Graduiertenkolleg: Methods for Discrete Structures

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Monday Lecture and Colloquium


Monday, November 30, 2015

Freie Universität Berlin
Institut für Informatik
Takustr. 9
14195 Berlin
room 005


Lecture - 14:15

Tim Römer - Universität Osnabrück

Commutative Algebra up to Symmetry

Abstract:
Ideal theory over a polynomial ring in infinitely many variables is rather complicated which is (beside other things) due to the fact that this ring is not Noetherian. Since very recently one is interested in ideals in such a ring which are invariant under certain well-behaved monoid actions. We present some new results and open questions on algebraic properties of these ideals and associated objects of interest.
The talk is based on joint work with Uwe Nagel.




Colloquium - 16:00

Matthias Henze - FU Berlin


On dual Minkowski-inequalities via covering minima

Abstract:
The subject of the Geometry of Numbers was started by the seminal and foundational work of Minkowski in the late 19th century. Based on natural geometric ideas he proved beautiful criteria for 0-symmetric convex bodies to contain non-trivial integral points, and he applied his findings to solve number theoretic questions. This theory developed into a lonestanding branch of mathematics that found applications also in integer programming, functional analysis, additive combinatorics, and many more.

Up until today, besides the efforts of many eminent mathematicians there is no satisfying dual theory to Minkowski's original theorems. We pick up on the duality between packing and covering arrangements, and in particular we study the so called covering minima of convex bodies. These minima were introduced by Kannan & Lovász (1988) to study flatness-theorems for integer programming and diophantine approximation.

Generalizing a problem posed by Makai Jr., we formulate precise conjectures on dual Minkowski-inequalities relating the volume of a convex body to its covering minima. We solve the conjectures for the special yet important class of unconditional convex bodies and we further explore asymptotic estimates for the general case. The occuring extremal examples give rise to an interesting family of convex bodies that may be thought of as analogs to the well-studied parallelohedra, that is, convex bodies tiling space by translation.



Letzte Aktualisierung: 16.11.2015