September 16, Brian Conrad - How to find real solutions to real equations

Work of Julia Robinson and others on Hilbert's 10th problem proved that there does not exist an algorithm to determine if an arbitrary system of several polynomial equations in several variables over the integers has an integral solution. It is generally believed (but not proved) that the analogous problem over the rationals also has a negative answer (thereby keeping number theorists gainfully employed).

Over the real numbers, the situation is completely different. A theorem of Alfred Tarski ("elimination of quantifiers") gives an algorithm to do the following remarkable thing: for an arbitrary system of finitely many polynomial equations and inequalities in several variables over the real numbers, there exists a real solution if and only if the collection of coefficients satisfies an auxiliary finite system of equations and inequalities (that can be constructed by an explicit algorithm, depending on the "shape" of the initial system to be solved). The most elementary interesting example of the theorem is this: a quadratic polynomial over the real numbers has a real solution if and only if its discriminant is non-negative.

We will discuss the framework in which Tarski's theorem takes place (the theory of semi-algebraic sets), we shall state the geometric form of the theorem he proved, and we shall present one detailed example that illustrates the method of the proof.

September 23, Stephen DeBacker - What's on the lobby floor anyway?

When you enter the math department through the first floor lobby, you walk on a design in the tiles which looks like part of the second picture here. What is it? Is it significant? Why is it there?

In this talk, we will not be so ambitious as to attack the latter questions; we will leave that for other speakers. Instead, we provide an elementary introduction to tilings and some of the mathematics connected to them. We will then discuss the tiling on the floor (an example of a Penrose tiling).

September 30, Kevin Tucker -- Flip-Turns in the Mathematical Swimming Pool: Symmetries and the Rubik's Cube

Assuming nothing but enthusiasm, interest, and the intuition you already have, we will formalize the notion of SYMMETRY by developing the language of Groups. After elucidating some pertinent examples, we will investigate the relation of Groups to the RUBIK'S CUBE. Without giving an explicit construction, we present all the necesary tools and outline a powerful method for solving the Rubik's cube AND ALL SIMILAR PUZZLES!

October 14, Thomas Fiore -- Beethoven and the Torus

One of the central concerns of music theory is to find a good way of hearing a piece of music and to communicate that way of hearing to others. Music theory gives us tools to turn our chaotic aural impressions into organized, tangible ideas. In the past thirty years, music theorists have come to rely on the formidable powers that pure mathematics has to offer. Group theory, graph theory, and topology provide the working music theorist with means to find new ways of hearing old pieces and to make sense of seemingly chaotic modern music. Composers and performers also draw upon the richness of these ideas. In this talk, I will describe the musical neo-Riemannian group and its associated graph. We will use this group to organize our chaotic aural impressions of the Beatles and Beethoven into tangible ideas, and we will find a torus in Beethoven's Ninth Symphony along the way! I will not assume any knowledge of the above subjects, so the talk should be accessible to anyone who enjoys music.

October 21, Paul Siegel -- Beyond the Rubik's Cube - Permutation Puzzles Based on Sporadic Groups

Group theory is infamous for being among the most abstract areas of mathematics. Fortunately, group theory has the advantage over other fields erroneously stigmatized by this reputation in that a good deal of insight can be gained simply by playing with toys, such as the popular Rubik's cube. I will discuss ways in which groups that possess a more complicated and less intuitive structure than the group underlying the Rubik's cube can be used to develop similar puzzles, and I will present my own computer implementation of some examples of such puzzles. I will also give a rough idea of how these games can be solved and what it means to do so, and then use the visual implementation to discuss important group-theoretic concepts like subgroups, orbits, transitivity, group action, etc. in a way that appeals to intuition rather than raw formalism. I am myself a novice in the areas of group theory and algebra, so no background is required or expected.

October 28, Leo Goldmakher -- On oranges and suns: The Banach-Tarski Paradox

The Banach-Tarski Paradox is commonly explained as saying that one can take an orange, cut it into a finite number of pieces, and reassemble these pieces into a solid sphere the size of the sun! In fact, the actual statement of the Banach-Tarski Theorem is even weirder than this. Surprisingly, the proof, although long, does not involve any advanced techniques; prerequisites are knowing just a bit about matrices, and (more importantly) having the patience to sit through an hour-long math lecture!

November 4, Zach Teitler -- Knots, DNA, and Singularities

Knots are fun to study and easy to think about. Yet, despite over a hundred years of intensive study, there are still many things we don't know about knots, and knot theory is a very active area of research. One exciting recent development is the connection between knot theory and biology, where DNA strands are studied in terms of knot theory. In addition, knots relate to many areas of mathematics.

We will talk about knots and see lots of examples. Then we will discuss how mathematicians were able to help biologists understand DNA a little better! One question concerns how enzymes can copy a strand of DNA while it is knotted and then separate the two strands from each other without getting them all tangled up. Another question asks how the knottedness of a piece of DNA affects the speed at which it can be pulled through fluid, such as cellular fluid or an electrophoretic gel. Also, understanding the shape of DNA strands helps us understand how viruses like HIV affect the DNA.

One place that knots turn up within mathematics is in the singularities of plane algebraic curves. For example, the curve given by $y^2 = x^3$ has a cusp (a sharp corner), and the associated knot is a trefoil; for two lines that cross, the associated knot is two circles linked together. We can use the knot to help visualize the singularity. These are fun to work out---and if you're interested, I have a couple of projects we could work on together.

November 11, Ravi Vakil -- The mathematics of doodling

Doodling has many mathematical aspects: patterns, shapes, numbers, and more. Not surprisingly, there is often some sophisticated and fun mathematics buried inside common doodles. I'll begin by doodling, and see where it takes us.