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 two individual reports (as explained below) and ideally a research paper, however that is not expected to achieve given the time constraints. In addition, during the semester there will be opportunities to present your work as well.

Participating in the research course 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.

TOPICS PROPOSED — Spring 2020.



  • Title: Combinatorial problems motivated by coding theory

    Description: A family of subsets of an n-set is r-cover-free, if no set is covered by the union of r others. To determine the maximum size f(n,r) of such a family seems to be hard, it is an opem problem since five decades. On the other hand, this problem has many applications in coding theory, geometry and theoretical computer science, so even a tiny improvement would be of great interest. In 1994 I gave a combinatorial proof for the best known upper bound and after 26 years I decided to get back to it. I am inviting students who like adventures, for this project.
    Professor: dr Miklos Ruszinko
    Prerequisits: basic combinatorics and probability theory.
    Assignment for the first week:: read and understand the following 3 papers article 1, article 2, article 3




    1. Title: Helly-type Theorems for Ellipses

      Description: click here

      Prerequisites: Basic arithmetic (linear combinations, scalar product) of vectors in R^d. Basic knowledge of matrices (interpretation as linear map, determinant) is a plus, but is not necessary.
      Professor: dr Marton Naszodi
      Assignment for the first week:: see the practice questions



    2. Title: On the Number of Edge-Disjoint Triangles in K4-Free Graphs

      Description: Together with Keszegh, we proved a 25 year old conjecture that every K4-free graph with n vertices and ?n2/4? + k edges contains k pairwise edge disjoint triangles. However several generalizations are open, though probably very difficult. What happens if the condition K4-free is deleted or weakened? Some other versions (not edge disjoint triangles) are studied extensively recently, but there is still a lot of directions to investigate.

      Prerequisites: graph theory and combinatorics
      Professor: dr Ervin Gyori
      Assignment for the first week:: read and understand the following papers:
      1. Ervin Györi, Balázs Keszegh: On the Number of Edge-Disjoint Triangles in K4-Free Graphs. Combinatorica 37(2017): 1113-1124.
      2. Ervin Györi: On the number of edge disjoint cliques in graphs of given size. Combinatorica 11(1991): 231-243
      3. E. Győri, On the number of edge disjoint triangles in graphs of given size, Combinatorics,Proceedings of the 7-th Hungarian Combinatorial Colloquium, 1987, Eger, 267–276.
      (available in Renyi Institute library or prof Gyori can give you a copy)



    3. Title: Peeling the onion -- The convex layer process

      Description: click here

      Prerequisites: some geometry and combinatory knowledge would be preferred, but we will cover the topics needed.
      Professor: dr Gergely Ambrus
      Assignment for the first week:: the Introductory Problem 1 & 2 in the description


    4. Title: Semiring Theory in Algebraic Dynamic Programming

      Description: Click here

      Prerequisites: Algorithm theory (dynamic programming), basic abstract algebra
      Professor: dr Istvan Miklos
      Assignment for the first week:: Click here



    5. 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