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We present an overview of the goals, structure, and results of a program of vertical integration of both research and training in the Mathematical Sciences at Rice University. We place particular emphasis on our key structural element, the PFUG - pronounced fugue in order to stress the analogy with a work propelled by interacting voices.


The mathematical sciences departments at Rice University have been engaged, since 2003, in a program of vertical learning and research. Nudged by grants from the National Science Foundation, we have adapted to mathematics (and our university) the type of "learning communities" common in other disciplines. The purpose of this collection is to encourage and equip, by example, other departments and universities to embark on the vertical integration of the their education and research efforts.


There are many goals to our VIGRE program, most of them in response to issues we saw in mathematics education at the undergraduate or graduate level. In general, we felt that mathematics research began too late, both for the development of an appreciation of research in the mathematical sciences and also as a recruitment method for the mathematical sciences. A cartoon of the model might be this. A sophomore interested in mathematics, biology and chemical engineering, but not sure which one to really focus on, asks a professor in each subject what it's like to work in their field. The bioscientist and the engineer offer the student lab space and a sliver of a research project that might lead to an authorship of a paper in a year; the mathematician assures the student that there's a great problem waiting for them in their third year of graduate school - they just have to be patient. The ambitious, impatient 19-year old, despite a possibly greater interest in the mathematical sciences, goes elsewhere.
Even the graduate students have to wait, and don't really get used to the model of doing research, with its mode of failing in ever more sophisticated ways until one succeeds, until the middle of their graduate education. Even then, they typically only get exposed to one area at the research depth, and one faculty member's ways of approaching problems.
The postdocs are at the most fragile stage of their professional career, with limited experience doing research problem, little formal outside assistance and the most pressure to create the most research product within the shortest of time intervals. And the whole system is quite stratified, with the faculty member interacting with each stratum (undergraduate, graduate, and postdoc) independently, but rarely having two adjacent strata teach each other in any but the most formal, e.g., grader vs. gradee, of settings.
Our goals were to introduce a variety of research experiences to all of these types of students, and even to different levels within the strata, in a way that eroded the barriers between the different academic stages.


Our basic organizational element is called a PFUG. The acronym refers to a group of Postdocs, Faculty, Undergraduates and Graduates working together on a problem; the term is meant to recall the musical fugue with its many different voices, all supporting the theme, none subordinate to the others.
In its ideal form, the PFUG involves all four ages of mathematical experience. The postdoc tends to guide the day-to-day research by the graduate students and the undergraduates (with the graduate students passing on to the undergrads the basic vocabulary and established techniques of the field, which the undergrads incorporate into their parts of the project). The faculty member has chosen the project for the group s/he has, and gives either gentle guidance and a few ideas/reactions or much more active participation, depending on the circumstances. As with most research, the project lurches forward, occasionally gets stuck (at which point the more senior participants get more actively involved), and eventually (and occasionally rapidly, given the number of fresh eyes) happens upon a direction in which substantial progress is made.
At Rice University, we do have PFUGs that run just like the ideal. On the other hand, there are many variations on these PFUGs. Many of our PFUGs have a substantial learning and presentation component: often the most junior members present material mostly known by the older participants, who help the presenters with their presentation and the material, if needed. We tend to require some sort of written summary, either as a poster or a paper - that has real value, both in introducing the writing process and in solidifying the progress. Some of our PFUGs are really PUGs, PFGs, or FUGs: it all depends on the participants and the topics - but many of the effects are the same. The U and G participants enroll in PFUGs for variable credit, from 1 for the engaged consumer to 3 for the regular contributor, during the academic year. The best undergraduates are then selected to conduct directed PFUG summer research for a modest stipend. One of the PFUGs has even had a consistentcontingent of High School students, receiving High School credit for their work.

Other structural components

Our PFUGs are nourished by our participation in a campus wide Research Info Fair held the first week of Fall classes as well as at two dedicated Math Poster/Info Sessions. The latter are held each November, during registration spring courses, and each April, during registration for fall courses and application forsummer internships. The larger student body learns of these events from Ambassadors (undergraduate PFUG veterans) that, in addition to creating fliers and facebook events, visit targeted class rooms and share with their peers their enthusiasm for research.

The results

We've been very pleased with the outcomes of our programs. Our undergraduates are much more excited about the mathematical sciences, and our numbers of majors have increased slightly. Moreover, we see a lot more enthusiasm for mathematics in the participants; they are more interested in graduate school in the mathematical sciences, and they see how their courses fit together in ways they didn't seem to appreciate earlier.
Our graduate students and postdocs have gotten experience with multiple research topics and with teaching in ways besides the traditional formal classroom instruction. They are more comfortable with each other, and seek each other out for help. Our faculty have found new ways to engage students and new resources and modes for their own research programs.
The foremost cost to the program is the time of the postdocs. Universities that try this model will either have to divert some faculty time into it or find resources to hire a cohort of postdocs to engage with it.

The topics

We are, of course, very flexible with respect to topic. Many of our most successful projects are quite interdisciplinary; this attracts mathematical scientists with other interests into the subject and displays the wide applicability of mathematics. But there is also room for strictly disciplinary topics, as there are students with those interests as well. One trap is to worry excessively about the quality of the research. It certainly is important to do new things, and the novelty of the topic and the creation of knowledge adds immeasurably to the excitement and appeal of the group, but it is also worth remembering that the real product here is young mathematicians; often the theorem proved or the study completed is of secondary importance.
In the other documents in this Connexions Collection, you can read reports by some of our PFUGs and trace their progress. We hope you find it inspiring and stimulating - inspiring in that you are impressed by what inexperienced researchers can accomplish in groups, and stimulating in that you see ways to adapt these approaches and topics to your own university.

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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