## 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, 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.
• Who can participate? Most professors 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 as much of these as you can by the Welcome Party and discuss your progress with the professor. First enrollment 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 (at least around 3) would like to sign up and are accepted by the Professor based on discussions at the Welcome party and/or the first week.
• Course work: weekly meetings.
Class will meet twice weekly, 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.
• Course work: presentations.
Week 3 - Milestone 1: The first three weeks of the course are spent discussing, gathering and studying necessary background information for your problem. At the end of the 3rd week, you (and your group) will have to present the problem you are working on in 20 minutes at a "mini workshop" organized for all BSM-TDK participants, their professors and everyone else interested. At this point professors evaluate progress and final enrollment decision is made. Ideally, the size of research groups is low, at most three.
Week 9 - Milestone 2:: Work continuous thrughout the semester. At the 9th week each group should present their results at a "Preliminary report session" organized for all BSM-TDK participants, their professors and everyone else interested.
Week 9/10-14:: Write up of results continuous and papers are produced.

## TOPICS PROPOSED — FALL 2014

1. Title: Beating the Delsarte bound

Description: Several famous problems in mathematics fit into the following general scheme: given an Abelian group G and a "forbidden subset" F in G, what is the maximal number of elements that we can select from G such that no two of them have a difference in F. For example, how many numbers can you select from 1 to N such that no two of them differ by a square? Or, the cyclic analogue of this problem: in the cyclic group Z_p (where the order p is a prime) what is the maximal number of elements such that no difference is a quadratic residue. Also, in geometry: what is the maximal density of a sphere-packing in the Euclidean space R^n? We will explore several other examples. There is a general method to give upper bounds in such problems. It originates from coding theory from the work of Delsarte. We will discuss this general bound, and make attempts to beat it.

Prerequisites: combinatorics (and some familiarity with the discrete Fourier transform and some minimal knowledge of programming can be useful)
Best for: students with strong problem solving skills
Professor: Dr. Máté Matolcsi
ASSIGNEMENT FOR THE FIRST WEEK: exercises (please solve 1-2-3 and try 4), related reading (please read the proof of Theorem 1.4 in Section 3)

2. Title: Comparing random networks and human brain networks

Description: The human brain has a given spiking activity as a result of the interactions of the neurons in the brain. An appropriately constructed and trained artificial neural network is able to generate the same spiking pattern. On the other hand, if one generate a random network with same/similar degree sequence, then this network will not show this spiking activity. The conclusion of this experience is that the degree sequence is not a sufficient descriptor of the human brain network.

We are going to compare the above mentioned trained artificial neural networks and random networks and try to find descriptors that separate the two types of networks. These descriptors are based on local patterns like 3 long paths, triangles, etc. The main goal of the project is to find sufficient descriptors for the spiking activity.

Prerequisites: basic graph theory, basic probability theory, knowledge of any programming language
Best for: students who are interested in applied mathematics and theoretical neural science
Professor: Dr. István Miklós
ASSIGNEMENT FOR THE FIRST WEEK: Read Chapter 12 from this electronic note: http://www.renyi.hu/~miklosi/AlgorithmsOfBioinformatics.pdf . This chapter gives the definition of degree sequences and also provides a randomization algorithm (the swap Markov chain) that generates a random network with prescribed degree sequence.

3. Title: Cubes and their centers

Description: The general question we would like to study is the following: How small the size of a set can be if it contains the vertices / the edges / the faces of an axis-parallel cube centered at every point of a set of given size? Here size can be the number of points or some fractal dimension. In our brand new paper http://arxiv.org/pdf/1408.1029v1.pdf we studied these problems in the plane and we obtained for example that if for every point p of a set S the finite set B contains the vertices of at least one axis-parallel square centered at p then S can contain at most (2|B|)^(4/3) points, and this is sharp up to constant multiple (Theorem 1.6). The generalization of some of our two-dimensional results to higher dimension seems to be straightforward, some others might be much harder. First we plan to attack the discrete results (Section 4) and then, after learning the fairly easy concept of box-dimension (see e.g. http://en.m.wikipedia.org/wiki/Minkowski.Bouligand_dimension ) we will also try to generalize the results of Sections 5 and 6. (To study Hausdorff dimension more background would be needed, so you can skip Section 3.)

Prerequisities: Combinatorics and Analysis
Best for: students who like analysis, combinatorics and would like to try research

Professor: Dr. Tamás Keleti
ASSIGNEMENT FOR THE FIRST WEEK: As a warm up, generalize Theorem 1.6 (b) of http://arxiv.org/pdf/1408.1029v1.pdf first to R^3 and then to general R^d by finding the right generalization of Lemma 4.3 and the proof of Theorem 1.6 (b), which can be found after Remark 4.4.
(should be done preferebly for the welcome party, or even better - before that - submit by e-mail)

4. Title: (Small) Forbidden Configurations

Description: The topic is extremal hypergraph theory formulated mostly in the language of 0-1 matrices, since that is the most convenient way. We say that a 0-1 matrix A has F as a configuration if there is a submatrix ofA, which is a row and column permutation of F. This concept is also called a trace or subhypergraph depending upon whether the context is set systems or hypergraphs.

The fundamental question is that for given F (or sometimes for given collection F of configurations) what is the largest possible number of columns of a simple mxn 0-1 matrix A on given number of rows without having F (or any member of F as configuration, in notation forb(m,F) or forb(m,F). A 0-1 matrix is simple if it has no repeated columns. Considering columns as characteristic vectors of subsets of an m element set matrix A describes a hypergraph, or set system.

In this project you will get acquainted with basic proof ideas of the area, such as standard induction'', shifting, applications of graph theory and linear algebra. The goal is to get results on particular forbidden configurations. For more info check
http://www.math.ubc.ca/~anstee/FCsurvey12.pdf

Prerequisites: basic combinatorics, possibly linear algebra;
Best for: students who intend to do research in combinatorics
Professor: Dr. Attila Sali

5. Title: Good Graph Hunting

Prerequisites: basic combinatorics;
Best for: students who intend to do research in combinatorics
Professor: Dr. András Gyárfás
ASSIGNEMENT FOR THE FIRST WEEK:Start working on the exercises given in the description.

6. Title: Parametric and nonparametric multiclass network models

Description:Networks can be modeled by unweighted or weighted, undirected or directed graphs. We want to find clusters of the nodes so that the information flow within the clusters and between the pairs of them is as homogeneous as possible. In the nonparametric scenario we use spectral clustering tools, whereas in the parametric one, variants of the stochastic block-model are considered (e.g., \beta-models and other loglinear type models). Students can compare the models and the related algorithms, prove some convergence facts, or make supplemental data analysis on real-life networks.

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

Assignment for the first week: start reading the first and last chapters of the following book, which is also available in the BSM library. Bolla, M., Spectral Clustering and Biclustering. Learning Large Graphs and Contingency Tables. Wiley, 2013.