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 at undergraduate research journals.

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.

• 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.
• 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 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 8 - Milestone 2:: Work continuous thrughout the semester. At the 8th/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 9-14:: Write up of results continuous and papers are produced. Week 13th a "BSM_TDK mini conference" is organized where groups present their work for cash prizes.

BASED ON PREREGISTRATION DATA, THE FOLLOWING PROBLEMS WILL BE OFFERED FALL 2012

1. Title: Besicovitch - Kakeya problem
   
   Description: A compact set in R^n is called a Besicovitch set if it contains unit segments in every direction. Besicovitch discovered about 100 years ago that there exist Besicovitch sets in the plane with Lebesgue measure zero, which easily implies that there are Besicovitch sets of zero measure in higher dimensions as well. The famous and surely extremely hard Kakeya conjecture states that the Hausdorff dimension of a Besicovitch set in R^n must be n. The conjecture is closely related to famous conjectures in harmonic analysis and in some other areas of mathematics, and this is one of the favorite problems of Terrence Tao. Of course, we won't even try to attack the Kakeya conjecture itself, but we may attack some related much less famous perhaps much easier problems.
   
   Prerequisites: measure theory; and it is useful if students learned about Hausdorff measure before, otherwise it has to be learned at the very beginning
   Best for: advanced students who likes geometric measure theory and intend to do research in analysis
   Professor: Dr. Tamás Keleti
   



2. 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
   Best for: students who intend to do research in combinatorics
   Professor: Dr. András Gyárfás
   



3. • Title: The number of most parsimonious SCJ scenarios
     
     Description: Consider two directed, edge labelled graphs, $G_1$ and $G_2$. The labels are the$
     - take a vertex with degree 2 and break it into two, degree 1 vertices
     - take two degree 1 vertices and merge them into a single vertex.
     
     It is easy to see that $G_1$ can always be transformed into $G_2$ with S$ But what happens if we put this problem onto a binary tree? Each leaf of the bin$
     
     Prerequisites: basic combinatorics.
     Best for: students interested in computer science and combinatorics
     Professor: Dr. István Miklós
     
   
   OR
   
   
   • Title: The pressing game on black and white graphs
     
     Description: Let $G(V,E)$ be a vertex colored graph, whose vertices are colored with black a$
     
     The main aim of the project is to develop fast computational methods for approxi$
     
     The problem has an interesting connection to bioinformatics and genome rearrange$
     
     Prerequisites: basic combinatorics and linear algebra.
     Best for: students who intend to do research in computer science and comb$
     Professor: Dr. István Miklós
     



4. Title: Spectral Clustering of Networks
   
   Description: Click here for pdf
   
   Prerequisites: basic combinatorics and linear algebra.
   Best for: students who intend to do research in networks
   Professor: Dr. Marianna Bolla
   



5. Title: Triangulations for sum sets in the plane
   
   Description: For bounds on the cardinality of the sum of two finite sets, there are "annoying$ for a finite non-collinear set A in the plane R^2, we consider the common number$ Prerequisites:Linear algebra and basic Eucledian geometry. No knowledge o$
   Best for: Students interested in the geometric aspects of additive number theory
   Professor: Dr. Károly Böröczky
   

TOPICS PROPOSED FOR RESEARCH, FOR OTHER SEMESTERS

• Title: Cardinality of sum sets in Euclidean space
  
  Description: Let |A| denote the cardinality of a finite set. It is not hard to see that if A and B are finite subsets of R^n, then |A+B| is at least |A|+|B|-1. But the equality |A+B|=|A|+|B|-1 implies that there is a vector v such that both A and B are arithmetic progressions with difference v. For example, if A=B, then |A+A|=2|A|-1 implies that A is subset of a line.
  Freiman proved in the 1960's that better bounds exist if A is "n-dimensional", say if A is not subset of a line (n is at least two) then |A+A| is at least 3|A| - 3. Observe that there is a jump in the factor, as if A is not subset of a line, then the general lower bound 2|A|-1 becomes 3|A| - 3. You may exprolore extensions of the result, or equality conditions for the Freiman bound.
  
  Prerequisites:linear algebra
  Best for: students who intend to do research in the geometric aspects of additive number theory
  Professor: Dr. Károly Böröczky
  



• 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. Miklós Dezsô
  



• Title: First passage times
  
  Description: Many phenomenon 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 PDE, random walks (in continuous media or on graphs), numerical methods
  Professor: Dr. Árpád Tóth
  



• Title: Miscellaneous simple open questions
  
  Description: Students can choose from a few very concrete and simple open questions of the instructor. These are simple in the sense that it is easy to state and understand them but since they are unsolved it is not clear how hard they are, what methods can be useful, or even what area of mathematics can provide promising methods. The motivation of the questions originally came from geometric measure theory but at their present form the questions are elementary.
  
  Prerequisites: strong problem solving skills
  Best for: students who would like to try research on simple looking unsolved questions Professor: Dr. Tamás Keleti
  



• Title: Number theory over finite function fields
  
  Description: The polynomial ring F_p[t] shares many similarities with the integers. In this polynomial setting most of the classical arithmetic questions become simpler and for many difficult conjectures their analogue over F_p[t] are theorems. In this project you would learn the basic relationships between the the integers Z and F_p[t], some more advanced arithmetic theorems that are much easier to state and prove over F_p[t] and you would explore some arithmetic conjecture (of your choice or mine) in this finite setting.
  
  Prerequisites: basic number theory and algebra
  Best for: students who intend to do research in algebraic or analytic number theory
  Professor: Dr. Árpád Tóth