COURSE DESCRIPTION
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 individual reports (as explained below) and
ideally a research paper, however that is not expected to achieve given the time constraints.
In addition, during the semester there will be opportunities
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
COURSE LOGISTICS
-
Who can participate?
The list of research topics and professors proposing them is posted below.
In addition to the problem description, each professor gives introductory
problems for interested students to solve under "Qualifying problems".
If you are considering participating, submit your solutions to the qualifying problems of the research problem that you would like to work
on by August 15th, directly to the professor whose problem you are interested in.
Enrollment will be based on solutions submitted to qualifying problems to professors and your transcript and reference letter submitted as your application materials to the American office. Decisions on admittance to the research group will be made by August 24th. - Which topics will actually be offered ("stay alive")? Of the initially offered research problems those will be offered eventually, for which a group of students sign up and are admitted to the program. Ideally, a group would consist of 3-5 students.
- Course work: weekly meetings & presentation. The research groups will meet at least 2 times weekly, two hours each.
(At least) one meeting is 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 case-by-case, depending on progress.
The last week each research group might present their results at a "Preliminary report session" organized for all participants, their professors and possible invited guests. - Course work: writing a paper. Depending on results obtained all work will be summarized in a paper/research report.
RESEARCH PROBLEMS PROPOSED — FALL 2020.
-
Title:
Generalized Turan Number of Graphs
Description: The subject is a new, very fast developing area of extremal graph
theory. The basic problem is to determine/estimate the maximum number
ex(n,H,F) of copies of H in a graph of n vertices not containing F as a subgraph.
(If H is just one edge then we get classical extremal graph theory problems.)
There are cases that are particularly interesting:
- The classical problem of Erdos solved half a century later: H is the 5-cycle, F
is the triangle,
- H is a fixed path, F is your favorite graph
- H is a triangle, F is your favorite graph again (see [1], [2] for the 5-cycle)
- Started just in the last couple of years: the same questions in planar graphs
(e.g., see [3])
Prerequisites: graph theory and combinatorics
Professor: dr Ervin Gyori
Contact: 
Qualifying problems:: read and understand some of the following papers and solve the two problems below:
[1] B. Bollobas and E. Gyori. Pentagons vs. triangles. Discrete Mathematics, 308(2008), 4332–4336
[2] B. Ergemlidze, A. Methuku, N. Salia, and E. Gyori. A note on the maximum number of triangles in a C5-free graph. J. of Graph Theory, 90(2019) 227–230.
[3] E. Gyori, N. Salia, C. Tompkins, and O. Zamora. The maximum number of Pℓ copies in a Pk-free graph. Discrete Mathematics & Theoretical Computer Science, 21(1), 2018. or arXiv:1803.03240
[4] D. Ghosh, E. Győri, R. R. Martin, A. Paulos, N.Salia, C. Xiao, O. Zamora, The Maximum Number of Paths of Length Four in a Planar Graph, arXiv:2004.09207
Problem 1. What is the maximum number of paths of three vertices in a bipartite graph G of n vertices? What if G is just triangle-free, but might be non-bipartite?
Problem 2. Show that the number of triangles in a planar graph of n vertices not containing any 4-cycle is at most 12(n-2)/5.
-
Title:
Transversal properties of families of convex sets
Description: click here
Prerequisites:
Although some background in convex geometry and combinatorics is helpful, we will start the semester with learning all the necessary tools.
Thus, it is going to be an excellent opportunity to study convexity, apply combinatorial ideas, and possibly use analytical and probabilistic tools.
Professor: dr Gergely Ambrus
Contact: 
Qualifying problems:: See the Qualifying Problems 1, 2 and 3
in the description
-
Title:
The solution space of genome rearrangement scenarios: a graph
theoretical and linear algebraic approach
Description: Click here
Prerequisites: basic combinatorics (elementary graph theory,
combinatorics) and algebra (elementary linear algebra, finite fields)
Best suitable for students who are interested in computer science and
combinatroics and are comfortable with using algebraic methods to
solve combinatorial and computer science problems.
Professor: dr Istvan Miklos
Contact: 
Qualifying problems::
Click here
-
Title:
Forbidden Configurations
Description: Click here
Prerequisites: basic graph theory, linear algebra
Professor: dr Attila Sali
Contact: 
Qualifying problems:: see within the
description
-
Title:
Research Problems in Knot Theory
Description:
We will go over background material, then turn to open problems. Here is the plan:
-
Week 1: Basic concepts of knot theory, isotopy, concordance. The
Reidemeister moves, examples of knots. First invariants of knots,
three-colorability, n-colorability. Methods for distinguishing knots.
-
Week 2: Polynomial invariants I: the Alexander polynomial. Kauffman
states, the black graph. Knot genus, its relation to the degree of
the Alexander polynomial. The genus of alternating knots. Fibered
knots.
-
Week 3: Polynomial invariants II: the Jones polynomial. Cube of
resolution. Crossing number, its relation to the Jones
polynomial. The crossing number of alternating knots.
-
Week 4: Manifolds, examples of three-manifolds. Dehn surgery.
Some three-manifold invariants.
