# 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:

- TBD (Pure)
- Prof. Steven Johnson (Applied)

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:

### 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

### Combinatorial Algebra and Pure O-Sequences.

**Project supervisor: **Prof. Fabrizio Zanello

**Term: **Fall 2017

#### Project description:

Combinatorics is the art of counting, hence the goal of combinatorial algebra is to count algebraic objects, or sometimes, to use algebraic methods to count combinatorial objects.

One of my recent interests in this area is called "pure O-sequences," a topic that finds (often surprising) applications to a number of different mathematical disciplines. Pure O-sequences can simply be defined as the integer vectors of the form h = (1, h_1, ..., h_N = T), where h_d counts the (monic) monomials of degree d dividing (any one of) T given monomials of degree N. There are several interesting open problems that can also be approached at the undergraduate level, and a good research project on this topic will likely be publishable in a fine international journal. A couple of useful reference are this AMS memoir I cowrote in 2012: https://arxiv.org/abs/1003.3825 , and this survey article: https://arxiv.org/abs/1204.5247 .

This is just one of a number of possible topics. Often, a pure math research project naturally arises from one-to-one discussions between the student and the potential advisor, and it also reflects the tastes of the student in a given area. Therefore, a priori, no combinatorial or ring-theoretic topic is off-limits for this project, and I encourage any interested student to get in touch with me for an informal discussion.

#### Qualifications:

At least one course in algebra (familiarity with some ring theory is necessary), and possibly one in combinatorics; lots of creativity and eagerness to learn new mathematics.

**Contact: **
Fabrizio Zanello zanello@mit.edu

**Posting Date:** 9/6/2017

### 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

### Efficient tensor decomposition using frame theory

**Project supervisor: **Dr. Elina Robeva, Prof. Philippe Rigollet

#### Project description:

Like the singular value decomposition for matrices, tensor decomposition is an increasingly useful tool especially because most data does not come in the shape of a matrix but rather in the shape of a multi-dimensional array, or tensor. Unlike matrix decompositions, tensor decompositions are very hard to compute. There are efficient methods to decompose a special class of tensors that are widely used in machine learning (see for example Sections 1.1 and 1.3 in https://arxiv.org/pdf/1409.6685.pdf). However, this class of tensors contains a very small portion of all tensors. The goal of this research project will be to find much larger classes of tensors that can also be decomposed efficiently. This will be an important contribution to theory of tensor decompositions.

Through the course of the project, the student will learn much about existing techniques regarding tensor decompositions. The final goal will be to develop a new method for decomposing tensors which performs much faster than known methods, and can be applied to a wide variety of tensors. More specifically, these tensors will have a decomposition in terns of what is called a tight frame, or in other words, a set of m > n vectors that span R^n and possess some important additional properties. There has been evidence that tight frames, and more specifically equiangular tight frames, are the exact way to extend efficient tensor methods to a larger class of tensors, providing the missing link between the small set of efficiently decomposable tensors and the set of all tensors. For additional reading, see https://link.springer.com/article/10.1007/s10958-009-9366-6 and https://arxiv.org/pdf/1504.08049.pdf.

The student will be working primarily with the direct supervisor Elina Robeva.

#### Qualifications:

Strong background in linear algebra. Basic algorithms complexity knowledge. Programming experience with computer language of choice, e.g. C++, Python, R, Matlab, Julia, or any others.

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

**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

### Combinatorial aspects of algebraic geometry and graph theory

**Project supervisor: **Dr. Dhruv Ranganathan

#### Project description:

There are a range of possibilities, but I’ve outlined one below.

**Title:** Graph associahedra and connections to algebraic geometry

**Description:** Graph associahedra are a large and interesting family of polytopes with a number of remarkable properties. These polytopes are associated to connected finite graphs and are constructed from a simplex by a truncation procedure. In addition to having interesting combinatorics, they arise naturally in fields that include Floer homology, moduli spaces, and knot invariants. In particular, the machinery of toric varieties provide an interesting mechanism to link finite graphs to certain algebraic varieties dubbed “toric graph associahedra". This goal of this project will be to learn, develop, and exploit a dictionary between these two objects. For instance, can one give an algorithm to compute the Betti numbers of a toric graph associahedron in a natural manner from the graph itself? Do natural numerical invariants of the graph encode particular geometric properties of the toric graph associahedron?

#### Qualifications:

Abstract algebra and basic analysis are the bare minimum requirements, as the rest can be learned "on the go". Courses in topology, algebraic topology, differential geometry, and algebraic geometry will all be useful and relevant, as will some basic knowledge of python/SAGE. A willingness to compute lots of examples will be paramount.

**Contact: **
Dhruv Ranganathan, dhruvr@mit.edu

**Posting Date:** 9/15/2016

### Wave Equations, Shock Waves, Fluid Mechanics, General Relativity

**Project supervisor: **Prof. Jared Speck

#### Project description:

Projects are arranged on an individual basis

#### Qualifications:

Mastery of 18.100 and 18.152

**Contact: **
Jared Speck, jspeck@math.mit.edu

**Posting Date:** 9/15/2016

### Spreadsheets, Big Tables, and the Abstract Algebra of Associative Arrays

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

#### Project description

Spreadsheets are used by nearly 100M people every day. Triple store databases (e.g., Google Big Table, Amazon Dynamo, and Hadoop HBase) store a large fraction of the analyzed data in the world and are the backbone of modern web companies. Both spreadsheets and big tables can hold diverse data (e.g., strings, dates, integers, and reals) and lend themselves to diverse representations (e.g., matrices, functions, hash tables, and databases). D4M (Dynamic Distributed Dimensional Data Model) has been developed to provide a mathematically rich interface to triple stores. The spreadsheets, triple stores, sparse linear algebra, and fuzzy algebra. This projects seeks to strengthen the abstract algebraic foundations of associative arrays. The student will work with the faculty advisor to develop the basic theorems of associative arrays by building on existing work on fuzzy algebra and linear algebra. Participants will be paid.

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

#### Qualifications:

Strong mathematical background (the student should have completed 18.701 and 18.702). Experience with Matlab is helpful, but not a requirement.

**Term:** IAP and Summer 2017

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

**Posting Date:** 2/01/2016

### Wave propagation in reactive flows

**Project supervisor: **Luiz Faria

#### Project description

Mathematics of combustion. In particular, trying to understand what types of traveling wave solutions a certain system of PDEs (derived from combustion theory) admits, and trying to compute the stability properties of such waves.

#### Qualifications:

Basic knowledge of partial differential equations. Basic knowledge of numerical methods. Knowledge of dynamical systems and stability theory would be a plus. Familiarity with Matlab (or a similar software) would be helpful, but the student can learn about it as he/she goes

**Contact: **
Luiz Faria

**Posting Date:** 09/24/2015

### 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.