This course is designed in the style of
the Hungarian "TDK" system, allowing advanced undergraduates to
become acquainted with research methods 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 two presentations (as explained below) and
ideally a research paper, however that is not expected to be achieved, given the time constraints.
Participating in the research course 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 and
several 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
Professors evaluate progress and final enrollment decisions are made, based on the written summary (if available), oral presentation and work done during the first 2.5 weeks.
Please, note that some research groups may die out or be discontinued after the 3rd week, so plan accordingly. Also, the research class is the only class where a student wishing to take the course may not be able to, since it is at the discretion of the professor to let students become members of their research group.
Week 7 - Milestone 2: students receive "midterm evaluation grades" (MAG) and continuance is determined. Grading is done on an A-F scale.
The MAG depends on all work up to that point. A self/group evaluation may be sent out to all group members (individually). If necessary, each student may be required to submit a (relatively short) report on the work in progress to their professor. (Thus the report should include a eg description of the problem, as well as the methods used in tackling the it and a write-up of results, if any.)
Note that only students meeting each of the following
criteria may continue working on research after week 7 (all other students will have to drop research
or will receive an "Audit" for the course):
|
Week 13 - Milestone 3:: Work continues throughout the semester. At the 13th week each group should present their results at a "Preliminary report session" organized for all RES participants, their professors and everyone else interested.
Grading is done on an A-F scale.
Write up of results is continuous and oftentimes streches to after the semester is over.Description: For graphs, all realizations of a degree sequence are connected by simple
switch operations—but this is no longer true for hypergraphs.
This project studies for which degree sequence the t-uniform hypergraph
realizations of it can be transformed into one another using simple local operations,
and when hidden structural barriers exist. We explore this question in case of uniform hypergraphs,
and investigate when “easy existence” also implies “easy transformation.”
For more details, click here
Prerequisites: Basic combinatorics and graph theory.
Professor: Istvan Miklos
Contact: miklos.istvan@renyi.hu
Qualifying problems: Solve the first 3 exercises and either exercise 4 or
exercise 5 given here, by January 27th.
The maximum number of students in the class is 4, in case of more than 4 students solving sufficient number of exercises,
solutions to the remaining exercises will be considered.
Description: We plan to study the following problem and its variants.
A unit line segment is moved continuously in the plane for a finite amount of time and
we are considering the (two-dimensional) Lebesgue measure of the trajectory of each point of the unit segment.
By parameterizing the unit line segment, we obtain a function f:[0,1] -> [0, ∞).
What functions can we get? What can we say about their continuity? What can the zero set of such a function be?
The existence of a Peano curve at least implies that this function f is not always identically zero.
It is easy to show that f is always upper semicontinuous and it is not hard to give a construction for which the zero set
of f is a singleton (see the Qualifying problems).
It seems to be much harder to show a construction for which f is not continuous.
The following two variants seem to be interesting.
In the plane one can also move continuously not just a line segment but a square.
Then, we can consider again the Lebesgue measure of the trajectory of each point to
get a non-negative valued function g on [0,1]×[0,1], and we can ask the same questions.
But we can also consider the Lebesgue measure of the region swept by each originally vertical
line segment of the square to obtain a function h on [0,1], and we can ask the same questions as in the first paragraph.
A related classical result of Cunningham concerns a certain type of line segment, which he calls a bird:
the segment is divided into three parts, with the two outer parts of length w referred to as the wings,
and the short middle part of length b referred to as the body.
Cunningham proved that given a bird as described above and any bounded set S,
and ε > 0, there exists a continuous motion of the bird such that its body passes over every point of S
while both its wings stay in a set K of area less than ε.
An other related result is the even more classical solution of Besicovitch to the Kakeya needle problem: there exist subsets of the plane with arbitrarily small area in which a unit line segment can be rotated continuously through 180 degrees within it, returning to its original position with reversed orientation.
There are also very recent related results by my former students, but some of them are unpublished.
Márk Kökényesi proved that the unit square can be fully rotated continuously in the plane in such a way
that each initially vertical line segment sweeps a set of arbitrarily small area, see the preprint at
https://arxiv.org/abs/2411.11083.
Zsigmond Fleiner in his recent BSc thesis
https://www.math.elte.hu/thesisupload/thesisfiles/2025bsc_mat3y-am57lq.pdf
gave a construction for which the above g is not continuous: in fact, it is
positive only at the center of the square [0,1]x[0,1]. His construction is quite involved.
It would be nice to give a simpler construction for a non-continuous f or g.
As far as I know it is not known whether the above function h can also be non-continuous.
Prerequisites: analysis and measure theory.
Best for: advanced students who are interested in geometric measure theory and intend to do research
Professor: Tamás Keleti
Contact: tamas.keleti@gmail.com
Qualifying problems:Solve as much as you can of the
preliminary assignment and send me your solutions by email.
Don’t hesitate to ask me if you need clarification or if you have any questions.