# Undergraduate Research

As a student in Mathematics, you have the option to engage in undergraduate research which complements your classroom learning. Undergraduate research involves a student exploring and obtaining on his/her own with faculty support results original to the student, although not necessarily new to mathematics. Several courses, such as Math 3000 or advanced undergrad electives, may have a required or optional paper, that could be a research project. In addition, you can devote an entire course to research at anytime during your time at Northeastern by taking a directed study course with a Mathematics faculty member.

A particular opportunity occurs in conjunction with one of the Department’s capstone courses, all of which involve student projects. Possible projects can involve any mixture of pure and applied mathematics and may incorporate tools such as programming languages, statistical packages, data bases. In addition to projects proposed by faculty, projects may be posed by students resulting from learning during their coop experiences, or through community partners in the University Service-Learning program, or through contact with their laboratory or course work in related subjects. If your math research project is particularly successful, it could form the basis for a junior/senior honors project and graduation with Honors in Mathematics. There are also other opportunities to present your research, or publish it, both at the university and outside. Mathematics students have been among presenters at the university’s annual Research and Scholarship Expo and also at national undergraduate conferences.

Your work with mathematics faculty together with your classroom work can form a strong foundation for a wide range of summer programs in mathematics and science, such as Research Experiences for Undergraduates (REU) programs that are offered throughout the country, with some international locations as well. These programs, and as well a Senior Thesis may lead to authoring or coauthoring journal articles, and can provide a stronger background for applying to graduate school in mathematics.

The following students received the distinction “Honors in Mathematics” for their Junior-Senior Honors project, upon graduating May 2015

- Annie Dai: Title: “Quantitative Thermo-acoustics Tomography”, Advisor: Professor Ting Zhou
- Kyle Myer: Title: “Colorful Analogues of Associahedra-like Polytopes”, Advisor: Professor Egon Schulte
- John Riestenberg: Title: “The longitudinal Hall invariant of a knot”, Advisor: Professor Alexandru Suciu

If you are interested in pursuing undergraduate research projects, please feel free to consult a faculty member, advisor, or an undergraduate research contact person. You can also consult our webpage on Research Topics and Problems.

## Faculty Advisors

The following may be consulted for problems on specific topics.

- Prof. Beasley: TBA
- Prof. Eigen: Ergodic Theory Does 2-mixing imply 3-mixing
- Prof. Gaffney: Force Fields of Point Charges
- Prof. Iarrobino: Linear Algebra and Combinatorics Jordan Types of Commuting Nilpotent Matrices
- Prof. Malioutov: Applied Statistics
- Prof. Massey: Topology of Singular Spaces

## Contact Faculty

The following are available to discuss a research project option not posted, or advise about undergraduate research and find a potential advisor.

- Prof. A. Iarrobino – instructor of Math 4020

**Office:**526 Nightingale Hall

**E-mail:**Send Mail - Prof. D. King

**Office:**447 Lake Hall

**E-mail:**Send Mail - Prof. R. McOwen – Instructor of Math 4025 in Spring 2015

**Office:**445 Lake Hall

**E-mail:**Send Mail

# Research Related Courses

**Directed Study:**

You can take a course in which you engage in individual research under the supervision of a faculty member by registering for one of the three courses, MATH 4991, 4992, or 4993. This requires the agreement of a faculty member and strong preparation, often a 3.4 or above average in major courses and prerequisite courses for your topic. Feel free to contact a potential advisor, or an undergrad math research contact person if you are interested in this.

**Capstone:**

A Mathematics Department Capstone can take the form of Research (pure Math 4020, or applied, Math 4020 or Math 4025), Modeling (Math 5131), or be a combination of the two (Math 4020 or 4025 or 5131). All capstones have a reflection component. In a pure math or applied research capstone the student will use or develop a theory to solve or contribute to the understanding of a mathematical problem. Math 5131 uses actual case studies which exclusively apply ordinary and differential equations. The case studies are a starting point for students’ own projects, which is the main focus of the course. The projects in Math 5131 may involve further research, or they may involve coming up with a good idea motivated by the case studies, and then doing the work to explore and explain them.

In the Math 4020 or Math 4025 Capstones a student develops an original theory or application, new to the student.

**Junior/Senior Honors Project and graduation with Honors in Mathematics:**

Successful completion of a junior/senior honors project means that the mathematics major will graduate with Honors in Mathematics; no prior participation in the honors program is required. Students must complete at least 8 hours of supervised research resulting in a written thesis. The 8 hours may occur through a superior research project in Math 4020 or Math 4025 or Math 5131 followed by Math 4970 (Junior/Senior Honors Project 1). Another possibility for 8 hours of supervised research is Math 4970 followed by Math 4971 (Junior/Senior Honors Project 2). Directed study (Math 4991, 4992, or 4993) may also be used as the first 4 hours of supervised research. Supervision is by a Math Department faculty project advisor and the Undergraduate Research Committee of the Mathematics Department.

