Project supervisors: Dr. Jeremy Kepner and Prof. Alan Edelman
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/
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
Project supervisor: Prof. Vadim Gorin
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.
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
Project supervisor: Prof. Gil Strang
Term: Fall 2017
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
Project supervisor: Prof. Vadim Gorin, Prof. Leonid Petrov
Term: Fall 2017, IAP, and Spring 2018
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.
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
Project supervisors: Dr. David Spivak
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
Project supervisor: Philippe Rigollet
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.
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
Project supervisor: Prof. Laurent Demanet
Application deadline: First Friday of each term. Summer UROPs may also be possible.
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.
Project supervisor: Prof. John Bush
See Prof. Bush's webpage for information on his current research projects and interests.
Contact: Prof. John Bush, bush@math.mit.edu
Project supervisor: Dr. Dhruv Ranganathan
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?
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
Project supervisor: Prof. Jared Speck
Projects are arranged on an individual basis
Mastery of 18.100 and 18.152
Contact: Jared Speck, jspeck@math.mit.edu
Posting Date: 9/15/2016
Project supervisor: Luiz Faria
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.
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
Project supervisor: Prof. Fabrizio Zanello
Term: Fall 2017
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.
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
Project supervisor: Dr. Elina Robeva, Prof. Philippe Rigollet
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.
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
Project supervisor: Dr. Kyler Siegel
Something related to computing holomorphic curve invariants in low dimensional symplectic and contact geometry.
Ideally have some basic familiarity with smooth topology (smooth manifolds, differential forms) and abstract algebra (groups, rings).
Contact: Kyler Siegel kyler@mit.edu
Posting Date: 9/21/2016
Project supervisor: Prof. Gil Strang
I have data on Oxford University final exams and want to analyze it to see which colleges and which subjects were best. This is an application of linear algebra.
Good work in 18.06 or equivalent and interest in applying it.
Contact: Gil Strang, gs@math.mit.edu
Posting Date: 9/15/2016
Project supervisor: Dr. Spencer Becker-Kahn
Some context: The study of minimal surfaces is a venerable line of inquiry that started sometime in the 18th century and still entertains many fundamental open questions. The ubiquity and utility of minimal surfaces in geometry is partly explained by the fact that there are so many different characterisations of what it means for a surface to be minimal (see the ‘Definitions’ section of the wikipedia page to get an idea of this: https://en.wikipedia.org/wiki/Minimal_surface ). And here at MIT, we have two experts in minimal surfaces, Professors Minicozzi and Colding, and consequently many junior faculty members and graduate students who do work in this kind of mathematics.
Possible Projects: Often one can test out a more advanced idea by seeing if it works in a simple setting first. A project in this area will likely take this form. There are ideas, arguments and computations which would be interesting to understand in the setting of classical minimal surfaces (such as the catenoid, the helicoid or Scherk’s surface) but which could have more far-reaching consequences.
The student will know the basics of the differential geometry of surfaces and real analysis.
Contact: Dr. Spencer Becker-Kahn (né Hughes) stbeckerkahn@mit.edu
Posting Date: 8/22/2016
Project supervisor: Philippe Rigollet
Additive models have proved to be a powerful alternative to linear regression models. In the high-dimensional setting several sparse variants of this model have been proposed and analyzed. The purpose of this project is to study the optimality a remarkably simple procedure to fit such models and its numerical performance. This project will blend many of the basic tools employed in nonparametric and high-dimensional statistics such as thresholding, wavelets, Fourier analysis, minimax optimality and approximation theory.
The student should have taken a course on introductory probability and statistics and real analysis. Experience with coding is mandatory, preferably with Matlab or R.
Contact: Professor Philippe Rigollet, rigollet@math.mit.edu
Posting Date: 1/29/2015
Project supervisor: David Spivak
Title: Categorical Information Theory
Category theory is often considered a very abstract branch of mathematics, but it also has practical applications. In fact, there is a strong link between categories and databases, which allows real-world situations to be modeled using categories. Communication between disparate entities, each having a base of knowledge (category), can be maintained by functors. This UROP project starts by understanding the fundamental connection between categories and information. There are many directions in which to proceed after that, depending on the students interest. These subjects range from theoretical to applied
The student should have seen categories and functors before (though if the student is willing to do extra work, he or she can pick them up as we go), and preferably be excited about them. The student should also be hard-working, have a desire to learn, and be able to ask good questions.
Contact: Dr. David Spivak, dspivak@math.mit.edu
Project supervisor: Andre Nachbin
In this "Waves in Fluids" project two main problems are addressed: (a) tsunami (long wave) propagation in complicated domains (coastal regions and also branching channels); (b) bouncing droplets on a vibrating fluid.
Both problems deal with mathematical modelling, theory of differential equations, as well as numerical simulations. Mathematically speaking both problems consider partial differential equations in intricate domains, or with highly variable coefficients. These physically motivated themes lead to new problems in Applied Mathematics.
Basic knowledge of partial differential equations. Basic knowledge of numerical methods. Some knowledge of programming. Familiarity with Matlab would be helpful. But of course the interested student can be guided through this background and quickly acquire experience.
Contact: Andre Nachbin
Posting Date:09/28/2015
Project supervisor: Semyon Dyatlov
Project descriptions:
In quantum chaos, there are several interesting questions regarding additive energy and other combinatorial quantities associated to trapped sets of noncompact hyperbolic surfaces. These quantities can be used to capture quantitatively the interferences between waves localized on different trapped trajectories for fractal trapped sets. One particular problem is to write an algorithm to compute the additive energy of Schottky limit sets (which are the hyperbolic version of Cantor sets) and give a proof that this algorithm gives a rigorous bound on additive energy. (See e.g., in particular slide 9.)
