In the Alexander group, we worked on visualizations of the "unit sphere" in flat space-time. In space-time, or Minkowski space, the "unit sphere" consists of vectors of square norm 1 and -1. This gives the two parts of the sphere in space-time, the deSitter space, and two copies of hyperbolic space. This unit sphere has interesting properties including arc length with imaginary terms which "do the bookkeeping" for how many 'infinities' away one point is from another. Pictured is a plot of the sphere, a plot of a geodesic segment which crosses infinity once, and a plot of orthonormal bases in 2 dimensional space-time.

Polyrhythms feature prominently in many genres of music and involve two seemingly conflicting but related rhythms being played at the same time, such as triplets and sixteenth notes.
In advanced polyrhythms, even the measure length for each rhythm may be different.

This can be visualized as a path on a torus: looping around the outside of the torus denotes completing
a measure for one of the rhythms, and looping through the torus denotes completing a measure in the other rhythm.
The full closed path represents the total length of time passing before the two rhythms start their measure on the same beat again.

We are studying the fine distribution properties of saddle connections on the octagon. We consider a regular octagon with opposite sides identified by translations and define a saddle connection to be a straight line path that connects corners. These trajectories are of great interest in the study of billiard paths in the octagon and associated right triangles. We are interested in doing both numerical experiments and theoretical work to understand the distribution of the gaps in slopes (equivalently, angles) between saddle connections of length R (as R approaches infinity). The same question has already been studied extensively for the Golden L.

Wigner matrices are irreducible representations of SU(2). These matrices have a large amount of both rough and fine structure; specifically, the maxima of a Wigner matrix form an ellipse related to the angle of rotation matrix, and outside of this boundary the elements approach zero (frozen regions). The results about the boundary shape have been proven, but the origin of the frozen regions remains a mystery. We explored the Wigner matrix in several different dimensions in an attempt to understand these frozen regions. In the future we hope to apply our work to other Lie groups.

Number fields associated to hyperbolic 3-manifolds: Given a finite collection of high precision complex numbers related to the invariants of hyperbolic 3-manifold, the goal of the project is to recover the number fields generated by those complex numbers. To realize this, we need to find the primitive polynomial, i.e., the irreducible polynomial of integer coefficients whose roots generate the number field we want to find. Even though such problem is generally NP-hard, LLL algorithm could solve it in polynomial time with high precision. After understanding the principle of LLL algorithm, we are now trying to implement it using Sage, a mathematical programming language based on Python, starting from the simple functions like Gram-Schmidt reduction. After further implementation of LLL algorithm, we may implement other algorithms like PSLQ and compare them.

The Mathematics Department at UIUC holds one of the world's largest collections of mathematical models dating to the late 19th and early 20th century.
These date to a time when there was no other way to adequately illustrate mathematical formulas and concepts for instructional and research purposes.
While many of the models at UIUC have been photographed, they are not currently documented and displayed in a manner useful to mathematicians.
As of the end of Spring 2012, we have documented 169 of the estimated 380 plaster cast, string, metal, glass, plastic, and paper models in the collection.
We have found that Ludwig Brill (Mathematical Institute of the Royal Poltechnicum, Munich), Martin Schilling (Halle and Leipzig), and Arnold Emch (University of Illinois)
published most of our models.

We plan to continue documentation of the balance of the collection, followed by identifying the maker, title, and mathematics behind each piece.
We will determine a proper arrangement, and provide interpretation in a manner that will make them relevant for today's researchers.
Later in the project, the development of a website is anticipated, including images of the models, their history and significance, paired with virtual versions that may be manipulated.

We study the distribution of parameters for the best approximation of the area under a curve.
In particular, we relax the condition that the lengths of the bases of the shapes used in a Riemann
sum must be the same. More precisely, take a continuous non-negative function and let be a partition of
a finite interval [a,b] so that . We say Pkis optimal for left Riemann sums, when
is a partition in which the left
Riemann sum has the maximum value for a fixed k over all possible distributions of the partition
points. In other words,

maximizes . As a starting point, we considered only the unit interval [0,1] and a simple function f that acts nicely, namely, . We succeeded in not only finding the optimal distribution of the partition points, but also finding a nice recursive formula that allows us to identify the behavior of the partition points as the number of them increase. Also, the techniques we used allowed us to generalized to ; yet, the techniques were not applicable to more general functions. We also succeeded in finding the optimal partition for trapezoidal Riemann sums for .

Our project involved visualizing the fractals generated from systems of contraction mappings known as an iterated function system (IFS). If we iterate the family of functions starting with an arbitary set, we get a sequence of sets that converges to a unique fixed point set, which is the fractal associated with the IFS. An example of this is the Sierpinski triangle, which turns out to be the fractal associated with three maps, the maps being contractions by a factor of two towards three points in the plane. The easiest way to generate the Sierpinski triangle is to:

1. Select a random point in the plane.

2. Select a random function from the system and apply it to the point.

3. Plot the point.

4. Repeat.

As a trial run, we decided to create morphings between different IFS's. For example, if F is the IFS for the dragon curve and G is the IFS for the Sierpinski triangle, we can linearly interpolate these maps to obtain a smooth morphing between the two fractals. This should give us an animation of the dragon curve morphing into the Sierpinski triangle.