# UROP

The Mathematics Department has hosted a wide diversity of Undergraduate Research Opportunities Program (UROP) experiences. Every year the Department hosts around 70 different UROP projects.

Here are some recent titles:

- "Modeling Droplet Dispersion to understand Disease transmission"
- "Quantum Groups and Hecke Algebras"
- "Modular Representations of Cherednik Algebras"
- "Combinatorics of the Bruhat Order"
- "Eigenvalues and Eigenfunctions of the Laplacian and Schrodinger Operators"

### Faculty UROP Coordinators:

More of these projects arose from conversations between the student and the advisor than as a project or idea proposed by the advisor. In almost every case the project represents individual research carried out by the student under the guidance of the advisor.

For information, visit the MIT UROP website.

### Advertised Opportunities:

### Mathematical Foundations of Big Data & Machine Learning

**Project supervisors: **
Dr. Jeremy Kepner and
Prof. Alan Edelman

#### Project description

Big Data describes a new era in the digital age where the volume, velocity, and variety of data created across a wide range of fields (e.g., internet search, healthcare, finance, social media, defense, ...) is increasing at a rate well beyond our ability to analyze the data. Machine Learning has emerged as a powerful tool for transforming this data into usable information. Many technologies (e.g., spreadsheets, databases, graphs, linear algebra, deep neural networks, ...) have been developed to address these challenges. The common theme amongst these technologies is the need to store and operate on data as whole collections instead of as individual data elements. This research explore the common mathematical foundation of these data collections (associative arrays) that apply across a wide range of applications and technologies. Associative arrays unify and simplify Big Data and Machine Learning. Understanding these mathematical foundations allows the user to see past the differences that lie on the surface of Big Data and Machine Learning applications and technologies and leverage their core mathematical similarities to solve the hardest Big Data and Machine Learning challenges. This projects seeks to strengthen the mathematical foundations of Big Data and Machine Learning. Participants will be paid.

**Website: **www.mit.edu/~kepner/

#### Qualifications:

Strong mathematical background. Experience with Matlab is helpful, but not a requirement.

**Term:** Summer 2018

**Contact: **
Dr. Jeremy Kepner,
kepner@ll.mit.edu

**Posting Date:** 11/13/2017

### Integrable Probability

**Project supervisor: ** Prof. Vadim Gorin

#### Project description:

Integrable probability studies the asymptotic behavior of large stochastic systems by essentially algebraic methods. My webpage http://www.mccme.ru/~vadicgor/research.html contains some descriptions and nice looking pictures.

Specific problems would largely depend on the level and interest of students.

#### Qualifications:

Basic understanding of probability theory, e.g. 18.600 class. Understanding of mathematical proofs (e.g. 18.100B/Q or similar proof-based class)

Knowledge of basics of group theory/representation theory is a plus, but not required.

**Contact: **
Vadim Gorin vadicgor@math.mit.edu

**Posting Date:** 9/7/2017

### Learning From Data

**Project supervisor: **Prof. Gil Strang

**Term: **Fall 2017

#### Project description:

The goal is to understand the better success of neural nets for deep learning. With small experiments in machine learning (deep learning), we will start with the open software on playground.tensorflow.org. We need more data on the accuracy as the number of layers and neurons per layer are changed.

**Contact: **
Gil Strang gilstrang@gmail.com

**Posting Date:** 8/30/2017

### Perfect sampling in 2d statistical mechanics

**Project supervisor: **Prof. Vadim Gorin, Prof. Leonid Petrov

**Term: **Fall 2017, IAP, and Spring 2018

#### Project description:

In the recent years integrable random systems (i.e., systems which can be analyzed by means of exact formulas) have been successful in analyzing complicated real-world phenomena ranging from energy spectra of heavy nuclei to shapes of melted crystals and growing bacteria colonies. However, the applicability of exact formulas is (and will remain) limited to special systems, and in order to understand more general models one could try to simulate and visualize them.The goal of this project is to implement existing and develop new methods for perfect sampling (simulations) of large random systems such as the six-vertex (square ice) model and random lozenge tilings. Examples of such systems can be seen in galleries at http://lpetrov.cc/research/gallery/

The student will work with the faculty advisors to learn about integrable random systems and their simulation, and will develop publicly available software for visualization of large random systems. Participants will be paid.

#### Qualifications:

The student should have taken at least one course in probability, and should be familiar with Markov chains and processes. Coding ability in a language good for fast simulations (such as C/C++ or modern alternatives) is mandatory.

**Contact: **
Vadim Gorin vadicgor@gmail.com

**Posting Date:** 8/30/2017

### Applied category theory

**Project supervisors: **
Dr. David Spivak

#### Project description

Category theory is an abstract language for composition---building new things from a collection of already-given things. It is used throughout mathematics and computer science, as well as in other areas of academia, building bridges between these domains. For example, one can use category theory to model information, communication, and interaction between agents.

Undergraduate research projects in applied category theory vary based on the mathematical (and categorical) background of the individual student. Some projects are more applied, others are more abstract. However, all of the projects are real research in the sense that the supervisor (me) doesn't have the 'answer', or even the 'right question' at the outset.

**Contact: **
Dr. David Spivak

**Posting Date:** 1/09/2017

### Revisiting the network scale-up method

**Project supervisor: **
Philippe Rigollet

#### Project description

The network scale-up method has been successfully employed by sociologists to estimate hidden or hard to reach populations (drug injectors, sex workers,...). This method consists in sampling a population by asking the question "How many people do you know in population X?" rather than "do you belong to population X?". Surprisingly, this problem has connection to the matrix completion problem that arises in recommender systems (e.g. the Netflix problem). The goal of this project is to understand and simulate new methods for this kind of data in light of this connection. Other statistical applications, beyond estimation of hidden populations are foreseeable.

#### Qualifications:

The student should have taken a course on introductory probability and statistics and linear algebra. Interest in graph theory is a plus (for secondary goals) but is not required. Experience with coding is desirable, preferably with Matlab or R.

**Contact: **
Professor Philippe Rigollet,
rigollet@math.mit.edu

**Posting Date:** 1/29/2015

### Projects in the Imaging and Computing Group

**Project supervisor: **
Prof. Laurent Demanet

**Application deadline: **
First Friday of each term. Summer UROPs may also be possible.

#### Project description

For information please see the attached PDF file.

**Contact: **
Apply directly with Prof. Laurent Demanet,
laurent@math.mit.edu.
State your interest and qualifications in the application.

### Fluid Dynamics

**Project supervisor: **
Prof. John Bush

#### Project description

See Prof. Bush's webpage for information on his current research projects and interests.

**Contact: **Prof. John Bush,
bush@math.mit.edu

Please note that UROP opportunities are not limited to those advertised above. Students are encouraged to speak to faculty to find out about possible projects.