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 RoseHulman 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: written and oral 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. By the end of the 3rd week, each student wishing to participate in a research group has to write a summary of the status of the research project at that point — on their own. The summary should consist of stating the problem/aims of the research group, plans on tackling the problem, as well as an outline of work done during the first 2.5 weeks with a write up of results (if any) achieved by that time.
In addition, each research group will have to present the problem they are working on in a 20minute talk, at a "mini workshop" organized for all research participants, their professors and everyone else interested.At this point professors evaluate progress and final enrollment decisions are made, based on the written summary, oral presentation and work done during the first 2.5 weeks.
Please, note that some research groups may die out or be discontinued after the 3rd week, so plan accordingly. Also, the research class is the oly class where a student wishing to take the course may not be able due, since it is at the discretion of the professor to let students become members of their research group.
Week 7  Milestone 2: at week 7 (just like at week 3), each student is required to submit a (relatively short) report on the work in progress, to their professor. (Thus the report should include a eg description of the problem, as well as the methods used in tackling the it and a writeup of results, if any.)Based on all work up to that point and the written report, students receive "midterm evaluation grades".
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.Grading is done on an AF scale.
Write up of results is continuous and oftentimes streches to after the semester is over.
TOPICS PROPOSED — Spring 2020.
Description:
A family of subsets of an nset is rcoverfree, 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
 Title:
Hellytype 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

Title:
On the Number of EdgeDisjoint Triangles in K4Free Graphs
Description: Together with Keszegh, we proved a 25 year old conjecture that every K4free 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 K4free 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 EdgeDisjoint Triangles in K4Free Graphs. Combinatorica 37(2017): 11131124.
2. Ervin Györi: On the number of edge disjoint cliques in graphs of given size. Combinatorica 11(1991): 231243
3. E. Győri, On the number of edge disjoint triangles in graphs of given size, Combinatorics,Proceedings of the 7th Hungarian Combinatorial Colloquium, 1987, Eger, 267–276.
(available in Renyi Institute library or prof Gyori can give you a copy)

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

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

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