BSM-TDK — Research Opportunities

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 or by email). Final enrollment will be based on your 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 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 Semester. 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.

PROBLEMS PROPOSED FOR SUMMER 2017

Title: Rigid and Globally Rigid Graphs Description: see here Prerequisites:graph theory Best for:students interested in graph theory and applications Professor: Professor Tibor Jordan Assignment for the first week: see the warm-up exercises in here

Title: Packing sparse degree sequences Description: The edge packing or graph factorization problem asks the question if an ensemble of edge disjoint graphs exists with prescribed degrees. The problem in general is a hard computational problem, however, it is easy for special cases. One special case is when the graph is very sparse, the sum of the degrees is less than or equal to 2n - 1, where n is the number of vertices. There are results when each degree sequence is a tree degree sequence, that is, each degree is positive, and the sum of the degrees is 2n-2. The proposed research project is to give necessary and/or sufficient conditions for sparse degree sequences to have edge disjoint realizations. More detailed description can be found here: http://www.renyi.hu/~miklosi/2017SummerRES/Packing.pdf Prerequisites: basic combinatorics and graph theory Best for: students interested in combinatorics, discrete mathematics and computer science Professor: Dr. István Miklós Assignment for the first week: http://www.renyi.hu/~miklosi/2017SummerRES/Packing.pdf

Title: Finding sails for summer 2017 Description:see here Prerequisites: basic combinatorics; Best for: students who intend to do research in combinatorics Professor: dr Andras Gyarfas Assignment for the first week: read article and do and email exercise 1& 2 in here