# Research Experiences for Undergraduates Program

- 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 4th through July 27, 2018: Research Project Descriptions
**Chris Connell****Description:**Simplices play an important role in both geometry and topology. In the former context they provide convenient building blocks for spaces, and in the latter context they arise in combinatorial descriptions of objects such as those used in homology and cohomology. For computations in both realms, knowing explicit volumes of simplices in terms of their geometric data become very useful. The volume of Euclidean simplices has been computed from different descriptive data, one important example of this is the Cayley-Menger determinant. On the other hand, sharp computations of the volumes of hyperbolic (constant negatively curved) simplices in high dimension has been much more elusive.In this project we will be interested in both sharp estimates for the volume of hyperbolic simplices and applications of these estimates to the volume of hyperbolic manifolds and explicit values of an important topological invariant called simplicial volume. For calculation purposes we will be interested in extending Cayley-Menger type formulas to the hyperbolic setting and also estimating existing Schläfli type iterated integral formulas.

**Prerequisites:**Nothing more than vector calculus and basic linear algebra is necessary, but familiarity with elementary group theory would be helpful for the topological applications. Familiarity with a computer algebra platform such as Mathematica would also be helpful.**Alex Kruckman**### Incidence Structures and Model Theory

**Description:**An incidence structure consists of a set of elements called "points", a set of elements called "lines", and a relation called "incidence" between them: p and l are incident when the point p lies on the line l. These structures are fundamental in the field of combinatorial geometry. A projective plane is an incidence structure such that every pair of distinct points are incident with exactly one line, and every pair of distinct lines are incident with exactly one point. The projective planes admit a natural generalization: say an (m,n)-pseudoplane is an incidence structure such that every m points are incident with exactly (n-1) lines, and every n lines are incident with exactly (m-1) points. So projective planes are (2,2)-pseudoplanes.Model theory is a subfield of logic which studies definability in mathematical structures and seeks to classify mathematical theories by measures of complexity related to definability. In a recent paper, Gabriel Conant and I studied the model theory of projective planes and (m,n)-pseudoplanes. In this project, we will attempt to resolve some open questions raised in that paper concerning the combinatorial structure of (m,n)-pseudoplanes. Additionally, depending on the interests of the student, we may 1. study the model theory of planar ternary rings, a class of algebraic structures which coordinatize projective planes, or 2. explore other classes of incidence structures omitting certain configurations.

**Prerequisites:**Some experience with graph theory and/or abstract algebra. Experience with logic (especially first-order/predicate logic or model theory) would be welcome, but not necessary.**Nick Miller**### Searching for Congruence Surfaces

**Description:**A congruence surface is a hyperbolic Riemann surface whose fundamental group is a congruence subgroup of an arithmetic group. In the non-compact setting, there are many examples of such surfaces, some of which have been shown by Schmutz to maximize the length of systole in their respective genus. However in the compact setting, it is unknown whether one can even produce a congruence surface for each possible genus.In this project, we will first learn some basics of hyperbolic geometry with an emphasis on Riemann surfaces and then learn about arithmetic constructions of hyperbolic surfaces. We will then search for congruence surfaces in low genus, with the eventual goal of constructing infinitely many congruence surfaces, one for each genus.

**Prerequisites:**Some knowledge of elementary group theory and number theory would be useful but is not necessary. Some experience with SAGE and/or python would also be useful but again is not necessary.**Carmen Rovi**### Cutting and Pasting of Manifolds and Group Actions

**Description:**### REU proposal.pdf

**Prerequisites:**Some familiarity with basic algebraic topology would be desirable.**Amr Sabry**### Homotopy Type Theory

**Description:**Homotopy Type Theory is a relatively 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 with a focus on computational interpretations. The exact topics will depend on the student's interest and background.**Prerequisites:**Good background in abstract algebra and logic; excellent programming skills; experience with proof assistants and dependent types is a plus; knowledge of some topology is a plus.**Noah Snyder**### Simple Skein Theories in Finite Characteristic

**Description:**A string can be tied into a knot in many ways, but it can be hard to see whether you can turn one knot into another one. In order to tell knots apart we need to find invariants that distinguish. Several very interesting invariants like the Jones polynomial are defined using "skein relations", which let you cut out a small part of a knot (called a tangle) and replace it with a sum of simpler tangles. This procedure has a very algebraic flavor and you can think of each of these invariants as giving a kind of ``algebra" where you can multiply pictures by connecting up loose ends of string. Links aren't the only kind of objects you can study skein theoretically. Another great example is planar 3-valent graphs. For both knots and planar 3-valent graphs there are theorems classifying the "simplest" skein theories. In this classification a lot of fascinating examples come up related to things called quantum groups.Usually the skein relations have coefficients that are complex numbers. In this project we will try to classify simple skein theories but where the coefficients instead live in a finite field like the integers modulo p.

**Prerequisites:**The main pre-requisite is a strong understanding of modular arithmetic, and enough ring and field theory to understand finite fields. Some familiarity with knot theory or graph theory is helpful but not required.**Graham White**### Markov Chain Mixing Times

**Description:**It is a commonly-quoted fact that when shuffling a deck of cards, seven riffle shuffles are `enough'. That is, after fewer than seven shuffles, many possible states of the deck are much less likely than they would be in a `perfectly random' deck, while after more than seven shuffles the probabilities of most states are close to what they should be. More precisely, after seven shuffles the distance between the two probability distributions is small.This sort of question may be asked for other Markov chains. If you have an interesting (and nice enough) random process, and you run it for long enough, it will approach a limiting distribution. How long is `long enough'? For instance, if one shuffles a deck of n cards by repeatedly swapping random pairs of cards, then about n * log(n) steps are required. If instead one swaps random pairs of adjacent cards, then this changes to n^3 * log(n) steps. Of course, both of these shuffling procedures are much slower than riffle shuffling, which only takes a number of steps proportional to log(n).

We will explore techniques which can be used to prove bounds of this sort, and investigate what bounds can be obtained for a variety of random processes. Many of these methods are combinatorial, for instance some involve carefully constructing a bijection between various sets of paths in the chain. Some possibilities for study could include random walks on graphs, groups, or combinatorial objects, depending on your areas of interest.

**Prerequisites:**Familiarity with probability will be helpful. There may be opportunities to steer the project toward graph theory, group theory, or combinatorics, so interest in any of these is a plus, but is not required. - 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 facilities. Mathematical 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.

- Frequently Asked Questions
**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 The National Science Foundation maintains acomprehensive listof REU sites.
** This year’s undergraduate research experience was funded through the generous support of Indiana University and the Department of Mathematics.

- Application and Contact Info
(APPLICATION FORM)

**Deadline: February 23**Be sure to read the FAQ.

Letters of recommendation may be emailed to Mandie McCarty, amm3308@indiana.edu.

If you have more questions, you can send an email to

Chris Connell,

Indiana University Mathematics Director REU program

connell@indiana.edu