Student research

Interested in pursuing an individual research project in mathematics? We encourage you to pursue independent research under the supervision of a faculty mentor. Student research activities include:

Honors theses
Inter- or intradisciplinary collaborative research on cutting-edge problems from a specific field of computer science
Reading courses on advanced topics
Qualified self-designed projects

Recent research projects

Differential Geometry and Brain Images

Franklin Feingold

Mentor: Gideon Maschler

In recent years, there have been multiple studies that use techniques in differential geometry to analyze brain MRIs. Franklin’s project focused on exploring some of the methods used and understanding how differential geometry can be used in modern-day medicine.
Topics in Differential Geometry

Emma Kirkman-Davis, Tenzing Gurung, Wenwen Shen

Mentor: Gideon Maschler

Emma, Tenzing, and Wenwen all worked on topics in differential geometry. Their work focused on curves and surfaces, lengths on these, intrinsic and extrinsic curvature, and the relation between curvature and topology (i.e., global geometry, the Gauss-Bonnet Theorem).
Counting Complicated Combinatorial Sets using Markov Chain Monte-Carlo Algorithms

Trung Ngo

Mentor: Michael Satz

Can you imagine a set that is finite but impossible to practically count even with a supercomputer? One way to build such a set is to define it as a subset all permutations of the integers 1 through N. The hard-to-count set may not be particularly large, but the parent set (which has N! elements) may well be too large for an exhaustive check-and-count approach. Monte-Carlo Markov chain methods introduce randomness to estimate the sizes and other features of such complicated combinatorial sets. Trung Ngo applied four algorithms to one such counting problem. The algorithms were implemented in Python and tested and analyzed for performance, convergence, and accuracy.
Image Processing and Drosophila Embryos

Logan Bishop-Van Horn, Teodor Nicola-Antoniu

Mentor: Jacqueline Dresch

Using raw data from microscope images in modeling gene expression levels presents the researcher with many challenges; one must be able to remove any extraneous information from the image and be able to compare images prepared and taken from different embryos on different days. For this reason, Logan and Teodor’s projects focused on creating a pipeline for processing these images, including noise and background subtraction, normalization, spatial registration, and extraction of quantitative levels of gene expression.
Topics in Functional Analysis

Andrew Mezzi

Mentor: Gideon Maschler

Andrew worked on a project that encompassed many different topics from functional analysis: infinite-dimensional linear algebra, operators, and applications of solutions to ordinary differential equations.

During the summer, Andrew was also exposed to a variety of internet resources and professional tools of the working mathematician, including the arXiv, MathSciNet, LaTeX and TeX-related editors, as well as the MathOverflow and StackExchange websites.
Designing New Evolutionary Algorithms for Parameter Estimation

Mike Gaiewski

Mentor: Jacqueline Dresch

One of the most important steps in mathematical modeling of a biological system is fitting the parameters of your model to experimental data. There is an extremely large number of different parameter estimation algorithms available, and an important question to ask is: what parameter estimation method should be used on my particular problem? In an attempt to answer this question in the context of modeling gene regulation, Michael developed and implemented novel parameter estimation algorithms based on the principles of evolution.
Bioinformatic Analysis of Transcription Factor Binding Sites

Navid Al Hossain, Regan Conrad

Mentor: Jacqueline Dresch

Predicting the location of protein binding sites within a genome is a difficult task, but is an integral part of furthering our understanding of gene regulation.

Navid’s project focused on analyzing the correlation between the frequency of binding sites within raw sequence data and the bioinformatically predicted ‘strength’ of these binding sites. His work has been instrumental in the experimental design and continued collaboration between the Dresch group in the Mathematics and Computer Science Department, the Drewell lab in Biology, and the Spratt lab in Chemistry.

Regan’s project focused on analyzing bioinformatic predictions of binding sites within core promoter regions from Drosophila melanogaster to investigate potential nucleotide dependencies and improve the predictive power of these algorithms.
Our Universe as a Manifold

Mateo Gomez

Mentor: Amir Aazami

This research project gave a rigorous introduction to understanding spaces that locally look just like our familiar space, but globally may be curved. The perfect example of this is Earth: locally it looks flat, but globally it is a sphere. In fact, our very universe itself is assumed to have this local/global property, but being four-dimensional — three dimensions of space and one of time — we cannot “see” this as we can with Earth. Such spaces are called manifolds. The key question is: how does one do calculus on manifolds? How does one study their geometry, their curvature? Everything from the ground up has to be re-formulated: differentiability, vectors, the gradient, the Jacobian of a function, etc. All this needs to be understood before one can even begin to talk about geometry, or curvature, or gravity, or the universe and Einstein’s equations.

Diving into Research

In addition to independent research projects, our department periodically provides opportunities for first-year and upper-class students to work in groups with faculty members on research projects through the course Diving into Research (MATH110, MATH 111). MATH110 Diving into Research is a year-long opportunity for first-year students to work in groups with faculty members on research projects.

Sample course descriptions

Mathematics behind Plasmas

Plasma televisions, plasma lights, the heat around the space shuttle and communication blackout caused by plasma, laser treatments in medicine, and production of microchips for computers are just a few applications of plasmas that became a big part of our lives. Students will learn about plasmas by developing and studying mathematical models that explain the experiments and help to obtain plasmas with certain properties.

Instructor: Natalia Sternberg

Mathematics behind Gene Regulation

This course will introduce the idea of mathematically modeling gene regulation in a developing organism. Students will learn how mathematicians work with biologists to design simple experiments and derive equations to model gene expression. We will also explore some of the computational approaches currently being implemented in modern biology, including bioinformatics, data processing, and parameter estimation. This one-year course will be an interactive experience for students interested in learning more about the interface of mathematics, computer science, and molecular biology.

Instructor: Jacqueline Dresch

Geometry

Geometry is a branch of mathematics which closely relies on visual intuition. As such, parts of it are accessible even without obtaining extensive preliminary background, while still being deep and thought-provoking. In this seminar-style class, we explore the subject from a number of different perspectives, thus demonstrating its richness. Among the possible topics chosen are Projective and Differential Geometry, and symmetries and their relation to the mathematical concept of a group. Our guiding principle for these choices will be their accessibility to direct geometric intuition and imagination. The need for prior mathematical background will be kept at a minimum level. We will also be employing computer graphics and related software for visual exploration. The main purpose is to have fun while appreciating geometry.

Instructor: Gideon Maschler