# Dr. Patrick X. Rault

- Dr. George Haddix Community Chair of Mathematics

University of Nebraska Omaha

Dr. Patrick X. Rault has been a member of the University of Nebraska Omaha faculty since August, 2018. Before joining UNO, Dr. Rault spent eight years at the State University of New York at Geneseo and two years as Program Director of Mathematics for the distance college at the University of Arizona. You can find more information in his Curriculum Vitae.

For more information, please see below for the following.

- Current Projects
- Teaching
- Student Resources
- Research
- Publications
- Awards

## Current Projects

Here are a few of the activities which I am involved in:

- Research on open questions related primarily to my previous papers. See my CV in the link to the left for papers, workshops, and conferences.
- Writing course textbooks and e-books for Introduction to Proof (first edition published in 2017), Theory of Numbers, and Abstract Algebra.
- Training professors in Inquiry-Based Learning via the NSF PRODUCT grant, 2016-2020: national IBL Workshops.
- IBL Communities project, 2016-current. This stemmed off of the Upstate New York Inquiry-Based Learning Consortium. $100,000 grant for 2014-17. I am now also working on the IBL Iowa-Nebraska Community, which was formed in 2019.
- Council on Undergraduate Research. Elected nationally for a 2014-17 term on the governing council; re-elected in 2017 for a second 3-year term. Elected as division chair for 2017-2020.
- Center for Undergraduate Research in Mathematics, Co-PI and Co-Director for 2017-2022. NSF-DMS #1722563.
- Undergraduate Research Special Interest Group of the Mathematical Association of America, elected chair 2018-2020.

Click here to see a list of conferences that I have attended or plan to attend in the near future.

## Teaching

Here are a few useful resources for mathematics instructors.

__College Instructor Resources:__

IBL Iowa-Nebraska Community (IBLINC)

NSF funded national Inquiry-Based Learning (IBL) Workshops.

Academy for Inquiry Based Learning

Journal of Inquiry Based Learning in Mathematics

Priority sorting program for IBL courses, created by Fall 2011 Teaching Assistant Brian Knapp. **See below.**

Center for Undergraduate Research in Mathematics

Council for Undergraduate Research Math-CS Division

**Priority sorting program for Inquiry Based Learning Courses**

(Program by Brian Knapp, Fall 2011 Teaching Assistant for Math 239, Introduction to Mathematical Proof, at SUNY Geneseo).

This program has three pieces, and one "ReadMe" file with directions for use. There are two packages available here, one from 2011 and one from 2012. Each contains the following files:

* PrioritySystem.jar - the program, written in java (you may need to find and download java separately), for imputting grades, sorting students, and displaying the sorted list.

* ClassList.csv - a classlist template in the "commas separated value" format, readable by MS Excel.

* Options.txt - a file which lists parameters for changing the sorting algorithm based on frequency of class meetings and frequencey of presentations.

* ReadMe.rtf - a file which gives instructions for use of the program.

While this program is now being used at several other colleges (including Nazareth College in Rochester, NY and University of Toronto at Mississauga), the original version of this program was used in a 25 student sophomore-junior level Introduction to Mathematical Proof (Math 239) course at the State University of New York, College of Geneseo, using a textbook by Ron Taylor and me. This 3-credit class met for three 50-minute sessions per week, for 15 weeks.

Students were awarded 0-4 points for presenting proofs in class, and 0-3 points for presenting a calculation (see my article in the MAA Notes for more detail). These points were inputted into the computer program, which then displayed the re-sorted list on a smart-board screen; note that this Smartboard was not a primary board, but rather an "overflow" board for when the class ran out of space on the various blackboards in the room, and for when the instructor had something to present. The program also includes a "recentness" feature - in this version of the program students are given 125 temporary points on the day which they present, which is reduced to 25 the next day, 5 points the next, 1 point the next, and then 0 thereafter. This prioritizes students who have not presented recently, and prevents students from presenting on consecutive class meetings. You are encouraged to attend an AIBL PRODUCT workshop for more information about IBL in Mathematics.

By popular request of students, the program was initially designed to quickly and efficiently display to students "who was up next" in the order of presentations; now, at the end of each class, I press the spacebar to reorder the list to the next day's recentness values. The program has also had the effect of giving a more transparent and fluid grading scheme, where students can quickly understand why they are or are not being chosen for a presentation.