For general information about this program, see College Honors . An online application form is available here. For the specific requirements imposed by the math department, see Honors in Mathematics . The student must make a written proposal for a Junior-Senior honors thesis that is accepted by the Mathematics Department Undergraduate Research Committee preferably before beginning the two course sequence, but at least before enrolling in the second course and before indicating intent to apply for College Honors in Mathematics online. For possible research topics, see Research Topics and Problems .

#### Further Information

Students graduating with Honors in Mathematics, must have a course of study in mathematics approved by the Undergraduate Research Committee. This will normally include the following upperclass courses required for a BA or BS in Mathematics:

Math 2331 Linear Algebra, Math 3150 Real Analysis, or Math 4565 Topology, Math 3175 Group Theory, Math 3560 Geometry or Math 3527 Number Theory, also a Mathematics Capstone (Math 5331, Math 4020 or Math 4025). Exceptions must be approved by the math department’s Undergraduate Research Committee. For example, a dual major might request a one course exception.

The student must complete two mathematics department courses (8 hours) comprising a junior senior honors project accepted by the Undergrad Research Committee. With approval of the instructor and URC one of the math department Capstones may be accepted as the first of these.

Students are advised to contact the URC with a proposal in advance of the two course project sequence, if they plan to complete an honors project. However, a student with the support of an instructor of a directed study in progress or completed may apply to the Undergraduate Research Committee to have that course considered as the first of a two course honors project.

Students beginning a Junior-Senior Honors thesis are expected to have taken relevant math department undergrad courses before beginning the first course of a two course honors project. For example a student proposing a project in combinatorics, should have completed Math 3533 or material equivalent prior to beginning his/her project; a student proposing a project in finite fields or rings should have completed Math 3175 prior to beginning the first research project course, and is expected to complete Math 4576 Rings and Fields or an equivalent before graduation. A student studying traffic flow as a project might wish to have taken Math 4525 or 4545 before undertaking the project.

Acceptance into the second research course for a Junior-Senior Honors thesis must be preceded by approval by the URC of a research proposal and advisor. In addition, the student must submit evidence of having completed a first research course on this project, and as well appropriate Math Department and other courses to support work on the project, and expertise in the area of the project. The submission must be evaluated and approved by the math department URC before registration for the second course of a Junior-Senior honors project, and before completing the relevant online honors in the discipline intention form.

The URC will choose a three person committee including the thesis advisor to supervise the undergraduate thesis.

**Summary of Research Related Courses:**

- MATH 3000 Coop Reflection.
- MATH 4020 Research Capstone.
- MATH 4025 Applied Mathematics Capstone.
- MATH 4970 and MATH 4971 Junior/Senior Honors Project 1 and 2. Graduating with Honors in Mathematics (
*Note that this page is currently being updated. The course Math 4997 is outdated and the correct course is Math 4970.)* - Directed Study: MATH 4991, MATH 4992, MATH 4993.
- MATH 5131 Introduction to Mathematical Methods and Modeling (Capstone).

# Additional Information

## Journals for Undergraduate Mathematics Research

- SIAM Undergraduate Research Online Web-based devoted to undergraduate research in applied and computational mathematics.
- Rose-Hulman Undergraduate Math Journal
- Furman University Electronic Journal of Undergraduate Research
- Pi Mu Epsilon Journal
- American Journal of Undergraduate Research
- Mathematical Spectrum
- Undergraduate Journal of Mathematical Modeling
- Involve
- Journal of Young Investigators
- International Journal of Research in Undergraduate Mathematics Education

## Presenting Your Research

There are many regional conferences which encourage undergraduates to present a paper or a poser

- SHOW YOUR RESEARCH AT NORTHEASTERN’S RISE Expo
- Hudson River Undergraduate Mathematics Conference
- MAA Opportunities to Present

## Sources of Financial Support for Undergraduate Research

- Northeastern Provost Undergraduate Research
- Northeastern Funding Resource page
- NSF Research Experiences for Undergraduate
- MAA Undergraduate Research
- Some National Research Opportunities UROP page of University of California, Irvine.
- Northeastern Student Research Page
- AMS Resources for Undergraduates in Mathematics

## Other Links

- Association for Women in Mathematics
- IAS Mentoring program for women in math the Institute, together with Princeton University, has hosted an intensive 11-day mentoring program for undergraduate and graduate women in mathematics.
- Center for Women in Mathematics at Smith College Women in Mathematics in New England
- Nebraska Conference for Undergraduate Women in Mathematics
- Hispanic Scholarship Fund
- United Negro College Fund
- Gates Millennium Scholars
- Siemens Foundation
- Newsletters and Undergraduate Mthematical Journals
- Lists of off campus opportunities
- List of undergraduate research journals – all fields
- List of unsolved problems at Wiki These problems are not normally appropriate for undergraduate research projects – most are long-time conundrums. But they do give a sense of how much is unsolved in mathematics. Enjoy!