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401-5006-07L DR D-USYS , D-MAVT , D-MTEC , D-MATH , D-BIOL , D-CHAB

Integer Points, Polyhedra, and Complexity

Lecturers: Prof. Dr. Alexander Barvinok
VVZ CR n/a

Last Updated: 2026-02-05 15:19:32

Content

I plan to cover some classical results about integer points in polyhedra and convex bodies in general, such as Pick's formula, Minkowski's Convex Body Theorem(s), Ehrhart polynomial, reciprocity, as well as recent developments related to valuations on rational polyhedra, the Lawrence-Khovanski-Pukhlikov Theorem, Brion's Theorem, integer semigroups, and ''local'' formulas for the coefficients of the Ehrhart polynomial due to Berline and Vergne. Connections with other areas will be discussed as well, such as continued fractions and their extensions, relations to commutative algebra and Hilbert series, algorithmic applications and integer programming, including recent successes of practical counting of lattice points for a variety of purposes, from statistics to computing structural constants in the representation theory. The prerequisites are minimal: linear algebra and some, mostly complex, analysis. Time permitting, I may be able to discuss some probabilistic approaches to integer point counting in polytopes of interest, such as counting of ''magic squares''.

General Information

Language
English
Levels
DR

Examination

Type
no performance assessment

Course Components

Type Title Time & Place Hours
lecture Integer Points, Polyhedra, and Complexity
Beginn: 28.03.2007
  • Wed 13:15-15:00 (HG G 43)
2 h weekly

Offered In