## 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, Involve or many 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

## COURSE LOGISTICS

The list of research topics and professors proposing them can be seen below. Contact the professor whose problem you are interested in.**Who can participate?**Most professors 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 as much of these as you can by the Welcome Party and discuss your progress with the professor. First enrollment will be based on work on these problem sets/reading assignments. Final enrollment will be decided by the third week (as explained below).**Which topics will actually be offered ("stay alive")?**Of the initially offered research topics below those will be offered eventually, for which a group of students (at least around 3) would like to sign up and are accepted by the Professor based on discussions at the Welcome party and/or the first week.**Course work: weekly meetings.**

Class will meet twice weekly, for 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.**Course work: presentations.**

__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 in 20 minutes 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 7 - Milestone 2:__around week 7 students receive "midterm evaluation grades" in each course they are taking informing them of their course grade up to that point. A student with an insufficient overall performance (e.g. C's in all other classes) will have to finish doing research at that time and will receive an "Audit" on their BSM transcript.__Week 13 - Milestone 3:__: Work continues thrughout the semester. At the 13th week each group should present their results at a "Preliminary report session" organized for all BSM-TDK participants, their professors and everyone else interested.

Write up of results is continuous sometimes streches after the semester is over.

## TOPICS PROPOSED — FALL 2017

- Title:
**Vertices of the hypercube within a sphere**

**Description:**For given n and 0< r < sqrt n / 2 find the maximal number of vertices of an n-dimensional unit cube that can be covered by a ball of radius r (the vertices of the n-dimensional unit cube are all the 0-1 vectors of lenght n).

**Prerequisites:**basic combinatorics and analysis**Best for:**students interested in combinatorics and applications**Professors:**Dr. Dezso Miklos and Dr. Tamas Keleti**Assignment for the first week:**Solve the problem for n=7 and r=0,85 (the assigment is due by 4:30pm of Monday, the 4th of September, i.e., the welcome party) -
Title:
**Trying to pack even more trees and caterpillars**

**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 degree sequences are tree degree sequences, that is, all degrees are positive and their sum is twice the number of vertices minus 2. Those degree sequences can always be realized by trees and also by some special trees called caterpillars (a tree is called a caterpillar if the vertices with degree more than one form a path.). Although we achieved significant progress in this topic, there are still several open questions. The proposed research project is to explore further when tree degree sequences have edge disjoint tree and/or caterpillar realizations. More detailed description can be found here: http://www.renyi.hu/~miklosi/2017FallRES/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:**see within http://www.renyi.hu/~miklosi/2017FallRES/Packing.pdf -
Title:
**What is unavoidable - Forbidden Configurations****Description**: Click here

**Professor**: Dr. Attila Sali

**ASSIGNMENT FOR THE FIRST WEEK**: Click here