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 RoseHulman Undergraduate Mathematics Journal or
Involve.
In some US PhD programs, fruitful undergraduate
reserach activity is a prerequisite for admission.
At BSM 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 at the
Welcome Party, but read everything carefully below first.
 Who can participate? Each professor 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 these
by the first week (the exact deadline will be discussed with
the professor at the Welcome Party or by email). Final enrollment will be based
on your work on these problem sets/reading assignments.
 Which topics will actually be offered ("stay alive")?
Of the initially offered research topics those
will be offered eventually, for which
a group of students sign up and are accepted by the
Professor based on first week performance as outlined above.
 Course work: weekly meetings.
The research groups will meet 34 times weekly, two hours each.
Two meetings are devoted to group work, when you discuss the problem
and possible solutions with your student group without the professor.
The other meetings of the week
are spent with your professor who will monitor your group's progress.
Whether you meet once or twice with your Professor on a given week will
be decided casebycase, depending on progress.
 Course work: presentation. Work continuous thrughout the Summer Semester.
The 5th week each research group should present their results at a
"Preliminary report session" organized for all BSMTDK participants, their professors
and everyone else interested.
 Course work: writing a paper. Depending on results obtained all work will
be summarized in a paper/research report.
PROBLEMS PROPOSED FOR SUMMER 2018
 Title: When the YuleSimpson paradox can be avoided
Description:
In contingency table analysis (containing joint responses for categorical,
mainly binary variables) Yule (1903) and Simpson (1951)detected an effect
reversal between marginal and conditional tables in some examples. To
illustrate this effect, consider the paper of Range, The New York Times Magazine (1979). Their 2x2x2 table describes the
sentences (death or other) in murder cases in Florida (197378). The color
of the skin both of the murderer and the victim was registered. The marginal table (when the color of the victim is ignored) shows a greater proportion of
white murderers receiving death sentence than black;
whereas the conditional tables for given color of victim
show a different picture: both for black and white victims,
there is a much higher proportion of black murderers receiving death sentences.
In particular, none of the white murderers, killing black victims, were sentenced to death.
Later on, in the possession of loglinear models, statisticians were rather
interested in conditions of avoiding this effect reversal.
Cox and Wermuth, J. R. Statist. Soc. B (2003) gave conditions based
on conditional independences in the Gaussian case, and anticipate that
similar conditions can be proven in the discrete case.
The research task is to formulate such conditions, while understanding the notion of
conditional independence and loglinear models, at least for three variables.
Prerequisites:basic probability and graph theory
Best for:students who are interested in statistics
Professor: Dr. Marianna Bolla
Assignment for the first week: read the two cited papers
http://www.math.bme.hu/~marib/bsmeur/out.pdf and
http://www.math.bme.hu/~marib/bsmeur/wermutheffreversal.pdf
and solve the exercise
http://www.math.bme.hu/~marib/bsmeur/exer.pdf


Title:Packing caterpillars without common leaves
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 degree
sequences are tree degree sequences, that is, all degrees are positive and
their sum is twice the number of vertices minus 2. Those degree
sequences can always be realized by trees and also by some special
trees called caterpillars (a tree is called a caterpillar if the
vertices with degree more than one form a path.).
Although we achieved significant progress in this topic, there are
still several open
questions. The proposed research project is to prove or disprove that
tree degree
sequences without common leaves always have edge disjoint caterpillar
realizations.
More detailed description can be found here:
http://www.renyi.hu/~miklosi/2018SumRES/PackingCaterpillars.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: see
http://www.renyi.hu/~miklosi/2018SumRES/PackingCaterpillars.pdf

 Title: The rate of locally thin families
Description:A family of subsets of an nset is rlocally thin, if for every
r of its member sets the ground set has at least one element
contained in exactly one of them. Alon, Fachini and Körner proved fairly tight bounds for the maximum f(n,r) of such a family
in case r is even. It is annoying, that for r odd, the gap between the upper and lower bounds is huge. It would be desirable to
narrow the gap in this case, which is an open problem in the last almost 20 years.
Prerequisites: basic combinatorics and probability theory
Best for: students interested in combinatorics, discrete mathematics, computer science or
information theory
Professor:Dr. Miklós Ruszinkó
Assignment for the first week: read and try to digest the following papers
