# Illinois Geometry Lab

## IGL Projects, Spring 2016

• #### Introduction to Stochastic Simulation Algorithms

##### Faculty Mentor: Ikemefuna Agbanusi

The goal of this project is to answer questions like: - If a patient takes a certain drug or medication, how long, on average, will it take till they feel its effect? - DNA and RNA molecules often require the presence of specific catalysts before processes like transcription etc, can take place. How long, on average, does it take a protein molecule to find a specific target site? What if the target is moving? These two questions are closely related and the goal of the project will be to explore how to answer them using some basic mathematical tools and numerical simulations.

Weekly
• ##### Project Difficulty
Intermediate
Completion of Calculus 3. Familiarity with ODEs at the level of Math 285. Basic Probability. Familarity with software like Matlab.
• #### Brownian motion and diffusion processes

##### Faculty Mentor: Jing Wang

We will construct and visualize sample paths of 2-dimensional Brownian motion as well as Kolmogorov process, as the scaling limit of random walks. We observe and then analyze the essential difference between diffusion processes that are non-degenerate and strongly degenerate.

• ##### Team Meetings
once a week
• ##### Project Difficulty
Intermediate
Completion of Calculus 3. Good coding skills are preferred. Have taken and have a good understanding of probability are preferred but not required.
• #### Randomness in Number Theory

##### Faculty Mentor: A J Hildebrand

Random-like behavior is ubiquitous in number theory. For example, the primes, the digits of pi and other famous constants, and the Moebius function and other number-theoretic functions, all appear to behave much like appropriately defined "true" random sequences. In this ongoing project we seek to explore such random features experimentally - via large scale computations and geometric visualizations as random walks - and, if possible, also theoretically. In past years we have investigated randomness associated with quadratic residues, the Moebius function, number-theoretic Fourier series, and sum-of-digit functions. This year (2015/2016), we are focusing on the behavior of leading digits of various classes of arithmetic sequences. Our initial investigations have confirmed some of the expected random-like behavior, but also revealed some striking non-random features and fractal-like features that have yet to be fully explained. We plan to continue these investigations in the spring. For more information, desired skills and prerequisites, and availability of spots visit Professor Hildebrand's webpage.

• ##### Team Meetings
Twice a week for 1-2 hours each.
• ##### Project Difficulty
Math 453 or an equivalent course with a grade of A or better. Strong programming skills in C/C++ or Mathematica desired.
• #### Calculus, Geometry, and Probability in n Dimensions

##### Faculty Mentor: A J Hildebrand

This project is part of an ongoing program, begun in Fall 2012, aimed at seeking out and exploring interesting problems in n-dimensional calculus and geometry that are accessible at the calculus level, but rarely covered in standard calculus courses. These problems typically arise as natural generalizations of familiar problems in 2 or 3 dimensions, and they are often motivated by applications to probability and statistics or other areas. In past projects we have considered intersections of cylinders in n dimensions, generalizations of the Broken Stick Problem (if a stick is broken up randomly into 3 pieces, what is the probability that the pieces form a triangle?), and the Random Triangle problem (if a triangle is chosen at random, what is the probability that it is acute?). This year (2015/2016) our focus is on the "geometry of voting", an intriguing geometric approach to voting theory in which voter preferences and election outcomes are represented by points in the plane or in a higher-dimensional space. This approach allows one to visualize different voting methods, and it can help explain voting paradoxes. For more information, desired skills and prerequisites, and availability of spots visit Professor Hildebrand's webpage.

• ##### Team Meetings
Twice a week for 1-2 hours each.
• ##### Project Difficulty
Intermediate
In general, students should have taken Math 241, preferably in the honors version, with a grade of A+ or A. In exceptional cases, students who are concurrently enrolled in Math 241 Honors may also be considered if they have an otherwise strong math background, strong leadership and communication skills. Strong programming skills, especially with Mathematica, are highly desirable, though not required.
• #### Discrete Morse Theory, Vector Fields, and Materials Science

##### Faculty Mentor: Ruth Davidson and Rosemary Guzman

The goal of this project is to compare properties of various materials used in engineering applications using techniques of topological data analysis derived from Discrete Morse Theory. We will perform these comparisons by studying the actions of discrete gradient vector fields on cell complexes using a variety of tools such as Mathematica and command-line open source software.

