## COURSE DESCRIPTION

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 in undergraduate research journals. (such as The Rose-Hulman Undergraduate Mathematics Journal or Involve.

In some US PhD programs, fruitful undergraduate reserach activity is a prerequisite for admission.

At BSM student research is supervised by professors. Research topics are offered by them, but students can also propose topics of their own interest.

## COURSE LOGISTICS

The list of research topics and professors proposing them can be seen below. Contact the professor whose problem you are interested in at the Welcome Party, but read everything carefully below first.**Who can participate?**Each professor gave 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 these by the first week (the exact deadline will be discussed with the professor at the Welcome Party or by email). Final enrollment will be based on your work on these problem sets/reading assignments.**Which topics will actually be offered ("stay alive")?**Of the initially offered research topics those will be offered eventually, for which a group of students sign up and are accepted by the Professor based on first week performance as outlined above.**Course work: weekly meetings.**The research groups will meet 3-4 times weekly, two hours each.

Two meetings are devoted to group work, when you discuss the problem and possible solutions with your student group without the professor. The other meetings of the week are spent with your professor who will monitor your group's progress. Whether you meet once or twice with your Professor on a given week will be decided case-by-case, depending on progress.**Course work: presentation.**Work continuous thrughout the Summer Semester. The 5th week each research group should present their results at a "Preliminary report session" organized for all BSM-TDK participants, their professors and everyone else interested.**Course work: writing a paper.**Depending on results obtained all work will be summarized in a paper/research report.

## PROBLEMS PROPOSED FOR SUMMER 2016

Click on the title to read to problem.- Apparent singularities of Garnier systems and parabolic structures
- Edge sign balance of (uniform) hypergraphs
- How to travel across packed, colored forests?
- Ramsey problems on Steiner triple systems

- Title:
**Apparent singularities of Garnier systems and parabolic structures**

**Description:**The study of topological spaces formed by systems of linear differential equations with prescribed singularities of a certain kind is the subject of intense current research with history dating back (at least) to the 19th century. This project relates to a question concerning the geometry of such spaces.

Specifically, one may consider systems of rank $2$ over the complex projective line with a given number $n + 1 \geq 5$ of logarithmic singularities and with fixed numerical invariants --- so-called Garnier systems. On the space of Garnier systems, there exist two well-known maps: the first one, denoted $q$, maps such a system to the set of its apparent singularities with respect to a cyclic vector; the second one, denoted $Q$, maps it to the equivalence class of the underlying parabolic vector bundle. A thorough study of the cases $n=3$ by Loray, Saito and Simpson and $n=4$ by Loray and Saito showed that generically these maps are transversal to each other.

Later, we developed an alternative approach to this problem, which allowed us to rederive the same result in the case $n=3$. The method of this new point of view uses so-called elemetary modifications of vector bundles at points, and its advantage is that it may lend itself for generalization to the case of arbitrary $n\geq 4$. The aim of the current project is to carry out this generalization.

**Prerequisites:**rudiments of linear differential systems in one variable, classical and linear algebra, and group actions on vector spaces.

**Best for:**students interested in geometric applications of classical algebra.

**Professor:**dr Szilárd Szabó

**Assignment for the first week:**Click here.

- Title:
**Edge sign balance of (uniform) hypergraphs**

**Description:**In the research project two semesters ago we targeted the question: which graph has the MMS property (i.e., for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of the graph). More general, let the vertex sign balance of a (hyper)graph be the minimum number of nonnegative edges over all assignment of weights to the vertices of the (hyper)graph with nonnegative sum (where the weight of an edge is the sum of the weights of the vertices in the edge). We investigated the behaviour (value) of this new (hyper)graph parameter.

This term we will turn the question to the edges: let the edge balance of a (hyper)graph be the minimum number of nonnegative vertices over all assignment of weights to the edges of the (hyper)graph with nonnegative sum (where the weight of a vertex is the sum of the weights of the edges contaning the vertex). As it turns out (see the intial assigments) the edge sign balance of a graph is an easy question (unlike the vertex sign balance). However, it becomes interesting for other (uniform) hypergraphs, which is the target question of this project.

**Prerequisites:**basic combinatorics and graph theory

**Best for:**students who intend to do research in graph theory or combinatorics

**Professor:**Dr. Dezsô Miklós

**Assignment for the first week:**prior the first meeting of the course: think over the definition and try to answer the following questions:- what can be the edge sign balance of a graph?
- which are the graphs with edge sign balance = 1? And =2?
- Find the edge sign balamce of the complete 3 uniform hypergraph (that is, the edges are all the three elements subsets of an n element set)?
- What is the relation of the vertex sign balance and the edge sign balance?

- Title:
**How to travel across packed, colored forests?**

**Description:**The edge packing or graph factorization problem asks the question if an ensemble of edge disjoint graphs exists with prescribed degrees. The problem in general is a hard computational problem, however, it is easy for special cases. One special case is when the graph is very sparse, the sum of the degrees is less than or equal to 2n -2, where n is the number of vertices. Above the existance problem, we are also interested in the connectivity problem: what are the necessary and sufficient transformations to transform solutions into each other? More detailed description can be found here: http://renyi.hu/~miklosi/RES2016SUM/coloredforest.pdf

**Prerequisites:**basic combinatorics and graph theory

**Best for:**students interested in combinatorics, discrete mathematics and computer science

**Professor:**Dr. István Miklós

**Assignment for the first week:**see in http://renyi.hu/~miklosi/RES2016SUM/coloredforest.pdf

- Title:
**Ramsey problems on Steiner triple systems**

**Description:**click here

**Prerequisites:**basic combinatorics;

**Best for:**students who intend to do research in combinatorics

**Professor:**dr Andras Gyarfas

**Assignment for the first week:**see the problems in here