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 at undergraduate research journals.

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.


The list of research topics and professors proposing them can be seen below. Contact the professor whose problem you are interested in.


  1. Title: Besicovitch - Kakeya problem

    Description: A compact set in R^n is called a Besicovitch set if it contains unit segments in every direction. Besicovitch discovered about 100 years ago that there exist Besicovitch sets in the plane with Lebesgue measure zero, which easily implies that there are Besicovitch sets of zero measure in higher dimensions as well. The famous and surely extremely hard Kakeya conjecture states that the Hausdorff dimension of a Besicovitch set in R^n must be n. The conjecture is closely related to famous conjectures in harmonic analysis and in some other areas of mathematics, and this is one of the favorite problems of Terrence Tao. Of course, we won't even try to attack the Kakeya conjecture itself, but we may attack some related much less famous perhaps much easier problems.

    Prerequisites: measure theory; and it is useful if students learned about Hausdorff measure before, otherwise it has to be learned at the very beginning
    Best for: advanced students who likes geometric measure theory and intend to do research in analysis
    Professor: Dr. Tamás Keleti

  2. Title: Monochromatic connected pieces

    Description: A first exercise in Graph Theory says that either a graph or its complement is connected. This observation can be extended in many directions, summarized in a recent survey of the instructor. Hopefully ambitious students can make some advances among the many unsolved problems of this area.

    Prerequisites: basic combinatorics
    Best for: students who intend to do research in combinatorics
    Professor: Dr. András Gyárfás

    • Title: The number of most parsimonious SCJ scenarios

      Description: Consider two directed, edge labelled graphs, $G_1$ and $G_2$. The labels are the$
      - take a vertex with degree 2 and break it into two, degree 1 vertices
      - take two degree 1 vertices and merge them into a single vertex.

      It is easy to see that $G_1$ can always be transformed into $G_2$ with S$ But what happens if we put this problem onto a binary tree? Each leaf of the bin$

      Prerequisites: basic combinatorics.
      Best for: students interested in computer science and combinatorics
      Professor: Dr. István Miklós

    • OR

    • Title: The pressing game on black and white graphs

      Description: Let $G(V,E)$ be a vertex colored graph, whose vertices are colored with black a$

      The main aim of the project is to develop fast computational methods for approxi$

      The problem has an interesting connection to bioinformatics and genome rearrange$

      Prerequisites: basic combinatorics and linear algebra.
      Best for: students who intend to do research in computer science and comb$
      Professor: Dr. István Miklós

  3. Title: Spectral Clustering of Networks

    Description: Click here for pdf

    Prerequisites: basic combinatorics and linear algebra.
    Best for: students who intend to do research in networks
    Professor: Dr. Marianna Bolla

  4. Title: Triangulations for sum sets in the plane

    Description: For bounds on the cardinality of the sum of two finite sets, there are "annoying$ for a finite non-collinear set A in the plane R^2, we consider the common number$ Prerequisites:Linear algebra and basic Eucledian geometry. No knowledge o$
    Best for: Students interested in the geometric aspects of additive number theory
    Professor: Dr. Károly Böröczky