arXiv subjects

Martin Grötschel

Publications and source records attributed to Martin Grötschel.

3 recordsLinked to original sources

arXiv2023

The Mathematics of László Lovász

This is an exposition of the contributions of László Lovász to mathematics and computer science written on the occasion of the bestowal of the Abel Prize~2021 to him. Our survey, of course, cannot be exhaustive. We sketch remarkable results that solved well-known open and important problems and that -- in addition -- had lasting impact on the development of subsequent research and even started whole new theories. Although discrete mathematics is what one can call the Lovász home turf, his interests were, from the beginning of his academic career, much broader. He employed algebra, geometry, topology, analysis, stochastics, statistical physics, optimization, and complexity theory, to name a few, to contribute significantly to the explosive growth of combinatorics; but he also exported combinatorial techniques to many other fields, and thus built enduring bridges between several branches of mathematics and computer science. Topics such as computational convexity or topological combinatorics, for example, would not exist without his fundamental results. We also briefly mention his substantial influence on various developments in applied mathematics such as the optimization of real-world applications and cryptography.

math.HO↗

arXiv2021

On angular measures in axiomatic Euclidean planar geometry

We address the issue of angular measure, which is a contested issue for the International System of Units (SI). We provide a mathematically rigorous and axiomatic presentation of angular measure that leads to the traditional way of measuring a plane angle subtended by a circular arc as the length of the arc divided by the radius of the arc, a scalar quantity. We distinguish between the \emph{angular magnitude}, defined in terms of congruence classes of angles, and the (numerical) \emph{angular measure} that can be assigned to each congruence class in such a way that, e.g., the right angle has the numerical value $\frac\pi2$. We argue that angles are intrinsically different from lengths, as there are angles of special significance (such as the right angle, or the straight angle), while there is no distinguished length in Euclidean geometry. This is further underlined by the observation that, while units such as the metre and kilogram have been refined over time due to advances in metrology, no such refinement of the radian is conceivable. It is a mathematically defined unit, set in stone for eternity. We conclude that angular measures are numbers, and the current definition in SI should remain unaltered.

math.HO↗

arXiv2002

On the representation of polyhedra by polynomial inequalities

A beautiful result of Bröcker and Scheiderer on the stability index of basic closed semi-algebraic sets implies, as a very special case, that every $d$-dimensional polyhedron admits a representation as the set of solutions of at most $d(d+1)/2$ polynomial inequalities. Even in this polyhedral case, however, no constructive proof is known, even if the quadratic upper bound is replaced by any bound depending only on the dimension. Here we give, for simple polytopes, an explicit construction of polynomials describing such a polytope. The number of used polynomials is exponential in the dimension, but in the 2- and 3-dimensional case we get the expected number $d(d+1)/2$.

math.MG↗