**Who can participate?**The list of research topics and professors proposing them is posted below. In addition to the problem description, each professor gives introductory problems for interested students to solve under "Qualifying problems". If you are considering participating, submit your solutions to the qualifying problems of the research problem that you would like to work on**by June 1st**, directly to the professor whose problem you are interested in.

Enrollment will be based on solutions submitted to qualifying problems to professors and your transcript and reference letter submitted as your application materials to the American office. Decisions on admittance to the research group will be made**by June 8th**.**Which topics will actually be offered ("stay alive")?**Of the initially offered research problems those will be offered eventually, for which a group of students sign up and are admitted to the program. Ideally, a group will consist of 3-5 students.**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. Around week 6 or 7 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.**Evaluations and grading:**Grading for the course is done on an A-F scale.

Based on all work up to that point, students receive "midterm assessment grades" around week 4/5. As part of the midterm evaluation a student may be required to submit a (relatively short) report on the work in progress, to their professor. (Thus the report should include eg a description of the problem, as well as the methods used in tackling it and a write-up of results, if any.)

Note that a student may have to drop the course, depending on their midterm assessment grade received for the research course but also their midterm assessment grades received in their other courses. (That is, research students are expected to do well in their other course(s), too.)

Students also receive a course grade at the end of the summer term.

The transcript will contain only the course grade.

This course is designed in the style of
the Hungarian "TDK" system, allowing *advanced* undergraduates to
become acquainted with research methods 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 research paper, ideally, however that is not expected given the time constraints.
In addition, during the semester there will be an opportunity
to present your work as well.

Participating in the research course 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,
Involve and
several others.

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.

You can view
articles that were written under the auspices
of the BSM program

**Description:** Click here.

**Prerequisites:**We shall mostly use graph theoretic and combinatorial methods, so familiarity with
the basics of graph theory is useful.
Depending on the problems, in some cases
geometric intuition is also useful, as well as
familiarity with elementary linear algebra.

**Professor:**Tibor Jordan

**Contact:**tibor.jordan@ttk.elte.hu

**Qualifying problems:**Solve at least four out of the
five exercises at the end of this document.

Long time ago Paul Erdôs posed the natural question if there exists a universal infinite set. The long standing Erdôs similarity conjecture states that there exists no infinite universal set. Although the problem seems pretty simple it turned out to be surprisingly hard. It is known that any slowly convergent sequence forms a non-universal set but the conjecture is so wide open that it is not known for ANY exponentially quickly converging sequence if it is a counter-example or not.

Of course, it is not a realistic goal to prove or disprove the conjecture itself but perhaps we can find interesting related problems that we can answer. A natural, much more modest but probably still very hard goal might be to construct an exponentially quickly converging sequence that forms a non-universal set; that is, to construct an exponentially quickly converging sequence and a set of positive measure together such that the set does not contain any similar copy of the sequence. This - possibly much weaker - problem was posed by Mihalis Kolountzakis.

One can study a lot of variants in which we ask if large sets must contain some patterns. A concrete natural example is the following question, which seems to be open (but this needs to be checked): Does every measurable set of positive Lebesgue measure contain the translated copy of at least one infinite geometric progression? To get a negative answer is absolutely hopeless since it would clearly disprove the Erdôs similarity conjecture in an extremely strong sense. However, the far more likely positive answer can be much easier and perhaps still non-trivial.

The main task is to understand the nature of these problems, to learn different methods to attack them (techniques and ideas of analysis, probability theory, combinatorics and number theory can naturally show up) and to find some non-trivial and unknown variants that we can solve.