# Research Experiences for Undergraduates Program

For over four decades, the Department of Mathematics at Indiana University has offered experiences in mathematics research to students from all over the country. During the summer, a select group of undergraduates participates in research projects with individual faculty mentors on a wide variety of topics. The National Science Foundation supports the program via a Research Experiences for Undergraduates (REU) grant.
• General Program Description

The program emphasizes close relationships with faculty advisors: students typically will work work closely with a faculty member in small groups of 1-3. For this reason, we try to match each student with an appropriate project and advisor during the admissions process. Thus faculty advisors are able to send suggestions for background reading prior to the start of the program.

The program unofficially begins with a reception hosted by the mathematics department. This provides an opportunity for students to meet other students as well as their faculty mentors.

During the following eight weeks, students meet privately with their faculty advisors several times per week. During the first few weeks the student will work to further understand the details of the project. During the middle weeks, the student will be working full-time on the project. During the last two weeks, the student will be preparing both a written and oral report on his or her results. All along the way, the faculty mentor will be providing assistance and encouragement.

During the first week of the program, the students will be given an orientation of the Swain Hall Library, the library that houses an extensive collection of mathematics books and journals. Students will also be given a tour of available computer facilities.

Students are given a dedicated seminar room in which to study and socialize. At least once a week, faculty will give accessible lectures on topics of current research interest in mathematics. A graduate student will provide an introduction to the LaTeX typesetting system and serve as a consultant for preparation of the written report.

During the last week each student will give a lecture on his or her own work. Each student is also required to prepare a written report. These reports are bound together into a single volume that will duplicated and distributed to students and faculty mentors. Students are also encouraged to submit their research paper for publication although there is no requirement to do so.

Certainly one of the most important aspects of the program is student-to-student interaction. For some this may be their first opportunity to get to know other students with comparable mathematical interest and ability. We encourage this interaction in a number of ways. In particular, students live together in a single dormitory on campus. There are several organized social events. Each group usually develops its own unique character, often organizing some of its own activities.

• June 5 - July 28, 2017: Research Project Descriptions

### Heegaard Floer homology and torus bundles

Faculty Mentor:
Corrin Clarkson

Description:

Heegaard Floer homology is a topological invariant i.e. it is a way of measuring topological complexity. Given such a tool, it is natural to ask whether or not it is precise enough to distinguish between distinct topological spaces. In general, Heegaard Floer homology is good a distinguishing between spaces, but it is also known to have blind spots. In fact, there are infinitely many examples of distinct manifolds having the same Heegaard Floer homology. Determining whether this invariant distinguishes between a particular pair of manifolds can be challenging due to the difficult nature of computing Heegaard Floer homology. This is why we will focus on torus bundles.

Torus bundles are a particularly nice class of three-manifolds. There is a correspondence between these manifolds and elements of SL(2,Z) which allows us to use linear algebra to describe them. Moreover, there are relatively efficient algorithms for computing the Heegaard Floer homology of these manifolds. The question we will be exploring is the following. Is Heegaard Floer homology a complete invariant of torus bundles i.e. does it distinguish between any two distinct torus bundles?

Prerequisites:
Linear algebra, some programming experience ideally in Python, a basic understanding of group theory would also be useful, but is not necessary.

### Convex Hulls of Closed Curves

Faculty Mentor:
Chris Connell

Description:

For a curve c in Rn, its convex hull is the smallest convex set containing c. There are a number of interesting questions one can ask about the relationship of a curve c to its convex hull. Many of these fall into the mold of an "isoperimetric" type problem where one geometric quantity is being optimized subject to another geometric constraint. A number of authors, including I. Schoenberg and A. Weil, have made important contributions in this area. We will focus on one such question that was related to me by M. Ghomi in 2004. Namely, whether a round planar circle maximizes the surface area of the convex hull among all closed curves c in R3 of a fixed length L. (Here the surface area of the degenerate convex hull of a round circle should be interpreted as twice the area of the disk.) Our first plan will be to attack this problem from an existing approach built upon prior work of N. Krabbenhoft and myself. As is often the case, there are a number of analogues to this question in other settings and in higher dimensions which we can also explore depending on student interest.

