Proofs by example and numerical Nullstellensätze
Freitag, 3.12.21, 10:30-11:30, BBB room
We study the proof method "proof by example" in which a general statement can be proved by verifying it for a single example. This strategy can indeed work if the statement in question is an algebraic identity and the example is "generic". This talk addresses the problem of construction a practical example, which is sufficiently generic, for which the statement can be verified efficiently, and which even allows for a numerical margin of error.\nOur answer comes in the form of a numerical Nullstellensatz, which is based on Diophantine geomery, in particular an arithemetic Nullstellensatz and a new effective Liouville-Lojasiewicz type inequality.\n\nIf time permits we moreover consider "proofs by several examples", which in addition requires a conceptual notion of sufficient genericity of a set of points. Besides theoretical and algorithmic criteria for sufficient genericity, we obtain several new types of Nullstellensätze in the spirit of the combinatorial Nullstellensatz and the Schwartz-Zippel lemma, also for varieties.
C^1-triangulations of semi-algebraic sets
Freitag, 17.12.21, 10:30-11:30, Raum 404, Ernst-Zermelo-Str. 1
If one wants to treat integration of differential forms over semi-algebraic sets analogous to the case of smooth manifolds, it is desirable to have triangulations of semi-algebraic sets that are globally of class C^1.\nWe will present a proof of the existence of such triangulations using the 'panel beating' method introduced by Ohmoto-Shiota (2017) and discuss possible generalizations.