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. (such as The Rose-Hulman Undergraduate Mathematics Journal or Involve.

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

At BSM 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 at the Welcome Party, but read everything carefully below first.

• Who can participate? Each professor gave a list of problems and/or some reading and related tasks for those who are interested in working on their problem. If you are interested in participating, do these by the first week (the exact deadline will be discussed with the professor at the Welcome Party). Final enrollment will be based on work on these problem sets/reading assignments.
• Which topics will actually be offered ("stay alive")? Of the initially offered research topics those will be offered eventually, for which a group of students (at least around 3) sign up and are accepted by the Professor based on first week performance as outlined above.
• Course work: weekly meetings. The research groups will meet 3-4 times weekly, two hours each.
  Two meetings are devoted to group work, when you discuss the problem and possible solutions with your student group without the professor. The other meetings of the week are spent with your professor who will monitor your group's progress. Whether you meet once or twice with your Professor on a given week will be decided case-by-case, depending on progress.
• Course work: presentation. Work continuous thrughout the Summer Session. The 5th week each research group should present their results at a "Preliminary report session" organized for all BSM-TDK participants, their professors and everyone else interested.
• Course work: writing a paper. Depending on results obtained all work will be summarized in a paper/research report.

HERE IS A LIST OF PRELIMINARY RESEARCH TOPICS OFFERED FOR THE SUMMER SESSION, 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
   ASSIGNMENT FOR THE FIRST WEEK: exercises and related reading
   
   
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
   ASSIGNMENT FOR THE FIRST WEEK: exercises (do at least part (a), if you are interested in participating)



3. Title: Coloring t-intersecting hypergraphs
   
   Description: click here for the description of the problem
   
   Prerequisites: basic combinatorics;
   Best for: students who intend to do research in combinatorics
   Professor: Dr. András Gyárfás
   ASSIGNEMENT FOR THE FIRST WEEK: will be given at the Welcome Party



4. Title: Realizations of graphical degree sequences
   
   Description: Let D = d_1, d_2, ... d_n be a series of positive integers. We say that D is a graphical degree sequence, if there exists a simple graph G whose degrees are the given numbers. G is called a realization of the degree sequence. Erdos and Gallai gave sufficient and necessary conditions when a degree sequence is graphical, and Havel and Hakimi gave an algorithm how to construct one realization if exists (otherwise report that the sequence is not graphical).
   
   We are going to study the properties of the realizations of graphical degree sequences, extending it to mulipartite graphs, restricting the solutions to some subsets of solutions with prescribed properties, etc., based on our earlier results, see: article 1, article 2 and article 3.
   
   Prerequisites: basic combinatorics
   Best for: students who intend to do research in graph theory
   Professor: Dr. István Miklós
   ASSIGNMENT FOR THE FIRST WEEK: Read Chapter 15 and solve exercises 15.1-15.6.