More details about every facet of this program can be found in the ReadMe.rtf file (enclosed in either package), written by Brian Knapp.

Here are the files to download:

New package, with priority queue at the top of the screen. This is perfectly suited for rooms where the projector screen covers the blackboard or whiteboard, but one may pull down the screen just a few feet to allow for blackboard use "in the spotlight."

The original package, with the queue on the left side of the screen. This was designed for use on a Smartboard, where the left side is cannot be written on because of the presence of page tabs.

Note: if you are having trouble opening these zipped folders, please look into getting an unzipping program like Winzip.

## Student Resources

Here are some resources which may be useful to mathematics students.

__Student resources:__

LaTeX starter file, with many examples which you can edit and see the effect of.

Undergraduate Semesters in Mathematics

__Career resources:__

For those who ask "Who uses math?"

AMS Careers and Employment page

Career Options for Undergraduate Mathematics Majors: 2011 Pannel

__Undergraduate Semesters in Mathematics__:

Want to study abroad and learn a lot? Does full immersion in mathematics sound fun to you?

Study abroad programs combine the instruction of college with the allure of foreign cultures. More and more students study abroad to fulfill their college's requirements, through first-hand experience, on history and language.

Unless your major is History or a foreign language, there are few chances for upper-level courses in your major whose credits will transfer effectively. That is, until Budapest Semesters in Mathematics (BSM) became popular in 1985.

Hungarians have a long-history of smarts and make no reservations in telling you about it – more Nobel Prize winners than any other country – many chess champions and the inventor of the Rubik's Cube – and the lingering legends of study-places of great mathematical problem solvers like Paul Erdos and G. Polya.

**Budapest Semesters in Mathematics** puts North American Mathematics majors into a great study environment, allowing them to take full semesters in only mathematics (as well as courses in Hungarian culture, history, and language). Students from research institutions like MIT come for courses with renowned professors, while students from liberal arts colleges come to supplement their education and test whether Graduate School is right for them with a semester of only math. Independent studies with other renowned local mathematicians were possible (I studied Quadratic Forms with Joseph Pelikan). Inexpensive weekend trips to the many nearby destinations (Prague, Krakow, Vienna, Brasov, Croatia, etc) were common.

**MASS Program at Penn State. **More recently the Mathematics Advanced Studies Semester (MASS) program at Penn State University has opened up a similar opportunity for those who can't afford to go so far from home. MASS offers a seminar about the “must see” problems in math, a colloquium featuring such renowned speakers as John Conway (see my puzzles and game page), and three specialized graduate-style courses in the respective areas of Algebra, Topology/Geometry, and Analysis; these three courses vary widely: in 2001 Algebra concerned Combinatorics, Topology was Relativity Theory, and Analysis was Fluid Dynamics. In MASS one takes fewer courses than BSM (and has no choice of courses), but these well-taught courses give an excellent taste of graduate school. MASS adds to the anti with a well-payed summer Research Experience for Undergraduates (REU), and provides funding for those who cannot otherwise afford the semester program.

**Math in Moscow. **MASS was founded in part by a few Russian mathematicians who kept parts of the old system – the most daunting of which (though excellently done) are the oral exams. Their Russian colleagues in Moscow saw the success of BSM and MASS as a sign to start their own program: Math In Moscow (MIM). The Independent University of Moscow (IUM) was historically created to compete with bad political situations as the renowned Moscow State University (MSU). Excellent but persecuted professors and students chose the IUM for lack of other options; their friends and sympathizers joined them. Today many co-teach or co-study at MSU and IUM. Spring 2003 was one of the early years so my opinions may not be representative; with four students instead of today's 20, the courses were very small (1-3 students each) and student-teacher contact was extraordinary. The caliber of professors combined with the interactions was the best of all three programs. However, I would recommend against arriving in January as Russian Winters are renowned to be dreary (i.e. start your studies in the Fall).

To my knowledge, I am the first and only student to have attended all three programs. Would I recommend it? No. I recommend instead staying a year in one location and getting to know the professors well through research. In Mathematics, relationships with advisors are analogous to those with significant others: long-term relationships will get you further than short flings.

There was an article in the Notices of the American Math Society (AMS) comparing the programs, which can be found here.

To my knowledge, these are the only semester-long college math programs. There are, however, a number of advanced master's programs: Cambridge TRIPOS in England, Ecole Normale Superiur in France, and the Utrecht Master Class in the Netherlands.

