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 — SPRING 2019

• Title: Hausdorff dimension of the graphs of injective functions
  
  Description: It is well known that the graph of a continuous real function can be surprisingly large, it can even have Hausdorff dimension 2. These functions oscillate heavily and they are very far from being injective. In fact, continuous injective real functions on an interval are strictly increasing or decreasing and so it is not hard to prove that their graph cannot be large, they must have Hausdorff dimension 1.
  
  The situation is more interesting if we do not require continuity or we still require continuity but our function does not have to be defined on an interval. Are there injective functions with large graph then?
  
  In the Research Course "Lebesgue measure and Hausdorff dimension of unions of lines or planes" of the previous semester the above problem arose and we could construct a Cantor type compact set K of zero Lebesgue measure and a continuous injective real function f on K such that the graph of f has Hausdorff dimension 2.
  
  We would also need injective functions on compact subsets of the real line of given Hausdorff dimension 0 < s < 1 such that the graph is large. It is easy to show that the graph must have Hausdorff dimension at most s+1. Perhaps, modifying the above mentioned construction, for any 0< s< 1 we can construct a Cantor type set K of Hausdorff dimension s and a continuous injective function f on K such that the graph of f has Hausdorff dimension s+1. This would have applications in our previous project.
  
  In this course the first goal is to make such a construction, or a weaker one if this is impossible. Then we can try to find extensions, generalizations and applications of the results we find.
  
  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: (should be partly done before the Welcome Party!) 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