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 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, if you have any questions about the problem.

• Who can participate? Professors give 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 assignments as you can by the Welcome Party and discuss your progress with the professor. Initial participation 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 (around 3) would like to sign up and are accepted by the Professor based on discussions at the Welcome party and/or the first 2.5 weeks.
• 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. Note that this is a minimal reqiurement. Naturally, the group of students working on a problem can meet more (as often as necessary) than once a week.
• Course work: presentations.
  Week 3 - Milestone 1: The first 2.5 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 research 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 for the RES course.
  Week 13 - Milestone 3:: Work continues throughout the semester. At the 13th week each group should present their results at a "Preliminary report session" organized for all RES participants, their professors and everyone else interested.
  Write up of results is continuous sometimes streches after the semester is over.

TOPICS PROPOSED — FALL 2019

1. Title: Geometry and Mechanics of Convex Polyhedra
   
   Description: Click here
   
   Prerequisites: mathematical maturity, familiarity with convex geometry is a plus (but not required). Knowledge of at least one of MATLAB, C++ or MAPLE programming languages is required. (Other programming languages may be considered - contact prof Domokos to inquire.)
   Professors: Dr Gabor Domokos
   Assignment for the first week: See the questions here
   
   
2. Title: Skew Bases of Cubic Fields
   
   Description: Click here
   
   Prerequisites: classical algebraic number theory (finite extensions of Q, algebraic integers, notion of discriminant, etc.).
   Professor: Dr. Péter Maga
   Assignment for the first week: Click here
   
   
3. Title: Numerics in the Prime Geodesic Theorem
   
   Description: In 2 dimensions, take the modular group acting on the hyperbolic plane. Every hyperbolic element has a length, and this quantity is invariant under conjugation. Consider then the (hyperbolic) conjugacy classes [g] in PSL(2,Z). A conjugacy class [g] is called primitive (or by abuse of notation, prime) if g is not the power of another distinct element g' in PSL(2,Z). Then consider the function
   \pi(x) = #{primitive conjugacy classes [g] in PSL(2,Z) with length([g])<=x}.
   To be precise, it is more common to use the exponential of the length, but that only implies that x is to be replaced by e^x.
   Then one can prove an asymptotic formula for \pi(x) analogous to the prime number theorem, with main term going like Li(x). This result is called prime geodesic theorem (for PSL(2,Z)).
   The conjecture is that the remainder in the counting is at most of size sqrt(x), and the numerics seem to suppor this.
   Next one goes to the group PSL(2,Z[i]), where the entries are now Gaussian integers. This group acts discontinuously on the hyperbolic space H^3, and the quotient is called Picard manifold. One defines length and norm of a conjugacy class in a totally similar way as in two dimensions, and it is of interest to study the function \pi(x) in this new settings. The main asymptotic is Li(x^2), and there are some (recent) papers in the literature on how big the remainder should be. Numerics are not available and the research class aims to produce them. The computation involves class numbers of quadratic forms over Q(i) [so, morally, class numbers of quartic extensions of Q].
   Click here for the project description and preliminary problems.
   
   Prerequisites: basic of number theory, basic of analysis and complex analysis, basic of algebraic number theory (class groups of number fields), basic of program- ming.
   Professor: Dr. Giacomo Cherubini
   Assignment for the first week: Click here
   
   
4. Title: The diameter of large components in r-edge-colorings of K_n (Part II.)
   
   Description: Click here
   
   Prerequisites: basic combinatorics and graph theory, Ramsey theory
   Professor: Dr. Miklós Ruszinkó
   Assignment for the first week: Click here, and see these papers: Paper 1, Paper 2
   
   
5. Title: What is unavoidable - Forbidden Configurations

   Description: Click here
   Professor: Dr. Attila Sali
   Assignment for the first week:: Click here