The Kakeya problem was proposed in 1917, by the Japanese mathematician Soichi Kakeya. The problem states,
In the class of figures in which a segment of length 1 can be turned around through 360˚, remaining always within the figure, which one has the smallest area?
In this talk I will give a very brief introduction to the Kakeya problem. I will give connections with harmonic analysis and some other field. I will also talk about the recent progress on the problem.
Lill's Method is a geometric approach for finding roots of polynomials with real coefficients. We will prove Lill's method, with examples, and then we will prove some familiar polynomial results in this new unfamiliar way. If there is time, we will also prove a generalized Lill's Method.
Any two distinct points define a unique line, and any two distinct lines intersect in a unique point. Have you ever wished that this statement were true? Has the idiosyncratic behavior of parallel lines always troubled you? Then fret no longer! This brief introduction to projective geometry will put parallel lines in their proper place and set your mind at ease. [Warning: it may also raise questions equally as distressing or more so.
Imagine for a moment that you are sitting through a boring lecture and you start playing with your calculator. You input a number into the calculator and then hit the cosine key. Then you keep hitting the cosine key and look at what happens to the numbers. Do they converge to a ?finite number or do they tend to in?finity? Do the numbers start to repeat in a pattern, or do they continue to change without any pattern? This is an example of a discrete dynamical system. Discrete dynamical systems have important applications in biology and other sciences as well as being interesting on their own. In this talk, we will discuss the properties of discrete dynamical systems and some tools that can be used to determine the behavior of discrete dynamical systems. No previous knowledge of dynamical systems is required to understand the talk, and there will be only one theorem.
Self-dual codes show good error-correcting performance. The Gleason-Pierce-Ward Theorem is useful for telling us when self-dual codes may exist. We will present the theorem and outline its proof.
A magic square is a square matrix with nonnegative integer entries whose line sums are all equal. Finding magic squares for a fixed size and line sum makes for fun puzzles, but how do we know when we've found them all? This question will be explored, revealing deep underlying mathematics that can be applied in much more generality.