Prerequisites:
Multivariable caluculus and enthusiasm is a must. An introductory course in either differential geometry (curves and surfaces) or ODE's would be a welcome bonus, but not necessary.

### Vertical averaged velocity in Bénard convection

Faculty Mentor:
Michael Jolly

Description:

This project is suitable for a student with strong computational skills and a keen interest in partial differential equations from fluid dynamics. The question is whether the vertical average of a 3D fluid driven by a temperature imbalance on the boundary displays the features of 2D turbulence. Toward an answer for this, we would carry out direct numerical simulation of the 3D problem and then extract from that solution certain quantities which determine the nature of the body force in the equation for the vertically averaged velocity.

The simulation of the 3D problem can be done by Dedalus: http://dedalus-project.org/about.html a suite of Python scripts which call computational modules. In fact there is an example specifically for the problem of interest. The task then, is to construct the body force for the vertically averaged velocity. Since the relevant quantities are defined in terms of derivatives and integrals, and Dedalus is a spectral code, this should be straightforward. The student would then interpret the results in light of our previous work which explains the criteria for turbulence. We have already done analysis to determine an upper bound on this body force. Without a meaningful lower bound however, we cannot say if the force is strong enough to support 2D turbulence. This is why we turn to simulations.

Prerequisites:
Good computational skills, some exposure to PDEs and an open mind are essential.
Full Description

### Inference in Natural Language

Faculty Mentor:
Larry Moss

Description:

When computers carry out human-level reasoning, they do it in a number of ways. Frequently they do it by translating some natural language (NL) input into a very different form, some language or other that looks much more like formal logic (FL) than NL. This has the advantage of allowing one to use off-the-shelf tools having to do with FL, and this is good. But translation into FL comes with disadvantages: first of all, one is limited to very special kinds of reasoning; second, whatever one gets from FL then has to be re-translated back to NL if one wants to use it. This project is connected to Prof. Moss' project of Natural Logic, an attempt to do reasoning in something that looks more like NL than FL. It is a big interdisciplinary project and has people working on all aspects of it, publishing papers and writing programs.

Next summer would mark the fourth time that someone worked on a summer project related to Natural Logic. There are a number of projects available, mostly having to do with proving theorems about logical systems or about developing and implementing algorithms.

Prerequisites:
The ideal student would be one who has had courses in one or more of the following: logic, theoretical computer science, combinatorics, linguistics, algorithms, or programming. One would not need all of these (of course!), but the more the better.

### Conformal dimension and energy of graph maps

Faculty Mentor:
Kevin Pilgrim and Dylan Thurston

Description:

Prerequisites:

Full Description

### Cutting and pasting of manifolds and TQFTs

Faculty Mentor:
Carmen Rovi

Description:

Suppose M is a closed manifold which can be cut open along a codimension 1 submanifold. By doing this cutting operation we obtain two manifolds with the same boundary which we can now glue back together using an automorphism of the boundary which is not the identity. The new object that we obtain is said to be "cut and paste" or SK-equivalent to the manifold M we started with. It turns out that the cut and paste operation doesn't only define an equivalence relation, it defines certain groups called the SK-groups. The SK-groups were first defined some 40 years ago, but they haven't been developed or investigated in depth since then, even though they have interesting connections with very active areas of research. An interesting topic is to understand which topological invariants are invariants of the cut and paste operation. The goal of this project will be to describe which cut and paste invariants are partition functions of topological quantum field theories.

Prerequisites:
Some familiarity with basic notions of topology would be desirable.

### Homotopy Type Theory

Faculty Mentor:
Amr Sabry

Description:

Homotopy Type Theory is a new development that establishes surprising connections between logic, algebra, geometry, topology, computer science, and physics. This project will involve formalization of mathematical results in homotopy type theory. The exact topics will depend on the student's interest and background.

Prerequisites:
Good background in abstract algebra and logic; excellent programming skills; knowledge of some topology is a plus.

