# Fall 2019

During Fall 2019 MEGL ran a 7 project program with 19 participants (faculty, graduate, and undergraduate students). The six research/visualization groups were titled:

• Computations of test ideals of Big Cohen-Macaulay modules,
• Wave Fronts in DTDS Population Models,
• Decision in Finance from the Knapsack Point of View,
• Statistics in Deformations of Large Knots,
• The Geometry of Self-Driving Cars, and
• Visualizing Bruhat-Tits Buildings.

Additionally, there was one public engagement group which we refer to below as Outreach. The research/visualization groups engaged in experimental explorations involving faculty, graduate students, and undergraduates. Teams met weekly to conduct experiments generating data, to make conjectures from data, and to work on theory resulting from conjectures. The outreach group involved faculty, graduate students, and undergraduates to develop and implement activities for elementary and high school students that were presented at local schools and public libraries. We concluded with an end of term symposium and a poster session (scroll down for pictures).

In our research, we compute the test ideals of finitely-generated CM (MCM) modules of various rings. The rings we study include sub-rings and quotients of polynomial rings and power series rings. The test ideals of such rings yield information about the rings’ singularities and geometric properties: vaguely, the larger the test ideal, the less singular the ring, and vice versa. If the ring is non-singular, the test ideal coming from any MCM module is the whole ring.

The aim of this research is to better understand population growth models and describe the speed at which populations expand. For our project, we chose to focus on a discrete-time discrete-space (DTDS) model. Population sites are nodes on an infinite, one-dimensional lattice. After each generation, each population site undergoes migration, where a proportion of the population migrates to adjacent nodes, and reproduction, where the population grows by some factor.

If you drop some small amount of population in an empty lattice, the population will migrate and grow until some nodes hit their carrying capacity. Once this happens, a population “wave” forms and moves outward with a certain speed. Our project studies the structure and speed of these waves. So far, we have been able to describe waves with speed 1, 1/2, and all speed 1/n waves. In future research, we hope to be able to describe waves with any rational speed, and even irrational speed waves.

The knapsack problem is a type of optimization problem that has been widely studied and used in computer science and mathematics for at least a century. It has numerous applications in business and resource allocation, which leads us to a financial application. Namely, we are studying the knapsack problem as it applies to optimizing a financial portfolio. Additionally, we consider specific assumptions about our financial application that allow us to solve and optimize solutions in a reasonable time.

An example of a knapsack problem applied to finance is the following
investment-related problem: An investor is looking to buy from a collection of
companies $c_1, c_2, … , c_n$ with corresponding prices $p_1, p_2, …, p_n$. For
these prices, we assume the change in price, $\Delta p_i(t) = p_i(t+1) – p_i(t)$,
is constant around time $t$. With these assumptions, we want to maximize $\sum_{i}x_i \Delta p_i$, while respecting the constraint that $\sum_{i}x_ip_i \leq B$,
where $B$ is the budget of the investor and $x_i \in {0,1}$.

Three examples of specific cases of the knapsack problem that we analyzed
include: the 1-weight case, the 2-weight case, and the 3-weight case. In the
1-weight case of the $0-1$ knapsack problem, every $p_i$ of every item in the
constraint equation is equal. An algorithm to solve the 1-weight case could
simply sort the items in order of decreasing value and choose as many (of the
most valuable items) as possible. The 2-weight case allows 2 distinct values $p_1$
and $p_2$; each item must have a weight of $p_1$ or a weight of $p_2$. An
algorithm exists that can solve the 2-weight case in $O(n\log n)$ time. In fact
the sorting step of the algorithm requires $O(n \log n)$ time, while the rest
of the algorithm can be done in linear time. Finally, there is an algorithm for
the 3-weight case that is similar to that for the 2-weight case, but it solves
a 3-weight problem in $O(n^2)$ time.

We made graphs of the output of these algorithms. We observed the shapes
generated by the 2-weight algorithm’s output, which sometimes resembles a
parabola. We investigated the “jaggedness” of the output, which is
presumable caused by the restriction of variables to integers.

The goal of this project is to discover trends in knots based on a measure of “complexity.” Here complexity is measured by a specific knot invariant: the Krull dimension of the character variety of the knot group. For our project, we need only look at ‘large” knots as all “small” knots have dimension 1. Given a knot $K$, we look at its complement in 3-space, $K^C$. Then, we take the fundamental group of this space $\pi_1(K^C)$ and calculate the character variety, $\chi(\pi_1(K^C))$. Finally, we calculate the Krull dimension of the character variety. This dimension is a knot invariant; that is, if two knots yield different dimensions, they are fundamentally different.

Project Page

As Self-Driving cars are becoming more popular, there is a need to minimize the length or time in order to minimize cost. For this, we must first understand the geometry of Self-Driving cars. Since Self-Driving cars have a turning radius, we cannot use Euclidean distance to measure path-length. This problem resembles what Lexter Eli Dubins described in his 1957’s research paper, On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal Positions and Tangents, a paper in which he explained the possible optimal paths between two points given a constraint on curvature.

The Bruhat-Tits building associated to a connected reductive algebraic group G is a geometric realization of this group’s parabolic subgroup structure. They are used in the classi fication of maximal compact subgroups of G, but also in the study of smooth representations of G.
Aside of their mathematical relevance, they are visually attractive objects. The goal of this project was to develop some basic methods for visualizing them. This has been successful, and it is clear that the resulting images are compelling enough to be used in outreach. For example, they might serve as a tool to talk about symmetries in mathematics.
The building is realized by identifying its chambers with cosets of a Borel subgroup $B\subset G$. The most important part of the method then is finding canonical representatives for these cosets. We achieved this for a split reductive group by making use of both the Bruhat decomposition of G, and the root group factorization of B.

This semester MEGL Outreach ran 32 activities at 15 locations, reaching almost 1,000 students. We also grew our network and worked closely with Title I schools to get our programs to more under-served students. In addition, our outreach director was selected by the USA Science and Engineering Festival as a Nifty Fifty speaker, a group of science and engineering professionals that share their work at elementary, middle, and high schools around the country. We also, for the first time, announced MEGL Outreach internships for the Spring semester, so that undergraduates can get involved in the development and implementation of our outreach events.