BSM-TDK — Elective Undergraduate Research

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 — SPRING 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: Extremal sets of the vertices of the hypercube (over GDF(2))
   
   Description: We plan to investigate (cases of) the following general question: How many vertices (maybe of certain further property, like of fixed weight) of the n-dimensional hypercube can be picked such that subspace spanned by them - over GF(2) - does not contain or does not intersect certain configurations of the hypercube (vertices, vertices of given weight, subspaces, hyperplanes, etc.)
   
   In this project you will understand the structure of the hypercube over the reals and GF(2), develop algebraic methods to solve extremal set theoretical problems and establish constructions and will reach - in the worst case - some concrete results.
   
   Prerequisites: basic combinatorics and linear algebra
   Best for: students who intend to do research in algebra or combinatorics
   Professor: Dr. Dezsô Miklós
   ASSIGNMENT FOR THE FIRST WEEK: exercises and related reading
   
   
3. Title: Graph realizations with constraints
   
   Description: A degree sequence D = d_1, d_2, ... d_n is a series of non-negative integers. A degree sequence D is called graphical if there is a graph whose degrees are exactly D. Erdos and Gallai gave sufficient and necessary conditions when a degree sequence is graphical, Havel and Hakimi gave an algorithm which constructs a realization if such exists otherwise reports that the sequence is not graphical.
   
   There are several open questions whether or not a degree sequence is graphical when further constraints are given. For example, Let D = d_1, d_2, ... d_n and E = e_1, e_2, ... e_m two degree sequences for vertex sets V_1 and V_2. An open question if such a realization exists whose degree sequence is the union of D and E and there are exactly k number of edges between V_1 and V_2.
   
   In this research class, we are going to overview the classical results of Erdos, Gallai, Havel, Hakimi, Gale, Ryser, Tutte, and our previous results, see
   http://arxiv.org/abs/0905.4913 http://arxiv.org/abs/1302.3548 and then we start working on the open questions Prerequisites: There are no prerequisites for this research class, however, students with some prior knowledge on graph theory are more encouraged to$ Best for: This course is ideal for students who want to do research on graph theory, combinatorics and computer science.
   Professor: Dr. István Miklós
   ASSIGNEMENT FOR THE FIRST WEEK:
   
   
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
   ASSIGNMENT FOR THE FIRST WEEK: Click here




5. Title: The Zoo of Singular Varieties
   
   Description: We will study subsets of Euclidean spaces defined by polynomial equations from the point of view whether they are smooth (have no corners etc) or not. More specifically, we plan to study the giant zoo of singular varieties in terms of to what extent polynomials vanish on them.
   
   This is made possible by a nice correspondence between certain subsets of polymials on the one hand, and certain subsets of Euclidean spaces that is sometimes called the algebra-geometry dictionary. In particular, we will take the ideal of polynomials vanishing identically on our algebraic variety, and study how its powers behave.
   
   The kind of research one does can vary in a wide range from fun computer algebra computations to actual research papers in algebraic geometry.
   
   Prerequisites: a good relationship with polynomials
   Best for: students who plan to go on to graduate school and/or want to do research in algebra or geometry
   Professor: Dr. Alex Küronya
   ASSIGNMENT FOR THE FIRST WEEK: Familiarize yourself with the first vew pages of an introductory book an algebraic geometry (see the syllabus for the Intro Algebraic Geometry course). Alternatively, download and install the computer algebra package Singular.