# Quarterly Research Projects for Undergraduates

Please note that this is a dynamic list each quarter. Check back regularly to see if new research positions have been added. If you are interested in working on an area of research not represented on this list below, you are encouraged to contact faculty directly who are doing work in that area. A list of math faculty, including their research areas, is available here.

For more information about research units and the benefits of research, please visit the Math Undergraduate Research homepage.

**Apply for Winter 2019 Research: **The application for Winter 2019 research opportunities is available online. Please note that you will need the Application Code (from the section below) for the position(s) you are interested in. If you are interested in multiple research opportunities, submit a separate application for each research project.

Link to application: https://goo.gl/forms/944vyn9v8Nb0xjk52

For full consideration, please apply before Thursday, January 31.

### WINTER 2019 Research Projects

**KNOT INVARIANTS AND NEURAL NETWORKS**

**Principal Investigator (PI):**Prof. Allison Moore

**Description:**There is a lot of publicly available, numerical data about knots and knot invariants. The goal of this project is to predict the behavior of certain knot invariants using neural networks. More specifically, we will design and implement a feed-forward neural network to make predictions about three and four-dimensional knot invariants.

**Requirements:**Linear algebra and experience with programming in Python, Matlab or R. No knot theory or topology background is required. Student will be expected to implement a feed-forward neural network in Python and/or to write helper functions that manipulate data and "vectorize" knots. Prior experience with neural nets is helpful, but this is not a requirement either.

**Application Code:**moore

**CELLULAR AUTOMATA**

**Principal Investigator (PI):**Prof. Janko Gravner

**Description:**The students would find interestting structures in cellular automata evolutions, and investigate the role of randomness in such dynamics.

**Requirements:**Some knowledge of probability, and willingness to do computer programming.

**Application Code:**gravner

**IS THERE A PROJECTION OF A KNOT WHOSE NUMBER OF UNKNOTTED DIAGRAMS IS EXPONENTIAL ON THE NUMBER OF CROSSINGS?**

**Principal Investigator (PI):** Prof. Mariel Vazquez and Carolina Medina

**Description: **

It is known that for every regular knot diagram D with n crossings, the number of unknot diagrams that can be obtained from D by crossing exchanges is at least 2^{\sqrt[3]{n}}.

(Umin(n)>= 2^\sqrt[3]{n})

We are interested in estimating an upper bound. As our fist approach we would like to investigate the Gauss Codes arising from canonical projections of toroidal knots T(p,q) when p and q are small integers. This will help us understand the behavior of these projections for larger p and q. The interested student will be involved in the implementation of an efficient algorithm for the automated calculation of certain knot invariants.

**Requirements:** Programming background, preferably C++ and/or Python. Ability to conduct numerical simulations with care, learn existing tool suites, and interest in the underlying biological motivation.

**Application Code:**vazquez

**DEVELOPING A LIQUID CRYSTAL MODEL FOR DNA INSIDE VIRUSES**

**Principal Investigator (PI):**Prof. Javier Arsuaga

**Description:**In certain viruses DNA is highly condensed and it is believed that the DNA molecule is found in the form of a liquid crystal. Motivated by these experimental observations we are interested in developing a model of liquid crystal for DNA under confinement. The project is open to the computer implementation of continuous or discrete models of liquid crystalline phases.

**Requirements:**Some background in PDEs or Physics is desirable. Programming experience, preferably C++ and/or Python. Ability to conduct numerical simulations with care and interest in the underlying biological motivation.

**Application Code:**arsuaga

**IMPROVING ALGORITHMS FOR FIXED SMALL PATTERNS**

**Principal Investigator (PI):**Prof. Chaim Even Zohar

**Description:**

**permutation**of size n is an ordering of {1,2,…,n}, for example 38261457, of size 8. This particular permutation contains 824 as a sub-sequence. We say that this is an occurrence of the

**pattern**312, since the map from 8,2,4 to 3,1,2 respectively preserves the order relations. Indeed, it maps the largest to the largest, the second largest to the second largest, and so on.

**contains**an occurrence of the pattern 312 or not? A naive approach would have to exhaust all (n choose 3) sub-sequences. This would have a running time of O(n^3), and in general O(n^k) for a fixed pattern of size k. However, it was shown by Guillemot and Marx (2014) that this can actually be done in linear time, O(n), where only the implicit constant coefficient depends on the fixed parameter k.

**counting**the number of occurrences of a fixed pattern in a large permutation of size n. Again, the naive approach requires time O(n^k), for a fixed pattern of size k. One can somewhat improve on that in some simple special cases, such as the patterns 12, 123, or 321.

**Requirements:**Background in discrete math and algorithms, basic programming skills.

**Application Code:**evenzohar

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### FALL 2018 Research Projects

**DETECTING PERCOLATION IN RANDOM ENVIRONMENTS**

**Principal Investigator (PI):**Prof. Javier Arsuaga

**Description:**Certain morphogenesis processes in molecular biology seem to be subject to percolation. In previous work we have developed algorithms to detect percolation in planar lattices and have done some preliminary work on detecting percolation in random environments (see reference below). In this project, one or two, student(s) will work together to develop algorithms to detect percolation in random environments in the plane and in spheres. The applications of these models include kinetoplast DNA and the formation of some viral capsids.

**Reference:**

**Requirements:**GPA at least 3.6, MAT128A and B or C. Interested in applications of mathematical physics to molecular biology. Excellent programming skills, excitement for interdisciplinary work. At least 10 hour/week commitment.

**TAU FUNCTIONS OF DRINFELD-SOKOLOV HIERARCHIES**

**Principal Investigator (PI):**Prof. Martin Luu

**Description:**Drinfeld and Sokolov constructed in the 1980’s interesting dynamical systems attached to suitable Lie algebras (such as square matrices with vanishing trace). For each point in the phase space one expects to have a special function, the so-called tau function, encoding important and subtle information. Only recently a general definition of this function has emerged. In this project the student will write computer code that can (with suitable initial data) actually calculate these functions.

**Requirements:**Most important is a good grasp of linear algebra as covered in MAT 67 as well as a strong background in programming.

**RENEWABLE ENERGY STOCHASTICS**

**Principal Investigator (PI):**Prof. David L. Woodruff (Graduate School of Management)

**Description:**Paid internship: Help develop, test, document and experiment with software for characterizing uncertain energy production from renewable energy sources.

**Requirements:**Coursework in probably and statistics, demonstrated skill as a Python programmer.

**MATHEMATICAL MODELING PROJECTS IN NEUROBIOLOGY & CARDIAC ELECTROPHYSIOLOGY**

*[Note: Not looking for new students until Spring 2019]*

**Prof. Tim Lewis**

**Investigator (PI):****Principal****Description:**Topics include:

**Requirements:**MAT 22AB necessary; MAT 119A and/or MAT 124 highly preferred. Some experience in computer programming is required, and a willingness to learn mathematical modeling and biology is essential.

**Application Code:**will be available closer to Spring 2019