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.

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 2013

• Title: Modeling of Home Equity Position markets
  
  Description: A new market for the trade of equity positions in owner-occupied, residential real estate is evolving in the US. As such a market has never existed before and will differ from the known stock and real estate markets, it is an interestig task to model the growth and behaviour of such a market.
  
  During the semester, we will use existing theory from the fields of Stochastic Calculus, Financial Mathematics and Quantitative Finance to set up a computer simulation that models the dynamics of Home Equity Position markets.
  
  Professor: Dezsô Miklós, Rozi Miklós
  
• 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)



• Title: First passage times
  
  Description: Many phenomena including diffusion, neuron firing, or the triggering of stock options are driven by stochastic processes and their first hitting times (also called first passage times). These hitting times are the times when the random process reaches a certain threshold. The determination of these hitting times requires the solution of certain partial differential equations associated to the random process.
  
  The disctretization of these processes lead to interesting random walks on graphs. In this project you would develop methods to explore and exploit these connections and develop tools for fast computations of first passage times.
  
  Prerequisites: strong calculus and linear algebra skills, complex numbers
  Best for: students who would like to do research in random walks (in continuous media or on graphs), numerical methods
  Professor: Dr. Árpád Tóth
  ASSIGNEMENT FOR THE FIRST WEEK: will be given at the Welcome Party



• 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/ãnstee/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




• Title: Large set in the line without a given pattern
  
  Description: In the real line by a rectangle we mean points of the form x, x+a, x+b, x+a+b, and by the area of such a rectangle we mean the product ab. The question is how large a subset of the real line can be without having rectangle of area at least 1. One can get a more precise question in a number of different ways depending on the the assumption about the set and the measurement we use, but, as it can be easily seen, they are all equivalent, so students can choose the form they prefer. In the measure theoretic form we consider Lebesgue measurable sets and we use Lebesgue measure. In the combinatorial form the sets are finite unions of intervals and we consider the sum of the lengths of the intervals. In the number theory form the points of the sets are chosen from a grid and we consider the number of chosen points multiplied by the length of the grid. The methods of all the above areas can be used and it is not clear at all which will be useful. The motivation comes from a long standing very hard unsolved problem of A. Carbery. An arbitrarily large set without rectangle of area at least 1 would immediately answer Carbery's problem.
  
  Prerequisites: none
  Best for: Students with good problem solving skills who would like to work heavily on unsolved problems.
  Professor: Dr. Tamás Keleti
  ASSIGNMENT FOR THE FIRST WEEK: exercises (do at least part (a), if you are interested in participating)



• Title: Monochromatic connected pieces
  
  Description: A first exercise in Graph Theory says that either a graph or its complement is connected. This observation can be extended in many directions, summarized in a recent survey of the instructor. Hopefully ambitious students can make some advances among the many unsolved problems of this area.
  
  Prerequisites: basic combinatorics; read to help decide
  Best for: students who intend to do research in combinatorics
  Professor: Dr. András Gyárfás
  ASSIGNMENT FOR THE FIRST WEEK: Click here



• Title: Stochastic networks
  
  Description: Networks appear in many places in life sciences, especially in sociology (social networks) and neuroscience (network of neurons). These networks are built up in a stochastic way, however, they have typical properties like connectivity, emergence of certain components during the stochastic built up, etc.
  
  We are going to study the properties of several stochastic networks appearing in life sciences. We will use computer simulations as well as pure mathematics to prove theorems. The ideal research team consists of students having both the necessary programming and mathematics background, however, collaboration between programmers and theorists might also work pretty well.
  
  Prerequisites: basic combinatorics, graph theory
  Best for: students interested in computer science and combinatorics
  Professor: Dr. István Miklós
  ASSIGNEMENT FOR THE FIRST WEEK: