Packing of congruent convex sets in a big container and Covering of a set by congruent convex sets

 

Date:

Organizing Unit

3 July – 14 July

 

Institute of Information Technology

Language of Instruction

 Credits

English

 

 5 ECTS

 

Course location

 Course fee

On-campus

(University of Dunaújváros, Dunaújváros, Hungary)

 1300 EUR

 

 

 

 

 General information about UOD Summer Schools

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Summary

Packing and covering problems are special optimization problems concerning geometric objects in a given space or region. Many of the problems involve arranging geometric objects (usually identical) into the space or region as densely as possible with no overlap.

 

The set S is convex if the segment connecting any two points lying in S is contained in S. Let S_1,...,S_k be copies of the convex set S. We say that the sets S_1,...,S_k are packed in the container C if the sets S_1,...,S_k are pairwise disjoint and the union of the sets S_1,...,S_k lies in C. The question is the following. After fixing S, k and C what is the smallest 'a' such that the sets S_1,...,S_k can be packed in the container 'a'C. If we have found this number 'a', then what is the arrangement of the sets S_1,...,S_k in 'a'C. We will see some famous problems of packing of convex sets, the results and some open questions.

Let S be a convex set and let S_1,...,S_k be copies of the convex set S. The set C is covered by the sets S_1,...,S_k if the union of the sets S_1,...,S_k contains the set C. The question is the following. After fixing S, k and C what is the largest 'a' such that the sets S_1,...,S_k can cover the set 'a'C. If we have found this number 'a', then what is the arrangement of the sets S_1,...,S_k in order to cover 'a'C. We will see some famous covering problems, the results and some open questions.

 

Target group

This course would suit students strong in the field of mathematics, especially those with a background in geometry.

 

Course aim

 

After studying this course, you will:

  • develop knowledge and understanding of packing and covering,
  • understand some open questions in discrete geometry,
  • see some application of discrete geometry questions,
  • see connection between geometry and different branch of mathematics.

 

Topics

 

  • The definition of packing and covering
  • Simple examples of packing and covering, some applications
  • The definition of density of packing and covering
  • Dirichlet-Voronoi cells, Delanuay cells
  • Calculating of the density of packings of circles on the plane
  • Calculating of the density of packings of circles on the plane
  • Finding the best packing of congruent circles on the plane
  • Finding the best covering of the plane by congruent circles
  • Packing congruent balls in the space
  • Best packing of congruent balls in the space
  • Covering the space with balls
  • Packing congruent circles into a container. Packing congruent circles into a square
  • Packing congruent circles into a triangle. Packing congruent circles into a circle
  • Covering a square with congruent circles
  • Covering a triangle with congruent circles. Covering a square with congruent circles
  • Packing squares into a rectangle
  • Perfect packing of squares into a rectangle
  • Periodic packing of congruent circles on the plane
  • Periodic covering the plane with congruent circles
  • Periodic packing of congruent balls in the space 

 

The classes will be taught by highly experienced academics. Students will receive official Transcripts of Records upon completion of the program. The program includes various interesting and enjoyable social programs and excursions.

Course fee:       1300 EUR

 

This includes:

 

  • Application fee (15 EUR)
  • All tuition, including lectures, seminars, and tutorials.
  • Assessment, transcript of academic performance, and certificate.
  • Accommodation at the student hostel of the University
  • Breakfast and lunch during the summer course
  • Social activities, including two excursions to the Hungarian countryside

 

 For information about cancellation and refund policy and visa related information please visit the Summer School/General Information menu point.