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


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.


  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