Thesis supervisor: Gábor Domokos
belső konzulens: Zsolt Lángi
Location of studies (in Hungarian): Department of Geometry, Institute of Mathematics, BME Abbreviation of location of studies: BME
Description of the research topic:
A physically extremely interesting area of convex geometry deals with the description of convex solids with the aim to identify, categorize and track the evolution of natural shapes. In addition to convex geometry, mathematical tools include geometric partial differential equations, in particular, curvature-driven flows which are closely related to the heat equation. An equilibrium point of a convex solid is a stationary point of the distance function measured from the center of gravity, placing the solid on a horizontal plane it can be statically balanced at these points. We can distinguish between maximum, minimum and saddle points, the numbers of which we denote by S, U and H, respectively. In case of convex solids, the Poincaré-Hopf Theorem implies the relationship
S+U–H=2,
and based on this any convex solid can be assigned to an {S,U} equilibrium class. In addition to the number of equilibrium points, the topology of the integral curves in the gradient flow connecting these points also describes convex solids. Based on this aspect, within each equilibrium class we can distinguish topological subclasses. By the equilibrium class and subclasses, a very interesting and from geological point of view very useful classification system can be defined for shapes that can be found in nature (e.g. pebble shapes). Our former research verified that both the system of equilibrium and that of topological classes are complete in the sense there is neither empty class, nor empty subclass. This classification system, complemented with ideas from shape evolution led to the verification of the mathematical model with the aid of which a research team from Budapest, Philadelphia and NASA found compelling evidence of fluvial activity on ancient Mars, based alone on the pictures of Martian pebbles shapes, taken by NASA’s Curiosity rover.
In the present PhD research we investigate some particularly interesting geometric properties of the above defined classification system. Our goal, among other things, is to find out how robust these classes and subclasses are; that is, by what probability a convex solid can move from one class or subclass into another one by abrasion. We already have some inital results, but many questions are not yet answered which are essential from physical applications.
The topic esentially is geometrically motivated, within this knowledge of classical differential geometry is important.
Expertise in low-dimensional dynamical systems is an asset, and also familiarity with numeric computations and programming is very useful. The topic has also statistical aspects, we welcome applicants with such interest as well. Primarily we expect the applications of students with a degree in mathematics or physics.
Required language skills: English Further requirements: The topic esentially is geometrically motivated, within this knowledge of classical differential geometry is important.
Deadline for application: 2017-05-31
2024. IV. 17. ODT ülés Az ODT következő ülésére 2024. június 14-én, pénteken 10.00 órakor kerül sor a Semmelweis Egyetem Szenátusi termében (Bp. Üllői út 26. I. emelet).
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