Disclaimer: These are opinions and memories of the reader only. Facts may not be completely accurate. I appreciate any citations you may have.

## Research

I completed my Ph.D. at UW Madison in 2008. My areas of research are number theory (specifically, arithmetic geometry) and linear algebra (specifically the study of numerical ranges in matrix analysis). My thesis advisor was Jordan Ellenberg.

For more information about my background, you can see my CV at the top of the page.

### Undergraduate Research

Undergraduate Research is defined as "An inquiry or investigation conducted by an undergraduate student that makes an original intellectual or creative contribution to the discipline." (Council on Undergraduate Research). Here at the University of Arizona, we are exploring some official channels for bringing undergraduate research opportunities to more students. Some examples of successful programs which I have been involved in are individual Directed Research projects, Honors Theses, Research Weekend, summer NSF-supported REUs, and class taste-of-research experiences via Inquiry-Based Learning.

This page currently focuses on individual directed research projects.

My past students have presented their work at many internal and external forums in which to share and present their research. Here are a few examples of conferences I have some experience with, where mathematics students can present their research, and to which I may have funding to support your trips:

**Local Undergraduate Research Days**are forums for students to present results from projects, directed studies, or research.**Local conferences**. Students can have opportunities to present their research projects in MAA, ASA, NCTM or other professional meetings. Undergraduate students give talks at each of the meetings. Other meetings involve various activities, such as a poster competition, a game show, an ice cream social, or a scavenger hunt.**National conferences**. MathFest, the Joint Mathematics Meetings, the National Conference on Undergraduate Research, and especially Posters on the Hill are all very prestigious places to present your research.

**Why do research?**

Research projects are valuable experiences for undergraduate students. For instance, a research project:

- Enhances student learning through mentoring relationships with faculty
- Increases retention
- Increases enrollment in graduate education and provides effective career preparation
- Develops critical thinking, creativity, problem solving and intellectual independence
- Develops an understanding of research methodology
- Promotes an innovation-oriented culture

(cf. CUR)

**How to get started? **

__1. Ask a professor.__ Some professors have ideas for research projects or directed studies, so visit office hours and ask around. However, advising a student in research takes a lot of time, so be respectful when they decline. Take a look at each professor’s website to get an idea of what topics they are interested in. Some professors have sites specific to working with students, and we may list some links here. Others have specific semesters in which they get many students involved in research.

__2. Attend math talks or via online math talks.__ Sometimes questions discussed in talks can lead to interesting projects. If you go to a physical mathematics talk, such as a colloquium talk, since many professors are present so it can also lead to a natural person to work with.

__3. Math journals.__ One easy type of project to get involved in is to try solving a problem posed in a math journal. For example, you can find current problems from the following journals: Math Horizons, Math Magazine, College Math Journal, American Math Monthly, and the Pi Mu Epsilon Journal. Dr. Rault has access to each of these. These problems can be worked on with other students or even with a professor, and correct solutions submitted by their posted deadlines will be acknowledged in the next issue. Several of Dr. Rault's past students have been cited in these journals (see CV).

**Some recent student research projects: **

**Coons, Jane I; Knowles, Doug; Jenkins, Jack; Luke, Rayanne; Rault, Patrick X.**Numerical Ranges over Finite Fields. Linear Algebra and its Applications. 501, 2016, 37-47.**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.**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, 161-175.**Cheung, Wilson; Rault, Patrick X.**On uniform bounds for rational points on quadratic rational curves and thin sets. Journal for Algebra and Number Theory Academia. August 2012, 37-62.

**Expectations of the student:**

__a. Professionalism__. 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.

__b. Meetings__. Most research progress takes place outside of meeting times. However, it is expected that a faculty adviser may ask you to meet for 1 to 3 hours per week. 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.

__c. Agreement to disseminate__. Faculty receive credit for doing research when talks or publications arise as a result. Therefore, it is an expectation that, if the faculty adviser deems the project a success, the student (i) seek travel funding to attend a conference, (ii) create a talk using LaTeX and practice it under supervision of the faculty adviser, and (iii) help in writing any requested proof details for a paper for publication.

