Mathematics Research Experiences for Undergraduates at Iowa State University

Investigacion en Matematicas y Estadisticas
 
supported by the National Science Foundation through
DMS 0750986, DMS 0502354,
DMS 0353880
and the ISU Department of Mathematics


ISU MATH REU 2013    ISU MATH REU 2011     ISU Math/Stat REU10        ISU Math/Stat REU09      
ISU MATH REU 2006     
ISU Math REU05        ISU Math REU04         ISU Math REU03
ISU REU Publications                
         
This page is the genric ISU Math REU page.  For an individual summer, see links above.  The ISU Math REU will not be offered in 2014.  Please check back in 2015.


Participants spend eight weeks working on research projects. The projects are in a variety of mathematical areas, representing the diverse research interests of faculty in the ISU Mathematics Department, such as linear algebra, dynamical systems,
graph theory, numerical analysis, mathematical biology.   Students will work in teams as part of active research groups at ISU.  This is a research group based REU and all participants collaborate with others; if you prefer to work alone this REU is not a good fit for you.

At the beginning of the summer the mentors explain the necessary background to the students.  During most of the program, students conduct research, meeting daily with their faculty and graduate student mentors.  In addition to their own research, students attend weekly REU Seminars, where they hear faculty lectures on a variety of mathematical topics and on such topics as using LaTeX and attending graduate school, but the focus is on research and there is no workshop or class component to this REU.   The REU concludes with a symposium of student reports. Many projects submit papers for publication (list of papers) and students frequently present their REU research at conferences. 

Participants are provided a stipend, accommodation in University student housing, some travel reimbursement and some meals, and will have the opportunity to participate in social activities for REU students, both Math REU and campus-wide ISU REU activities.

More information about the ISU Math REU can be found in the 2012 article to appear in the a specila issue of Involve, Proceedings of the Trends in Undergraduate Research in Mathematical Sciences Conference, and an older 2006 article in the Proceedings of the Conference in Promoting Undergraduate Research in Mathematics, or on the webpages from prior years (linked above). 




Photo from Summer 2013 REU  


Math Project Descriptions 2013

 
Algebraic Graph Theory Group  Dr. Sung-Yell Song, Katy Nowak
 
Existence and Nonexistence of (Directed) Strongly Regular Graphs

A strongly-regular graph with parameters (v, k, λ, μ) is defined as an undirected simple graph G with v vertices satisfying the property:
“The number of common neighbors of vertices x and y is k if x = y, λ if x and y are adjacent, and μ if x and y are non-adjacent vertices.”
Let A denote the adjacency matrix of G, and let I and J  denote the v x v identity matrix and all-ones matrix, respectively. Then G is a strongly-regular graph with parameters (v, k, λ, μ) if and only if
(i) JA = AJ = kJ and
(ii) A^2 = kI + λA + μ(J-I-A)

A loopless directed graph D with v vertices and adjacency matrix A is called directed strongly-regular graph with parameters (v, k, t, λ, μ)  if and only if A satisfies the following conditions:
(i) JA = AJ = kJ and
(ii) A^2 = tI + λA + μ(J-I-A)

We are interested in constructing (directed) strongly-regular Cayley graphs of various classical groups with suitable generator sets. We are also interested in settling some problems related to characterization and classification of these graphs and related objects. So, the sample problems that we are tempted to
explore look like:
• Find (directed) strongly-regular graphs that can be obtained as Cayley graphs of classical groups.
• How many strongly regular graphs with parameters (64, 28, 12, 12) are there?
• Is there a directed strongly regular graph with parameters (24, 10, 5, 3, 5)?
• Characterize all tactical configurations that yield the directed strongly regular graphs with parameters ((n2 − 1)(n3 − 1)/(n − 1)2, n(n + 1), n, n − 1, n).


Analysis Group   
Dr. Justin R. Peters, Jiali Li
We will be investigating two unrelated problems. Both problems will require some background in introductory analysis. One of the problems will have a large computational compenent. Students may choose to be involved in one or both projects.

1) Given a transcendental function f(x), let p_n(x) be its nth Taylor polynomial. We will investigate the behavoir of the zeroes of pn in the complex plane. We will be asking questions such as, do the zeroes of p_n lie in some region we can determine? Do the zeroes of p_n accumulate on some curve, possibly after renormalization? How are the zeroes of p_n related to those p_1? Is there some way in which the zeroes of p_n approach the zeroes of f? Is there a minimum distance between the zeroes of p_n, which is independent of n? In the case f(x) = exp(x) this has been investigated and has led to some nice results.

2) Which Cauchy sequences {a(n)}_n>1 with a(n) > 0, n=1,2,... have the property that the sequence {b(n)}_n>1 is also Cauchy, where
    b(1)=a(1), b(2)=a(2)^b(1), ..., b(n+1)=a(n+1)^b(n)?
We note that the case where a(1) = a(2) = a(3) = ... = a > 0 has been solved, and the solution is both surprising and interesting, though the arguments involved use nothing more than elementary calculus.


