In Notes 0, we introduced the notion of a measure space , which includes as a special case the notion of a probability space… 6,973 more words

## Tags » Math.CA

#### 275A, Notes 0: Foundations of probability theory

Starting this week, I will be teaching an introductory graduate course (Math 275A) on probability theory here at UCLA. While I find myself *using* probabilistic methods routinely nowadays in my research (for instance, the probabilistic concept of… 7,754 more words

#### Equidistribution for multidimensional polynomial phases

The equidistribution theorem asserts that if is an irrational phase, then the sequence is equidistributed on the unit circle, or equivalently that

for any continuous (or equivalently, for any smooth) function . 2,002 more words

#### A differentiation identity

Here’s a cute identity I discovered by accident recently. Observe that

and so one can conjecture that one has

when is even, and

when is odd. 131 more words

#### Cancellation for the multilinear Hilbert transform

I’ve just uploaded to the arXiv my paper “Cancellation for the multilinear Hilbert transform“, submitted to Collectanea Mathematica. This paper uses methods from additive combinatorics (and more specifically, the arithmetic regularity and counting lemmas from… 417 more words

#### Failure of the L^1 pointwise and maximal ergodic theorems for the free group

I’ve just uploaded to the arXiv my paper “Failure of the pointwise and maximal ergodic theorems for the free group“, submitted to Forum of Mathematics, Sigma… 1,445 more words

#### A remark on the lonely runner conjecture

The lonely runner conjecture is the following open problem:

1,296 more wordsConjecture 1 Suppose one has runners on the unit circle , all starting at the origin and moving at different speeds.