**Suggested pre-requisites for extended projects:**

Seminars and Inquiry-Based Learning experiences can provide a taste-of-research, with many of the benefits described above. But if you are looking for the full experience:

- It is recommended that students have completed
*Math 2230, Introduction to Abstract Mathematics,*before seeking out research projects in mathematics. - In addition, many projects may require knowledge of advanced junior or senior level courses in mathematics, as well as computer programming and writing skills. However, a good mentor may be able to find some good problems for you regardless.
- Education research projects are also possible, with equivalent education department background requirements to those listed above. Please come chat with me if you're interested in doing a project about Inquiry-Based Learning or something similar.

**Can I get credit?**

Yes, it may be possible to enroll in a directed research or directed study course for credit. However, this may take some additional paperwork and may increase the expectations of you in the project.

**Can I get paid?**

Payment in mathematics research is rare, unless you attend a summer Research Experience for Undergraduates. The university does have some programs which offer some support for students doing research in other disciplines, so if your participation in the project is contingent on not working in a job then please let Dr. Rault know so that he has a documented need and can seek further funding.

**Resources **

- http://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044
- http://www.maa.org/programs/students/undergraduate-research
- http://www.ams.org/employment/reu.html for a list of REUs

## Publications

- Cushman, Jane R; Gantner, Ryan; George, C. Yousuf; Morrow, Margaret L; Rault, Patrick X. A Model for Expanding Active Learning Regionally: The Greater Upstate New York Inquiry-Based Learning Consortium,
*PRIMUS,*July 2018. DOI: 10.1080/10511970.2018.1424743. - Taylor, Ron; Rault, Patrick X. A TeXas Style Introduction to Proof. Mathematical Association of America (MAA) Textbook Series (July 2017).
- Rault, Patrick X. Teaching proofs via Inquiry-Based Learning. MAA Notes Volume on "Innovative Techniques for Teaching Proof-writing." Mathematics Association of America (MAA) Notes volume, 2016, 177-184.
- Coons, Jane I; Knowles, Doug; Jenkins, Jack; Luke, Rayanne; Rault, Patrick X. Numerical Ranges over Finite Fields. Linear Algebra and its Applications. 501, 2016, 37-47.
- 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.
- 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. On uniform bounds for rational points on rational curves and thin sets of arbitrary degree. Journal of Number Theory. 133 (9), 2013, 3112-3118.
- Rault, Patrick X; Sendova, Tsvetanka; Spitkovsky, Ilya. 3-by-3 matrices with elliptical numerical range revisited. Electron. J. Linear Algebra 26, 2013, 158-167.
- 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
- Rault, Patrick X. On uniform bounds for rational points on rational curves and thin sets. Ph.D. Thesis, 2008
- Johnson, Charles R.; Harel, Yonatan; Hillar, Christopher J.; Groves, Jonathan M.; Rault, Patrick X. Absolutely flat idempotents. Electron. J. Linear Algebra 10, 2003, 190-200

## Awards

- National Science Foundation grant to support the Center for Undergraduate Research in Mathematics (CURM). As Co-PI and CURM Co-Director, I am aiding in training national faculty applicants to work with undergraduates on research. NSF-DMS. $1,387,000 for 2017-2022.
- Council on Undergraduate Research Transformations Project, to support implementation of new curricula to include scaffolded undergraduate research experiences in two science departments. Principal Investigator; project includes over 15 colleagues & administrators.
**Henry L. Alder Award, presented by the Mathematics Association of America (MAA), 2015**- Center for Undergraduate Research in Mathematics (CURM) $22,000 mini-grant for a 2014-15 student research group.
- Elected to the Council on Undergraduate Research (CUR) for 2014-17
- Educational Advancement Foundation (EAF) grant to support training initiatives in Inquiry-Based Learning (IBL). PI for $102,210. 2014-17. Joint with St. John Fischer College, Nazareth College, SUNY Plattsburgh, and Buffalo State.
- Council on Undergraduate Research (CUR) Mathematics and Computer Sciences Division 2013 Faculty Mentoring (National) Award for Outstanding Mentoring of Undergraduate Students in Research.
- "Honoring Teachers" Award, by the Teaching and Learning Center and the SA Academic Affairs Committee, for "positively impacting students' experience." 2013-2014.
- Project NExT Fellow (New Experiences in Teaching), 2008-2009
- NSF VIGRE Fellow, 2004-2005
- Barry M. Goldwater Scholar, 2002-2004