Topics for BA-, MA-theses or Student Research

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Published on 2011. január 23. vasárnap, 11:47

This is a continuously changing, not restrictive list of mathematical topics proposed to students for their work for theses or research.

"People are generally better persuaded by the reasons which they have themselves discovered than by those which have come into the mind of others."  -  Blaise Pascal ("Pensées", 1670; "Thoughts")

Beside of reading the list of proposed topics below it may be worth to look at the works of my eralier students. Some theme of lower depth can be useful to prepare to a subject given for higher level, that makes the curricular delivery of these easier, but this is not compulsory, one can change subject anytime by wish.

I work happily on any other subject different from the themes given below, if I found it interesting and/or I can help the candidate, because, as Minarik the laundry said in the movie "Régi idők focija" (Sándor Pál, 1973), "Need a team".

Topics for BA theses

Fractals in the world

After choosing a special part of the world (stocks, plants, animals, bugs, ice etc.) the fractals appearing there .

Geometric constructions by Geogebra

The computer software Geogebra offers great ways to demonstrate contsructional procedures or to find locuses.

Characterizations of conic sections

Conic sections have exceedingly many characterizations by their various properties, one can select a bunch of it by interest.

Constructive geometry and its applications

To find nice curves and shapes for different conditions often need mathematical intuition, while, because the circumstances and the idols are allways changing, there numerous project to work on.

Reuleaux triangle in Minkowski plane

The Reuleaux triangle is is the simplest non circular convex set in plane with constant width. What properties it has in Minkowski plane?.

Topics for MA theses

Sufaces with planar geodesics.

John A. Little proved that these surfaces are projective. Study of this or any of its variation serve some interesting experiences.

Determining curves

Determination can be done by cardinalities of intersections, by set of tangents, by projections, or by some other not complete data or configurations.

Can you here the length of a string?

The question is whether striking a string, the resulting sound reveals the length of the string by some calculations. Well, what about when the string is not straight, but a loop? What about the triangle?

How small area of a plane set can be within which a needle can be reversed?

A Kakeya set is a set within which a unit line segment can be rotated continuously through 180 degrees, returning to its original position with reversed orientation. Besicovitch showed that there are Kakeya sets of arbitrarily small positive measure, but not with zero measure.

What kind of a plane shape is all of whose circumscribed $n$-gon are isometric?

Well, what is the answer if only similarity is requested in the question? Can we say something about such circumscribed $n$-gons? The same problems can be formulated for the inscribed $n$-gons...

Shapes with constant width in Minkowski plane

How much the bodies of constant width change if the underlying metric changed?.

Topics for student research

Special projectiv metrics.

A projektív metrikák létezésének és osztályozásának problémáját még Hilbert vetette fel. Vajon van e olyan nem Minkowski metrika, amelyben a súlypont harmadolja a súlyvonalakat? Más hasonló kérdések összefüggésekkel mi a helyzet?. This theme can be useful to prepare to the subject given for PhD research below.

Can you here the shape of a drum?

Mark Kac raised this question, and it is decided already in the negative by a counterexample, but in special (or one can say, more lifelike) cases the problem is almost completely open.

How small area of a plane set can be which intersects every straight line in set of positive measure?

To solve this question the Besicovich set helps, which, as Besicovitch showed, can have zero measure. The situation changes considerable if lines are substituted by circles and further questions arise by taking other geometric shapes.

What can be said about a manifold all of whose parametrizations map circles into circles?

Beltrami's theorem says that if two surfaces have a geodesic map between them, and one of them has constant curvature, then so does the other one. Is there a similar connection if the geodesics are replaced by circles?.

Topics for PhD

A good one can only be choosen by oral communications, but it, of course, should be in the theory of integral geometry, geometric analisys or differential geometry.

A more specific "official" subject offering can be found on my page at Hungarian Doctoral Council.

Further clue may be gotten by looking over my scientific publications, but I am open to work together on any topic in the above given theories.