In dynamical systems, Pollicott-Ruelle resonances describe exponential decay of correlations for hyperbolic/strongly chaotic systems (see e.g.). One way to define them is as viscosity limits, by adding a small constant times the Laplacian to the vector field generating the flow and understanding the behavior of eigenvalues as the constant goes to zero. (This amounts to adding a small amount of white noise to the system.) One can investigate viscosity limits for certain chaotic and non-chaotic models, obtaining a more quantitative understanding of the convergence.
In general relativity, quasi-normal modes (QNMs) are complex characteristic frequencies describing exponential decay of waves on black hole backgrounds. There are several questions related to the behavior of QNMs which one can attack numerically - while they boil down to solving an ordinary differential equation depending on some parameters, it is quite hard to analyse this equation, in particular at low frequency where standard approximations break down, and numerical evidence can provide understanding needed to make a mathematical argument. There are also questions regarding how well QNMs do describe decay of waves, which can again be investigated numerically using a one-dimensional toy model. (See example 1 and example 2 )
Most of the projects involve doing numerical experiments - these can often be used as shortcuts to understand the behavior of complicated systems when a rigorous mathematical argument is not known. Therefore, some programming skills are desired - however, no specialized numerical skills are needed. Familiarity with differential geometry will make a quicker start, but can also be obtained during the course of the project.
Contact: Semyon Dyatlov
Posting Date: 09/28/2015
Project supervisor: Vadim Gorin
It studies the asymptotic behavior of large stochastic systems by essentially algebraic methods. My webpage contains some descriptions; nice looking pictures are available in the gallery. Specific problems would largely depend on the level and interest of students, so I prefer to have one-on-one discussions with interested students first.
Basic understanding of probability theory, e.g. 18.600 class. Understanding of mathematical proofs (e.g. 18.100B/C or similar proof-based class) Knowledge of basics of group theory/representation theory is a plus, but not required.
Contact: Vadim Gorin
Posting Date: 09/24/2015
Project supervisor: Jared Speck
Projects arranged on an individual basis.
Students should have mastered analysis at the level of 18.100 and ideally should have some experience with PDEs at the level of 18.152. Students interested in general relativity or geometric analysis should have some experience in those subjects.
Contact: Jared Speck
Posting Date: 09/24/2015
Project supervisor: Homer Reid
The ideal scenario would be if the project could continue over the summer.
More specifically, the student would (a) learn from me how to create computer models of nanoparticle and nanosurface geometries, (b) use computer tools I have developed to predict Casimir forces and heat-transfer rates between them.
For more details, students should consult these slides from a talk I delivered ("The Upside of Noise") at the MIT physics colloquium in October 2015:
http://homerreid.dyndns.org/research/talks.shtml
In particular, students should look at the pictures of QED pinwheels and photon torpedoes at the very end of the PDF file. This is work that was done by Eric Tomlinson, an MIT undergraduate physics major whose work (as a UROP and then senior thesis) I supervised last year.
Familiarity with linux and command-line software tools. In particular, students should look at the examples described in the following software documentation and should be confident that they would be able to perform these operations on a computer or to learn from me how to do so quickly.
http://homerreid.github.io/scuff-em-documentation
Contact: Homer Reid
Posting Date: 2/2/2016
Project supervisors: Prof. Karen Willcox, Prof. Haynes R. Miller
Department: Aeronautics and Astronautics (Course 16), Mathematics (Course 18)
As an MIT undergrad, you've been there - scared when your 2.004 professor assumes you know how to compute eigenvalues and eigenvectors on the first day of class. Wouldn't it be nice if you could review all the prerequisite eigenvalue topics in one place? Crosslinks is that place - a wiki of linkages and learning resources for any topic, authored by students, for students. It's a place where you can find useful links to techniques for solving specific problems. It's a place where you can get links to videos that other MIT students have found helpful. Ultimately, it's a place where you can see how all the topics taught at MIT are dependent on each other. At least, it will be. That's why we are seeking an enthusiastic student interested in education to help seed Crosslinks with initial content. You are a good fit if you like to read up on classes, review learning material and are a rising sophomore or beyond.
You will be the initial seeder of Crosslinks and gain "first-person-there" glory when we launch Crosslinks (public beta) in Spring 2015. You will identify key topics that cut across subjects at MIT, pinpoint the relationships among these topics and map out where in the MIT curriculum they are taught and used. You will also search for and identify good learning resources for each topic. As Crosslinks progresses and gain user adoption, you will engage in UX (user experience) analysis: you will analyze usage data, asking questions to hone in on user personas, determine the reasons behind user activity patterns and draw conclusions from findings to help improve Crosslinks. You will work with Prof. Willcox, Prof. Miller and Crosslinks' technology lead (Luwen Huang) on content creation, direction and usability experiments.
Hours are flexible; 20 hours per week preferred for Summer and 10 hours per week preferred for Fall. Start date is flexible; preferred start in late July/early August with continuance through Fall 2014.
Contact: Interested students are asked to email Prof. Willcox (kwillcox@mit.edu) and Prof. Miller (hrm@math.mit.edu) with their resumes.
Posting Date: 12/23/2014