BSM-TDK — Research Opportunities — RES

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 2018

• Title: Partitioning subsets of [n] into triangle free parts
  
  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: work on the exercises given in the description.


• Title: Lebesgue measure and Hausdorff dimension of unions of lines or planes
  
  Description: We plan to study the following general principle:
  
  The union of an m-parameter family (or more generally an m Hausdorff dimensional family) of k-dimensional planes of R^n has positive Lebesgue measure if m+k>n and has Hausdorff dimension m+k otherwise.
  
  Unfortunately, this is too good to be true, there exist very simple counter-examples. On the other hand, in some restricted cases (for example if k=n-1 or m is at most 1) this is known to be true, and more importantly, famous conjectures would (essentially) follow from some other restricted cases: The most famous is the surely extremely hard Kakeya conjecture (one of the favorite problems of Terry Tao), which states that any compact set in R^n that has a unit line segment in every direction must have Hausdorff dimension n. An other example is the also famous and long standing conjecture that states that for any 1
  Of course, attacking these conjectures would be too ambitious. Instead, we have the following more realistic (but less well defined) goals:
  
  1. to find and prove sufficient conditions that guarantee that the above principle holds,
  2. to find and prove necessary conditions that are needed for the principle,
  3. to find conjecture for the exact (necessary and sufficient) condition and
  4. to find counter-examples for naive false conjectures.
  
  
  Prerequisites: measure theory (knowledge about Hausdorff measure and Hausdorff dimension is useful, but not necessary: this can be learned at the very beginning of the semester)
  Best for: advanced students who like geometric measure theory and intend to do research in analysis
  Professors: Dr. Tamas Keleti
  Assignment for the first week: Solve these problems (the assigment is due by 10am of Monday, the 10th of September, i.e., end of the academic orientation)
  
  
• Title: The solution space of sorting by block interchanges and sorting by reversals
  
  Description: click here
  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 click here
  
  
• Title: The diameter of large components in r-edge-colorings of K_n
  
  Description: click here
  
  Prerequisites: basic combinatorics and graph theory, Ramsey theory
  Best for: students interested in combinatorics, discrete mathematics, computer science or information theory
  Professor: Dr. Miklós Ruszinkó
  Assignment for the first week: read and try to digest the following two papers 1.pdf and 2.pdf
  
  
• Title: What is unavoidable - Forbidden Configurations

  Description: Click here
  Professor: Dr. Attila Sali
  ASSIGNMENT FOR THE FIRST WEEK: Click here