Weekly
• ##### Project Difficulty
Intermediate
Completion of Calculus 3, curiosity about topics such as those covered in materials science courses offered by majors such as AE, MSE, TAM, or CHEM, curiosity about the interplay between theoretical advances in mathematics and algorithm and software design, and willingness to use command-line, open source software and learn basic scripting skills.
• #### Plasticity and Synchrony for Coupled Oscillators

##### Faculty Mentor: Jared Bronski

The most basic principle of learning and pattern formation in neuroscience is the Hebbian paradigm, which can be stated simply as "Cells that fire together, wire together". More explicitly, when cells have similar firing patterns, the synaptic coupling between them grows stronger; when different, weaker. A simple model of this type of system is one in which we model each of the cells as an oscillator and the synapse by a nonlinear coupling. Thus, a mathematical description of this model is an ODE defined on a graph where the edge weights are allowed to evolve. Said yet another way, we want to understand the behaviour of nonlinearly coupled oscillators when the connection strengths between these oscillators are allowed to vary. These types of systems aren't very well-studied, so even some systematic numerical simulations of these could be interesting. Proving something analytic about them would be even more so.

biweekly
• ##### Project Difficulty
Intermediate
Completion of Calculus 3 and an ODE course (285/286/441). Familiarity with software for scientific computation (e.g. MATLAB) preferred.
• #### Combinatorial models of surfaces

##### Faculty Mentor: Mark Bell

Surfaces can be made by gluing together polygons. By restricting ourselves to different types of polygons, we obtain many different combinatorial models a surface - each of which has its own advantages and disadvantages. This project focuses on building techniques for translating objects on surface, in particular loops, from one model to another.

• ##### Team Meetings

Initially weekly, then biweekly

• ##### Project Difficulty

Intermediate

Completion of Calculus 3. Knowledge of Python may be beneficial.

This project is based upon work supported by the National Science Foundation under Grant Number DMS-1449269. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

• #### Visualizations of 3-manifolds

##### Faculty Mentor: Mark Bell

The primary aim of this project is to construct new tools for visualizing 3d spaces by taking advantage of hyperbolic geometry. We will be particularly interested in the case when the space locally looks like a stack of surfaces and how we can illustrate how these surfaces run through the space. An additional goal of this project is to add these visualization tools to SnapPy.

• ##### Team Meetings

Initially weekly, then biweekly

• ##### Project Difficulty

Completion of Calculus 3. Knowledge of MATH 402 - Non-Euclidean Geometry and OpenGL or a similar API may be beneficial.

This project is based upon work supported by the National Science Foundation under Grant Number DMS-1449269. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

• #### Distributions of loops on surfaces

##### Faculty Mentor: Mark Bell

There are many different ways of assigning a "length" to path or loop on a surface which determines the geometry of the surface. For any particular choice one can study the lengths that occur and ask questions such as what is the length of the shortest loop? or what is the distribution of the lengths of loops? This project will focus on computing these distributions when the underlying geometry is "flat". That is, when large pieces of the surface look like the Euclidean plane. This assumption will allow us to reduce a much of the work to studying Euclidean spaces. Time permitting, we may also try to produce 3d-printed modules of these distributions for particular choices of metrics.

• ##### Team Meetings

Initially weekly, then biweekly

• ##### Project Difficulty

Intermediate

Completion of Calculus 3. Students may benefit from having taken MATH 403 or MATH 525.

This project is based upon work supported by the National Science Foundation under Grant Number DMS-1449269. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

• #### Interactive Learning Tools for Linear Algebra

##### Faculty Mentor: Cary Malkiewich and Jenya Sapir

We are creating a sequence of games that give players an intuitive understanding of linear algebra concepts such as linear transformations, null and column space, linear independence, and eigenvalues and eigenspaces. We need students with knowledge of linear algebra to be part of an interdisciplinary game development team. Participants would be involved in coding interactive "toys," that give an understanding of key concepts through dynamic visuals and organic play, as opposed to direct instruction and mathematical formulas.

Weekly
• ##### Project Difficulty
Intermediate
Completion of MATH 415 or MATH 416, or equivalent.
• #### Dynamics of subterranean motion

##### Faculty Mentor: Richard Sowers

Widespread sensing will be a hallmark of the future. One place where such sensing is having noticeable impact is precision agriculture; widespread use of drones provides imagery which can can give new and meaningful insights.

HOWEVER, drones can't see much beneath the surface of the earth.

Taking a cue from nature, however, we wonder if sensing could be done by a mechanical worm. In this project, we would like to understand and visualize the dynamics of worm motion. This project will involve some research (since I myself don't fully understand the equations) and coding.

• ##### Team Meetings
once a week for 30 minutes.