is to comprise two main themes: motivation and history. I was a little surprised to find out – as I was preparing the introduction – that, looking from the eyes of a student, the history of subject provided little motivation. I also began to oscillate between two opposite (and equally silly) viewpoints. The first viewpoint is that functional analysis is a big and respectable field of mathematics, which needs no introduction; let us start with the subject matter immediately since there is so much to learn. The second viewpoint is that there is absolutely no point in studying (or teaching) a mathematical theory without understanding its context and roots, or without knowing how it applies to problems outside of the theory’s borders. Pondering these, I found that I had some things to say before the introduction, which may justify the introduction or give it the place I intend.

Continue reading Functional analysis – a preface to the introduction →