BSM-TDK — Research Opportunities — RES

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, for 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 7 - Milestone 2: around week 7 students receive "midterm evaluation grades" in each course they are taking informing them of their course grade up to that point. A student with an insufficient overall performance (e.g. C's in all other classes) will have to finish doing research at that time and will receive an "Audit" on their BSM transcript.
  Week 13 - Milestone 3:: Work continues thrughout the semester. At the 13th week each group should present their results at a "Preliminary report session" organized for all BSM-TDK participants, their professors and everyone else interested.
  Write up of results is continuous sometimes streches after the semester is over.

TOPICS PROPOSED — SPRING 2017

1. Title: Fibered knots and grid homology

   Description: Grid homology is a combinatorial description of knot Floer homology groups of knots and links in the 3-space. The importance of Heegaard Floer theoretic invariants puts grid homology in the forefront of low dimensional topological research. It is known that knot Floer detects fiberedness of a knot, but a proof of this fact within the combinatorial setting of grid homology is yet to be found.
   
   The aim of the project is to find grid diagrams of fibered knots with low crossing number for which there is a single generator in the approproate grading, and extend these examples to possible infinite families of fibered knots. A more ambitious goal might be to see that if a knot admits such a grid diagram, then it is fibered.
   
   Prerequisites: point-set topology, some algebraic topology (see reading assignment below)
   Best for: students who intend to go to grad school in pure math and/or are interested in knot theory
   Professor: Dr Andras Stipsicz
   Assignment for the first week: read Ozsvath-Stipsicz-Szabo: Grid homology for knots and links Sections: 2.1, 3.1, 3.2
   
   

2. Title: Hausdorff dimension of unions of lines

   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. Instead, we will study some closely related problems of the following type: Given a curve in the plane or in R^n, how small the Hausdorff dimension of a set can be if through every point of the curve the set contains a line (such that the line intersects the curve only in finitely many points). We will try to construct a curve for which the answer is 1 (or at least less than 2) and we will try to find smoothness conditions for the curve that guarantees that the answer is 2
   
   Prerequisites: measure theory (knowledge about Hausdorff measure and Hausdorff dimension is useful, but not necessary: this can be learned at the very beginning of the semester)
   Best for: advanced students who like geometric measure theory and intend to do research in analysis
   Professor: Dr. Tamas Keleti
   Assignment for the first week: (should be partly done by the Welcome Party) see here
   
   

3. Title: Mixed graphical models

   Description: Graphical models provide a framework for describing statistical dependencies in (possibly large) collections of random variables. At their core lie various correspondences between the conditional independence properties of a random vector and the structural properties of the graph used to represent interactions (directed or undirected) between the vertices assigned to the random variables. These so-called causality models have been investigated since the 1980s, the first steps were made by J. Pearl. However, it was S. L. Lauritzen who showed how loglinear models can be used to estimate joint, marginal, and conditional probabilities and make predictions, by taking into consideration the graph structure.
   
   Students, taking this research are assumed to master some routine in hierarchical and decomposable loglinear models, based on the book of S. L. Lauritzen (Graphical Models, Oxford Univ. Press, 1995), but it suffices to read the pdf file attached. Then the research task would be to develop the models and algorithms in the following. The underlying variables are usually categorical (e.g., symptoms, medical diagnoses), but so-called mixed models, incorporating continuously distributed random variables (mainly Gaussian, conditioned on the discrete ones) are also proposed in the above book. The estimation methods could be extended to these mixed types of models, which are applicable in machine learning for building artificial intelligence (e.g., in medical diagnostic systems), so testing the models on real-life data is also welcome.
   
   Prerequisites: Basic probability and graph theory
   Best for: students who intend to do research in machine learning
   Professor: Dr. Marianna Bolla
   Assignment for the first week: start reading this text
   

4. Title: Packing sparse degree sequences
   
   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. There are results when each degree sequence is a tree degree sequence, that is, all degree is positive, and the sum of the degrees is 2n-2. 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://www.renyi.hu/~miklosi/2017SpringRES/Packing.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: http://www.renyi.hu/~miklosi/2017SpringRES/Packing.pdf
   
   
5. Title: What is unavoidable - Forbidden Configurations

   Description: Click here
   Professor: Dr. Attila Sali
   ASSIGNMENT FOR THE FIRST WEEK: Click here