Almost disjoint sets

It is intuitive and easy to see that a disjoint collection of subsets of a countable set is countable. (Although one has to be careful with saying “intuitive” and “easy to see” in set theory.) A natural question is, what happens if we allow the sets to have nonempty but finite intersection? … More Almost disjoint sets


When you can’t see the forest for the trees

Sometimes it happens that we are so confused amidst the complex relations and convoluted ideas that some simple pattern eludes us. This post is dedicated to a few problems, where it is easy to get lost in details and miss these obvious (or not so obvious) patterns. Although the solutions are explained, I urge everyone to wander into the forest (and maybe get lost in the woods) before exploring the paths I laid out. … More When you can’t see the forest for the trees

The Amazing Baire Category Theorem

One of the most fundamental theorems in mathematics is the famous Baire category theorem, which never ceases to amaze me. If you are a modern analysis enthusiast like me you probably know the feeling when you are working on a seemingly difficult problem and the solution appears to elude you every time you think about it, and then all of a sudden, snap! “Oh, I can just use the Baire category theorem.”

This post is dedicated to this feeling. … More The Amazing Baire Category Theorem

On the nonemptiness of the spectrum

This time we shall explore a topic which really fits the name of the blog. It it a well known and elementary fact that every complex matrix has an eigenvalue and an eigenvector. But, if we think of the matrix as a linear transformation between finite dimensional vector spaces, then what about the spectrum for linear transformations acting on infinite dimensional spaces? The answer for this question is interesting and it utilizes a very beautiful extension of complex analysis for vector valued functions. … More On the nonemptiness of the spectrum

Matrices and graphs

Our goal for today is to introduce a connection between matrices and graphs and to illustrate the strength of graph theoretic methods in linear algebra and vice versa. What is shown here is only the tip of the iceberg. These methods turned out to be very fruitful and they have many applications inside and outside mathematics. … More Matrices and graphs

The Biologist, the Mathematician and the Stalker

The Biologist and I were waiting in a dirty bar, slowly sipping our cheap and bitter beer. The Stalker arrived early in the morning, the fog hasn’t even settled yet. We jumped inside our car (without roof, of course, as the Stalker requested), then slowly started our trip into the Zone. After sneaking past a few soldiers… wait, no. Scratch that. Here is the real story of our trip into the Chernobyl exclusion zone and to the ghost city Pripyat. … More The Biologist, the Mathematician and the Stalker

Potential theory and asymptotics of the zero distribution of orthogonal polynomials

This week I gave an introductory talk at the Tenth Summer School on Potential Theory about potential theoretic methods in the study of orthogonal polynomials. This post is based upon my notes for that lecture. The aim of that talk was to prove a theorem about the weak-* convergence of normalized counting measures for zeros … More Potential theory and asymptotics of the zero distribution of orthogonal polynomials