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Newton polyhedra connects the speculation of convex polyhedra with algebraic geometry of toric varieties. For instance, Dehn–Zommerville duality for easy polyhedra results in the computation of the cohomology ring of easy toric varieties. Riemann–Roch theorem for toric varieties offers valuable info on the variety of integral factors in convex polyhedra and surprising multidimensional generalization of the classical Euler–Maclaurin summation formulation. Newton polyhedra permit to compute many discrete invariants of generic full intersections. Newton–Okounkov bodies connect the theory of convex bodies with algebraic geometry.

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This class supplies an introduction to classical topics in ergodic principle, with functions to dynamical techniques. In the final twenty years there is a rising interest in the connection between these two fields analogous to the one discovered within the 1970’s by Furstenberg for the Szemeredi theorem but now with the roles reversed. The course will begin with the basic Model concept of Fraisse structures and their limits and then continue with research of Logic actions of those limits. The aim is to reach a degree where structural Ramsey principle might be used to review these actions. Set concept and its relations with different branches of arithmetic.

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