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 a research paper, ideally, however that is not expected given the time constraints.
In addition, during the semester there will be an opportunity
to present your work as well.
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
Students also receive a course grade at the end of the summer term. The transcript will contain only the course grade.
Description: Click here.
Prerequisites: We shall mostly use graph theoretic and combinatorial methods, so famil-
iarity with the basics of graph theory is useful. Depending on the problems,
in some cases geometric intuition is also useful, as well as familiarity with
elementary linear algebra.
Professor:Tibor Jordan
Contact:tibor.jordan@ttk.elte.hu
Qualifying problems:Solve at least four out of the
five warm up exercises at the end of this document.
The well known and well understood Analyst's Traveling Salesman Problem is to characterize those sets that can be covered by a Lipschitz image of [0,1]. Our goal is to study the problem we get by replacing the interval by the Cantor set. Another motivation comes from the well known classical result that the compact metric spaces are exactly the continuous images of the Cantor set, so it seems to be natural to ask which metric spaces can be obtained as a Lipschitz image of the Cantor set.
With my colleague Richárd Balka we started to work on this problem and some natural variants a few months ago and we proved that every compact metric space of upper box dimension less than log2/log3 can be obtained as the Lipschitz image of the Cantor set. We characterized those self-similar sets with the strong separation condition (see the definitions in Exercise 2 of the Preliminary Assignment) that can be obtained as the Lipschitz image of the Cantor set. In fact, we have proved more general results than these ones but it is still very far from being clear how much these results can be generalized.
The main task is to understand the nature of these problems, to learn different methods to attack them (techniques and ideas of geometric measure theory, analysis, metric geometry and combinatorics can naturally show up) and to prove or disprove natural generalizations and to find some non-trivial and unknown variants that we can solve.
Prerequisites: metric spaces and measure theory