• Research Environment

Indiana University has a large and active mathematics faculty that enjoys and supports research work with undergraduates. The breadth of mathematical interests in the department provides students in the program with a firsthand view of the richness of the field.

Students have access to a first-rate mathematics research library located next door in Swain Hall. The library subscribes to approximately 450 research periodicals many of which can be accessed on-line anywhere on campus. The mathematics librarian is experienced in conducting bibliographic instruction tailored to undergraduate mathematics students.

Students also have access to state-of-the-art computer facilitiesMathematical packages such as Maple, Mathematica, and Matlab are all readily available on a variety of platforms.

A seminar room in Rawles Hall becomes the "REU Room" in the summer and is available all day to REU students as a place to study and discuss mathematics.

• Housing, Stipend, and Travel Allowance

During the eight weeks, particiants will

• be housed in a university dormitory with an included meal allowance,
• be provided with a $400 travel allowance to defray the costs of travel to Bloomington, and • receive a$4,000 stipend.
• Diversions

Bloomington and the surrounding area offer a wide variety of diversions. Here are a few:

• Music: Each summer the School of Music hosts a summer music festival that includes acclaimed musicians from around the world. The Buskirk-Chumley theatre and the IU Auditorium also feature many popular musical performers from various genres.
• Museums: The University art museum is housed in a building designed by I.M. Pei, the architect who designed the Louvre pyramid. The Mathers Museum has a collection of over 20,000 objects and 10,000 photographs representing cultures from each of the world's inhabited continents. The Lilly Library catalogues the largest collection of mechanical puzzles ever assembled, the Slocum collection.
• Film and theatre: The Ryder Film Series the best in foreign-language, independent and classic American films. The Brown County Playhouse puts on broadway musicals and comedies. Bloomington also has 23 screens on which mainstream first run movies are played.
• The great outdoors: Indiana is not just one continuous corn field. Bloomington is situated in rolling wooded hills. One can hike and camp in several nearby parks including the Hoosier National Forest and canoe and fish in nearby Lake Monroe.

Q: Have I taken enough math courses in order to be eligible?
A: Most of our applicants have completed courses in one-variable calculus, a course in multivariable calculus, and one or more courses such as linear algebra, differential equations, and probability and statistics. Many of our applicants have completed at least one course in abstact algebra or a course in real analysis.

Q: I'm a foreign student. Am I eligible?
A: No, only US citizens and permanent residents are eligible.

Q: I'm a foreign student, and would pay my own way. Am I eligible?
A: No, only US citizens and permanent residents are eligible.

Q: I'm graduating this spring. Am I eligible?
A: No, only students who have not received their undergraduate degree are eligible.

Q: Can I submit letters of recommendation and unofficial transcripts electronically?
A: Yes--send them to Mandie McCarty, amm3308@indiana.edu

Q: I'm a part-time student. Am I eligible?
A: Per NSF guidelines: Undergraduate student participants supported with NSF funds in either REU Supplements or REU Sites must be citizens or permanent residents of the United States or its possessions. An undergraduate student is a student who is enrolled in a degree program (part-time or full-time) leading to a baccalaureate or associate degree. Students who are transferring from one college or university to another and are enrolled at neither institution during the intervening summer may participate. High school graduates who have been accepted at an undergraduate institution but who have not yet started their undergraduate study are also eligible to participate. Students who have received their bachelor's degrees and are no longer enrolled as undergraduates are generally not eligible to participate.

Q: Do you have a minimum GPA requirement?
A: Formally, no. However, our program is competitive, and most of our applicants have GPAs in excess of 3.5 on a 4.0 scale.

• Activities Calendar
• Past Programs
• Some alumni of the program and where they are now
• 2016
• 2015 **
• 2014
• 2013
• 2012
• 2011
• 2010
• 2009
• 2008
• some alumnae of the program and where they are now

The National Science Foundation maintains a
comprehensive list
of REU sites.

** This year’s undergraduate research experience was funded through the generous support of Indiana University and the Department of Mathematics.