114. Mathematical Colloquium

Sergei K. Lando

Center for Advanced Studies in Skolkovo and Higher School of Economics in Moscow


THE UNREASONABLE EFFECTIVENESS OF GEOMETRY IN COMBINATORIAL PROBLEMS

Tuesday February 18, 2020, 14:00

aula (refektar), 1st floor MFF UK, Malostranské nám. 25, Praha 1

Abstract

On May 11, 1959 Eugene Wigner gave a talk at New York University in honor of Richard Courant. The name of the talk was “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. In this talk Wigner argued that there are no visible reasons for mathematics to be a good instrument for describing physical reality, while in practice there is no other tool of comparable utility. Wigner’s talk and his paper based on the one and published a year later serve since then as an archetype for numerous comparisons of visibly far domains of science, allowing for an efficient applications of one to the other.

The talk will be devoted to examples of effective applications of geometry (and its relatively recently emerged branch topology) to solving combinatorial problems. In spite of the fact that geometry and combinatorics both belong to mathematics, it is difficult to find its branches that are more distant from one another. It suffices to mention that geometry became a domain of formal mathematics as early as in the 6th century BC, while the first mentioning of a combinatorial problem in the European literature (arguably) dates back only to 13th century AD. Geometry and, especially, topology are about continuity, while combinatorics deals exclusively with discrete objects.

It is true that combinatorics helps a lot in studying objects of topological nature. However, in the opposite direction the applications are rare. And if such an application exists, then it often proves to be extremely effective, sometimes leading to new results not achievable by other tools. The examples will include

  • E. Looijenga’s proof (1974) of the Caley theorem enumerating trees with numbered vertices;
  • the ELSV formula (1999) expressing Hurwitz numbers in terms of geometry of moduli spaces of algebraic curves;
  • J. Huh’s proof (2010) of Read’s conjecture stating that the sequence of coefficients of the chromatic polynomial of any graph is unimodal (that is, the absolute values of the coefficients first increase, and then decrease).

About the speaker

Sergei K. Lando studoval na Moskevské univerzitě, kde v roce 1986 získal doktorát pod vedením věhlasného V. I. Arnolda. Titul DrSc. mu byl udělen v roce 2005. S. K. Lando byl zaměstnán na různých univerzitách a pracovištích Akademie věd a od roku 2008 jako profesor Higher School of Economics v Moskvě, kde byl rovněž děkanem matematického ústavu. Od roku 2012 je víceprezidentem Moskevské matematické společnosti.

Prof. S. K. Lando je vynikajícím matematikem v oblastech algebraické geometrie, teorie singularit, topologie a kombinatoriky. Je autorem tří knih, které byly publikovány v nakladatelstvích AMS a Springer. Jeho práce o teorii uzlů (a diagramů a  meandrů) jsou známé a vyústily v jeho zvanou přednášku na Mezinárodním kongresu matematiků v Hajdarábádu v r. 2010.

Poznamenejme též, že prof. S. K. Lando přednese ve čtvrtek 20. 2. ve 12:30 další přednášku v rámci seminárního cyklu na téma Teorie uzlů.