Combinatorial Matrix Theory Group 
Dr. Leslie Hogben, Dr. Adam Berliner, Dr. Travis Peters, Dr. Michael Young, Nathan Warnberg
Minimum rank, maximum nullity, and zero forcing on a graph

The graph of a real symmetric matrix A=[a_ij] has an edge between i and j if and only if a_ij is nonzero.  Finding the maximum multiplicity of eigenvalue 0 among symmetric matrices having a given graph is the same as finding the maximum nullity and is equivalent to finding the minimum rank among symmetric matrices having the given graph.  Initially a subset Z of the vertices of a graph G are colored blue and the remaining vertices are colored white.  The color change rule is that if a blue vertex v has exactly one white neighbor w, then change the color of w to blue.   The set Z is a zero forcing set if after applying the color change rule until no more changes are possible, all the vertices of G are blue.  The zero forcing number is the minimum size of a zero forcing set.  The zero forcing number is an upper bound for the maximum nullity of a graph, and arose independently in the study of control of quantum systems in physics, where it is called graph infection or propagation.   This project will investigate problems related to minimum rank, maximum nullity, and zero forcing number.

Students involved in this project will be part the ISU Combinatorial Matrix Theory Research Group; more information is available on that page. This group regularly publishes its results (see list of papers).

Linear algebra is a prerequisite for this project, graph theory is an advantage, and a strong theoretical mathematics background (usually including abstract algebra or real analysis) is expected.  The software we use is Sage and Mathematica, so knowing one or both of these in advance is helpful, but you can learn one or both of here.

Automated Recursive Projected CS (ReProCS) for Real-time Video Layering and Performance Guarantees  Dr. Namrata Vaswani, Dr. Leslie Hogben, Brian Lois, Kevin Palmowski, Chassidy Bozeman

A large class of video sequences are composed of at least two layers - the foreground, which is a sparse image that often consists of one or more moving objects, and the background, which is a dense image, that is either constant or changes gradually over time and the changes are usually global. Thus the background sequence is well modeled as lying in a low dimensional subspace that can slowly change with time; while the foreground is well modeled as a sparse "outlier" that changes in a correlated fashion over time (e.g., due to objects' motion). Video layering can thus be posed as a robust principal components' analysis (PCA) problem, with the difference that the "outlier" for PCA is also a signal-of-interest and needs to be recovered too. Real-time video layering then becomes a recursive robust PCA problem. We will develop algorithms using a novel approach called Recursive Projected Compressive Sensing (ReProCS) to solve this problem and we will try to bound their performance under practically motivated assumptions. In particular, we will study how to handle temporal correlations in the low-dimensional part (background).

Advice to the Applicant

Students frequently ask what information is most helpful to us in making decisions and what they should do to improve their chances of selection.   Here is some advice based on what has happened previously (do not be discouraged if you can't follow all of it- not all successful applicants do) and some of it is my (Leslie's) opinion. 

Application
Follow directions. We require certain specified documents, NOT in your resume.  Answer the eligibility questions and what projects interest you question on the application site correctly- these are importnat and determin who reads the application.

Writing the personal statement
Our program has a strong project-fit element to the selection process: the project mentors have great influence on who is picked.  So the best advice I can give you specific to our program is
    read the project descriptions carefully
    don't ask for a project for which you do not have the necessary prerequisites
    do identify all projects that match your background and interests, and explain why you are interested in that project in you application
This is of course in addition to answering the rest of the questions we ask.

General preparation:
Take the hardest mathematics courses available.  We are looking for people who seek challenges and love math.  We are also looking for a strong foundation in proof-based (theoretical) courses in which you read and write proofs- in some colleges this is all math courses, in others the first proof course might be abstract or linear algebra.  Having taken more theoretical courses is always good, whatever area.  Work hard in your math courses- having a high GPA in math courses definitely helps (we are also somewhat interested in science courses, but are not interested in non math/science grades).  Other ways to demonstrate mathematical interest include taking the Putnam Exam, membership in Pi Mu Epsilon, or participation in a selective mathematics program (e.g., study abroad).

In addition to mathematics courses, some projects may have requirements in other fields of study (e.g., biology).

All of the following are useful skills (if you have them, say so), but are less important than your math background, as we can teach this if needed after you arrive: Matlab, Mathematica, Sage, ability to write in LaTeX.

Get to know at least one or two faculty members at your college well- letters of recommendation play an important role in selection.

Being a freshman is a disadvantage, but we will consider you (unlike some programs) provided you are at least a mathematical sophomore, i.e., will have completed 2 years of college mathematics (calculus and above) by summer 2013.

A final suggestion- apply to several REUs, as no applicant, however outstanding, can be certain of admission to one specific REU program.
Leslie

ISU Math Homepage  
Web page maintained by Leslie Hogben