BSM-TDK — Elective Undergraduate Research

COURSE DESCRIPTION

This course is designed in the style of the Hungarian "TDK" system, allowing advanced undergraduates to become acquainted with research methods and means in detail and acquire additional knowledge beyond their obligatory curriculum. (For a brief English description of the TDK system see a relevant ELTE University homepage.)

In this course, a student can choose from the topics/problems listed below and work with other students and the professor to solve the given problem. All work is summarized in a paper and during the semester there will be opportunities to present your work as well.

This may contribute to the successful beginning of a scientific career: depending on level, the results obtained can be presented at school, statewide or national undergraduate meetings ranging from a local Undergradute Seminar at your home school to MAA's MathFest. Papers may also be published in undergraduate research journals.

In some PhD programs, fruitful undergraduate reserach activity is a prerequisite for admission.

Student research is supervised by professors. Research topics are offered by them, but students can also propose topics of their own interest.

COURSE LOGISTICS

The list of research topics and professors proposing them can be seen below. Contact the professor whose problem you are interested in.

• Class will meet twice weekly, two hours each. One class time is devoted to group work, when you discuss the problem and possible solutions with your student group (without the professor). The other class time is spent with your professor who will monitor your group's progress.
• Week 3 - Milestone 1:: The first three weeks of the course are spent discussing, gathering and studying necessary background information for your problem. At the end of the 3rd week, you (and your group) will have to present the problem you are working on at a "mini workshop" organized for all BSM-TDK participants, their professors and everyone else interested. At this point professors evaluate progress and final enrollment decision is made. Ideally, the size of research groups is low, at most three.
• Week 9 - Milestone 2:: Work continuous thrughout the semester. At the 9th week each group should present their results at a "Preliminary report session" organized for all BSM-TDK participants, their professors and everyone else interested.
• Week 9/10-14:: Write up of results continuous and papers are produced.

BASED ON PREREGISTRATION DATA, THE FOLLOWING PROBLEMS WILL BE OFFERED Spring 2013

1. Title: Extremal sets of the vertices of the hypercube (over GDF(2))
   
   Description: We plan to investigate (cases of) the following general question: How many vertices (maybe of certain further property, like of fixed weight) of the n-dimensional hypercube can be picked such that subspace spanned by them - over GF(2) - does not contain or does not intersect certain configurations of the hypercube (vertices, vertices of given weight, subspaces, hyperplanes, etc.)
   
   In this project you will understand the structure of the hypercube over the reals and GF(2), develop algebraic methods to solve extremal set theoretical problems and establish constructions and will reach - in the worst case - some concrete results.
   
   Prerequisites: basic combinatorics and linear algebra
   Best for: students who intend to do research in algebra or combinatorics
   Professor: Dr. Dezsô Miklós
   
   
2. Title: Large set in the line without a given pattern
   
   Description: In the real line by a rectangle we mean points of the form x, x+a, x+b, x+a+b, and by the area of such a rectangle we mean the product ab. The question is how large a subset of the real line can be without having rectangle of area at least 1. One can get a more precise question in a number of different ways depending on the the assumption about the set and the measurement we use, but, as it can be easily seen, they are all equivalent, so students can choose the form they prefer. In the measure theoretic form we consider Lebesgue measurable sets and we use Lebesgue measure. In the combinatorial form the sets are finite unions of intervals and we consider the sum of the lengths of the intervals. In the number theory form the points of the sets are chosen from a grid and we consider the number of chosen points multiplied by the length of the grid. The methods of all the above areas can be used and it is not clear at all which will be useful. The motivation comes from a long standing very hard unsolved problem of A. Carbery. An arbitrarily large set without rectangle of area at least 1 would immediately answer Carbery's problem.
   
   Prerequisites: none
   Best for: Students with good problem solving skills who would like to work heavily on unsolved problems.
   Professor: Dr. Tamás Keleti
   



3. Title: Monochromatic connected pieces
   
   Description: A first exercise in Graph Theory says that either a graph or its complement is connected. This observation can be extended in many directions, summarized in a recent survey of the instructor. Hopefully ambitious students can make some advances among the many unsolved problems of this area.
   
   Prerequisites: basic combinatorics; read to help decide
   Best for: students who intend to do research in combinatorics
   Professor: Dr. András Gyárfás
   



4. Title: Spectral Clustering of Networks
   
   Description: Networks can be modeled by edge-weighted graphs, where edge-weights are pairwise similarities between the sites (vertices of the graph). We want to find clusters (in other words, communities) of the vertices such that the information flow between the cluster pairs and within the clusters is as homogeneous as possible; minimum and maximum multiway cuts are special cases. For this purpose, we define objective functions, for the minimization or maximization of which we use spectral relaxaton. To estimate minimum multiway cuts we use the smallest eigenvalues of the Laplacian or normalized Laplacian matrix assigned to the graph, whereas clusters are found by means of the corresponding eigenvectors. The methods are applicable to biological, social, or communication networks.
   
   Prerequisites: basic combinatorics and linear algebra.
   Best for: students who intend to do research in networks
   Professor: Dr. Marianna Bolla