# Seminars & Colloquia Calendar

## A Muddle Puddle Tweetle Poodle Needle Noodle Bottle Paddle Battle (Season 2)

#### Scott James

Location:  Hill 701
Date & time: Friday, 29 September 2023 at 2:00PM - 3:00PM

In 1917, Abram Besicovitch was considering the following question in Riemann integration: Suppose $$f: \mathbb{R}^2 \rightarrow \mathbb{R}$$ is a Riemann integrable function. Does it follow that there exists an orthogonal basis x,y such that $$\displaystyle\int f(x,y) dx$$ exists as a Riemann integral for all y, and is Riemann integrable as a function of y? What he noticed was that if he could construct a set of measure 0 that contains a unit interval in every direction, he could come up with a counterexample.

Also in 1917, Soichi Kakeya was considering the following question: What is the smallest (in measure) set in $$\mathbb{R}^2$$ that you can take a needle of length 1 and rotate it 180 degrees? In 1921, Gyula Pal showed that, if we require the set to be convex, then you can't do better than an equilateral traingle of height 1. When Besicovitch heard of this problem after he constructed his measure 0 set, he used his set to prove that you can make such a set to have arbitrarily small measure. Note that measure 0 is impossible as we are rotating the needle, and that will always sweep out positive area.

In this pizza seminar, I will go over a construction due to Oskar Perron, now called a Perron tree. From this, we will construct a set of measure 0 with a unit interval in every direction. We will also show that we can have a set with arbitrarily small measure in which we can rotate a needle 180 degrees. Then, given time, I will define the Hausdorff measure, state/prove some facts about it, and then state one of the most famous conjectures of harmonic analysis, the Kakeya Conjecture, along with the Kakeya Maximal Conjecture, which, if true, implies the Kakeya Conjecture.

Prerequisites to this talk include: area of a triangle; area of a slice of pizza; geometric sums; the ability to categorize all global sections of a line bundle on a reducible complex space of complex dimension 17.

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