Abstract.I will outline some results in Algebraic Geometry obtained in our works withT.Ziegler. Our proofs are based on Analysis over finite fields which leads tonew results even for complex varieties.
Seminar room 209, Manchester Building, Jerusalem, Israel.
How do algebras grow?
The question of `how do algebras grow?', or, which functions can be realized as growth functions of algebras (associative/Lie, or algebras having certain additional algebraic properties) is a major problem in the meeting point of several mathematical fields including algebra, combinatorics, symbolic dynamics and more.
Abstract: A circle packing is a canonical way of representing a planar graph. There is a deep connection between the geometry of the circle packing and the proababilistic property of recurrence/transience of the simple random walk on the underlying graph, as shown in the famous He-Schramm Theorem. The removal of one of the Theorem's assumptions - that of bounded degrees - can cause the theorem to fail. However, by using certain natural weights that arise from the circle packing for a weighted random walk, (at least) one of the directions of the He-Schramm Theorem remains true.