We have been awarded a minigrant by the Center for Undergraduate Research in Mathematics, which is itself funded by the NSF’s Division of Mathematical Sciences. This minigrant supports four SUNY Geneseo math majors to work with Dr. Rault on a project in pure mathematics.

The following is a taste of the type of topics we will be pursuing. You are not expected to have any familiarity with them in advance.

The numerical range of a matrix A is the complex set {<Ax; x> : x in C^n : ||x|| = 1}, and the product field of values is defined similarly. The numerical range is always a convex set with many interesting properties, as discussed in several recent papers:

- Camenga, Kristin; Rault, Patrick X; Sendova, Tsvetanka; Spitkovsky, Ilya. On the Gau–Wu number for some classes of matrices.
*Linear Algebra and its Applications*. 444 (2014), 254-262. DOI: 10.1016/j.laa.2013.11.045. - Camenga, Kristin; Rault, Patrick X; Rossi, Dan*; Sendova, Tsvetanka; Spitkovsky, Ilya. Numerical range of some doubly stochastic matrices.
*Applied Mathematics and Computation*. 221, September 2013, 40 – 47. - Rault, Patrick X; Sendova, Tsvetanka; Spitkovsky, Ilya. 3-by-3 matrices with elliptical numerical range revisited.
*Electron. J. Linear Algebra*26, 2013, 158-167. - Rossi, Dan.* The numerical range of 4x4 doubly stochastic matrices.
*Honors Thesis in Mathematics at SUNY Geneseo*, 2012.

(Student collaborators marked with *)

The product field is less studied and will have many interesting surprises for us.

**Suggested pre-requisites**: good comfort with linear algebra over the complex numbers. Willingness to work with the programming languages *Mathematica* and/or *Sage*.

Consider homogenous polynomials f(x,y) and g(x,y) with degree d and integer coefficients. Under certain conditions, we can count the number of integer points (u,v) satisfying |f(u,v)|<B and |g(u,v)|<C. This can in turn, after a series of deep steps, be used to count the number of rational points (p/r,q/r) on a curve for which max{|p|,|q|,|r||}<B gcd(p,q,r). This was done in the following papers, and there is room to explore similar problems.

- Rault, Patrick X. On uniform bounds for rational points on rational curves and thin sets of arbitrary degree. Journal of Number Theory. 133 (9), 2013, 3112-3118.
- Bennett, Mike*; Lazebnik, Kirill Y*; Rault, Patrick X; Singer, Jeffrey A.* On invariant area formulas and lattice point bounds for the intersection of hyperbolic and elliptic regions. Journal of Combinatorics and Number Theory 4 (3), 2012.
- Cheung, Wilson*; Rault, Patrick X. On uniform bounds for rational points on quadratic rational curves and thin sets. Journal of Algebra and Number Theory Academia, August 2012, 37-62.
- Rault, Patrick X. On uniform bounds for rational points on rational curves and thin sets. JP Journal of Algebra, Number Theory and Applications, 23 (2), 2011, 171—185.
- Rault, Patrick X. On uniform bounds for lattice points in intersections of hyperbolic plane regions. Journal of Combinatorics and Number Theory. 2 (3), 2010, 209--215

(Student collaborators marked with *)

**Suggested pre-requisites**: good comfort with linear algebra and either number theory or abstract algebra. Willingness to work with the programming language *Mathematica*.

To participate, you must at a minimum:

- be a US Citizen or Permanent Resident, as stipulated by the National Science Foundation (NSF) grant.
- be a declared Mathematics Major, because our grant is funded by the NSF’s Division of Mathematical Sciences.
- have a GPA of AT LEAST 3.3 in mathematics and 3.0 overall. Note that using this project to pursue an Honors Thesis will require an additional minimum of 3.7 in math and 3.4 overall.
- have completed Math 223 (Calculus III), Math 233 (Linear Algebra I), and Math 239 (Introduction to Mathematical Proof) before fall 2014.

In addition, your application will be strengthened if you:

- have completed one or more computer science courses, such as CS 119 or CS 120, or are extremely comfortable with programming languages like Mathematica or Sage.
- have completed relevant 300-level mathematics courses, such as Math 333 (Linear Algebra 2) or Math 330 (Abstract Algebra).

Research experiences often help us increase our self-efficacy and our motivation (see COEUR: A Comprehensive Framework for Institutional Self-Assessment) throughout our time as an undergraduate student. For those interested in pursuing a Ph.D. program, a research experience as a student can give us a taste of what graduate school would be like before committing to it. Employers are impressed by research experiences and/or internships, as they demonstrate deep problem solving ability, effective teamwork, and the ability to share one’s findings with a larger audience.

