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Summer Graduate School

Positivity Questions in Geometric Combinatorics July 10, 2017 - July 21, 2017
Parent Program: Geometric and Topological Combinatorics
Location: MSRI: Simons Auditorium, Atrium
Organizers Eran Nevo (Hebrew University), Raman Sanyal (Freie Universit├Ąt Berlin)
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Description

McMullen’s g-Conjecture from 1970 is a shining example of mathematical foresight that combined all results available at that time to conjure a complete characterization of face numbers of convex simple/simplicial polytopes. The key statement in its verification is that certain combinatorial numbers associated to geometric (or topological) objects are non-negative. The aim of this workshop is to introduce graduate students to selected contemporary topics in geometric combinatorics with an emphasis on positivity questions. It is fascinating that the dual notions of simple and simplicial polytopes lead to different but equally powerful algebraic frameworks to treat such questions. A key feature of the lectures will be the simultaneous development of these algebraic frameworks from complementary perspectives: combinatorial-topological and convex-geometric.  General concepts (such as Lefschetz elements, Hodge–Riemann–Minkowski inequalities) will be developed side-by-side, and analogies will be drawn to concepts in algebraic geometry, Fourier analysis, rigidity theory and measure theory. This allows for entry points for students with varying backgrounds.  The courses will be supplemented with guest lectures highlighting further connections to other fields.

Suggested prerequisites

For eligibility and how to apply, see the Summer Graduate Schools homepage

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