Activities This Week

Cut and project sets are aperiodic point patterns obtained by projecting an irrational slice of the integer lattice to a subspace. One way of classifying aperiodic sets is to study the number and repetition of finite patterns. Sets with patterns repeating linearly often, called linearly repetitive sets, can be viewed as the most ordered aperiodic sets. Repetitivity of a cut and project set depends on the slope and shape of the irrational slice. In an earlier work, joint with of Haynes and Walton, we showed that when the slice has a cube shape, linear repetitivity holds if and only if the following two conditions are satisfied: (i) the cut and project set has the minimal number of different finite patterns (minimal complexity), and (ii) the irrational slope satisfies a badly approximable condition. In a new joint work with Jamie Walton, we give a generalisation of this result to all convex polytopal shapes satisfying a mild geometric condition. A key step in the proof is a decomposition of the cut and project scheme, which allows us to make sense of condition (ii) for general polytopal windows.

Singularity Theory has originated (at the end of 19’th century) with the two basic questions:

• how does the graph of a function look locally?
• how does the zero set of a function look locally?

If the first derivative of a function does not vanish at a point then one can change the local coordinates to linearize the functions. Geometrically one “rectifies” the graph/the zero set. Accordingly the local geometry/topology/algebra are trivialized. The situation becomes much more involved when the first derivative vanishes. I will consider several simple (though non-trivial) examples, showing how the algebra/geometry/topology are involved.

(ההרצאה תתקיים בעברית)

I will report on recent work concerning the action of the motivic Galois group on Anabelian objects such as fundamental groups of algebraic varieties conveniently completed. I’ll sketch the proof of a motivic analog of a theorem of Pop (aka., the Ihara-Matsumoto-Oda conjecture) yielding several interpretations of the motivic Galois group as the automorphism group of some large diagrams of anabelian objects.