In addition, students are expected to gain:

- A $3,000 stipend from CURM, payable in increments as per their policy: “$1,000 paid at the beginning of the Fall and Winter semesters, $500 paid after the student presents his/her research at the Student Research Conference, and $500 paid after the student submits his/her final research report. Students are expected to spend ten hours per week during the academic year working on the research project.” In addition, “CURM will also cover the travel and lodging for each undergraduate student participant up to $650 per undergraduate student to attend the Student Research Conference at BYU.” See http://curm.byu.edu/mini-grants for full details.
- A speaking opportunity at both GREAT Day and at conferences outside of Geneseo, including the CURM conference in March in Utah (a requirement of all participants), and most likely one of the following: the MAA Seaway Section meeting in April, the Hudson River Undergraduate Mathematics Conference in April, and/or Mathfest in early August.
- If your talk is successful, then you will complete the mathematics department’s Oral Presentation and Research requirement (often completed via Math 348).
- If you qualify for an Honors in Mathematics, you are encouraged to follow the guidelines for an Honors Thesis in Mathematics. This will be extra work beyond our core project, as it includes a substantial thesis written independently by you and outlining your personal accomplishments and contributions.
- Hopefully, a publication in a professional mathematical journal. Such a publication would give you an Erdös number of 4.
- 3 credits of directed research or directed study enrollment (likely through Math 398, 399, or 393) in both the Fall 2014 and Spring 2015 semesters.

All students participating in this project will officially enroll in a 3-credit directed research class (likely Math 398, which should be newly available this fall -- possibly Math 393 or 399) in both the fall and spring semesters. As such, students should expect to work about 10 hours per week throughout the academic year.

This is not a course in the usual sense – but rather a working collaboration. A high level of professionalism is required: do not wait until the last moment to finish tasks, don’t be afraid to ask questions or look things up, come prepared to meetings, don’t waste time, give forewarning if you are going to cancel a meeting, and be sure to dress and act professionally at conferences where you will be representing SUNY Geneseo.

Most research progress takes place outside of meetings. However, you should expect to meet with your faculty adviser between 2 and 4 hours per week in both the fall and spring semesters. If one week you are unprepared or overwhelmed with exams, then be sure to cancel the meeting in advance: remember to avoid wasting time. Conversely, it is expected that the student leave extra time aside at high stress times like conference presentations - expect to meet extra.

Most of our work will be done outside of our meetings, either in submeetings between students or alone. This will require a significant drive and efficient time-management to make regular accomplishments relevant to the project.

Our finished product will hopefully be a publication in a professional mathematical journal. In mathematics we list all authors alphabetically (by last name), as the finished product would not be possible without everyone's contributions. Anyone who proved results given in the paper will be included as an author. We all need to work hard, together, to achieve the best possible theorems and writing, so strong collaboration is mandatory. It is therefore counterproductive to have a competitive mentality with others in our team: we all need to help each other out and rely on each other's strengths.

You are expected to prepare a professional talk, written professionally using LaTeX, and speak both at GREAT Day and at the CURM conference in March in Utah (a requirement of all participants, for which we have funds from CURM). These talks will require a significant amount of practice (in front of team members, friends, and alone) to make them presentable in a professional setting.

In addition, each participant is expected to apply for a SUNY Geneseo Undergraduate Travel grant for the spring semester, and possibly for the summer, while meeting all relevant deadlines (such as a **Student Proposal Workshop **and the subsequent grant deadline). Participants should speak at a second conference outside of Geneseo, most likely one of: the MAA Seaway Section meeting in April, the Hudson River Undergraduate Mathematics Conference in April, and/or Mathfest in early August (for which we have additional funds for members of Pi Mu Epsilon).

During the spring semester all results need to be written up in a professional, clear, and mathematically correct form, including all proof details for a paper for publication. This is a team collaboration, so effective communication and cooperation is essential. The writing should be complete enough that during the summer, after the collaboration is complete, your faculty mentor can put the finishing touches onto the paper and submit it for the team.

Our funding agency, CURM, has a number of requirements for each participant. These include:

- Complete our Training in the Responsible Conduct in Research course
- Each participating undergraduate student must complete a W9 form. Also, students may choose Direct Deposit for stipend payments by completing the form available online here and returning it to us with their W9. Using Direct Deposit can provide for faster receipt of funds.
- Coordinate directly with CURM for refunds and other needs.
- Anything else they may ask from you after the project begins. They will be provided with your direct contact information.

All of the above must be completed by the last day of classes in spring semester. The project is complete at the end of the spring semester, at which time your faculty mentor will have other obligations – including publication of the results.

There are many resources available to us in the academic community, but sometimes they take some time to make use of. Our library's math resources website has most of the essential tools we will need to look up what others have done. For example, MathSciNet is the primary tool nationwide for finding mathematics research papers, and we have free access when connected to the internet via Geneseo (e.g. on-campus or via VPN). MathSciNet also has direct links to our library's very quick article delivery system, IDS. Librarians are available for professional research consultations to discuss search tools and search tricks, which even Dr. Rault still learns a lot from. Participants should become comfortable with using these resources.

In a professional setting such as this, not everything can be foreseen in advance. Keep an open mind, don't hesitate to make suggestions, and be proactive.

- Complete the application form here.
- E-mail a
**resume**and**unofficial transcript**(copied from Knightweb is fine) to rault@geneseo.edu

Dr. Rault will begin reviewing applications near the end of spring break, but will continue interviewing candidates and accepting new applications until all decisions are made. We hope to finalize our team by mid-April.

If you have any questions about the project, please do not hesitate to contact Dr. Rault.