-
Week 5: Grid presentations of knots, grid homology. Basic notions
of homological algebra, the invariance of grid homology.
-
Week 6: Knot Floer homology; computations for (1,1)-knots.
The relation between knot Floer homology and grid homology.
-
Week 7: Problems related to Dehn surgery presentations of three-manifolds.
We will be using handouts and notes. Reference for the first few lectures can be found in
http://web.cs.elte.hu/summerschool/2019/egyeb/ssm2019.pdf (pages 62-78).
In addition, some chapters of Grid homology for knots and links (Ozsvath-Stipsicz-Szabo) (AMS Mathematical Surveys and Monographs, Volume 208) will be used.
Prerequisites:Basic topology and algebra, basic algebraic topology
(singular homology and the fundamental group).
Professor: dr Andras Stipsicz
Contact: 
Qualifying problems::
Click here
Description: The subject is a new, very fast developing area of extremal graph
theory. The basic problem is to determine/estimate the maximum number
ex(n,H,F) of copies of H in a graph of n vertices not containing F as a subgraph.
(If H is just one edge then we get classical extremal graph theory problems.)
There are cases that are particularly interesting:
- The classical problem of Erdos solved half a century later: H is the 5-cycle, F is the triangle,
- H is a fixed path, F is your favorite graph
- H is a triangle, F is your favorite graph again (see [1], [2] for the 5-cycle)
- Started just in the last couple of years: the same questions in planar graphs (e.g., see [3])
Prerequisites: graph theory and combinatorics
Professor: dr Ervin Gyori
Contact:
Qualifying problems:: read and understand some of the following papers and solve the two problems below:
[1] B. Bollobas and E. Gyori. Pentagons vs. triangles. Discrete Mathematics, 308(2008), 4332–4336
[2] B. Ergemlidze, A. Methuku, N. Salia, and E. Gyori. A note on the maximum number of triangles in a C5-free graph. J. of Graph Theory, 90(2019) 227–230.
[3] E. Gyori, N. Salia, C. Tompkins, and O. Zamora. The maximum number of Pℓ copies in a Pk-free graph. Discrete Mathematics & Theoretical Computer Science, 21(1), 2018. or arXiv:1803.03240
[4] D. Ghosh, E. Győri, R. R. Martin, A. Paulos, N.Salia, C. Xiao, O. Zamora, The Maximum Number of Paths of Length Four in a Planar Graph, arXiv:2004.09207
Problem 1. What is the maximum number of paths of three vertices in a bipartite graph G of n vertices? What if G is just triangle-free, but might be non-bipartite?
Problem 2. Show that the number of triangles in a planar graph of n vertices not containing any 4-cycle is at most 12(n-2)/5.
Description: click here
Prerequisites:
Although some background in convex geometry and combinatorics is helpful, we will start the semester with learning all the necessary tools.
Thus, it is going to be an excellent opportunity to study convexity, apply combinatorial ideas, and possibly use analytical and probabilistic tools.
Professor: dr Gergely Ambrus
Contact:
Qualifying problems:: See the Qualifying Problems 1, 2 and 3
in the description
Description: Click here
Prerequisites: basic combinatorics (elementary graph theory,
combinatorics) and algebra (elementary linear algebra, finite fields)
Best suitable for students who are interested in computer science and
combinatroics and are comfortable with using algebraic methods to
solve combinatorial and computer science problems.
Professor: dr Istvan Miklos
Contact:
Qualifying problems::
Click here
Description: Click here
Prerequisites: basic graph theory, linear algebra
Professor: dr Attila Sali
Contact:
Qualifying problems:: see within the
description
Description: We will go over background material, then turn to open problems. Here is the plan:
- Week 1: Basic concepts of knot theory, isotopy, concordance. The Reidemeister moves, examples of knots. First invariants of knots, three-colorability, n-colorability. Methods for distinguishing knots.
- Week 2: Polynomial invariants I: the Alexander polynomial. Kauffman states, the black graph. Knot genus, its relation to the degree of the Alexander polynomial. The genus of alternating knots. Fibered knots.
- Week 3: Polynomial invariants II: the Jones polynomial. Cube of resolution. Crossing number, its relation to the Jones polynomial. The crossing number of alternating knots.
- Week 4: Manifolds, examples of three-manifolds. Dehn surgery. Some three-manifold invariants.
- Week 5: Grid presentations of knots, grid homology. Basic notions of homological algebra, the invariance of grid homology.
- Week 6: Knot Floer homology; computations for (1,1)-knots. The relation between knot Floer homology and grid homology.
- Week 7: Problems related to Dehn surgery presentations of three-manifolds.
We will be using handouts and notes. Reference for the first few lectures can be found in http://web.cs.elte.hu/summerschool/2019/egyeb/ssm2019.pdf (pages 62-78). In addition, some chapters of Grid homology for knots and links (Ozsvath-Stipsicz-Szabo) (AMS Mathematical Surveys and Monographs, Volume 208) will be used.
Prerequisites:Basic topology and algebra, basic algebraic topology (singular homology and the fundamental group).
Professor: dr Andras Stipsicz
Contact:
Qualifying problems:: Click here