A projective toric scheme is specified by combinatorial data, viz., a polytope with integral vertex coordinates. I will show how the geometry of the polytope leads to a simple splitting result in the algebraic K-theory of the scheme. In the special case of projective space (given by a standard simplex) this reduces to the well-known splitting of K(P^n) into n+1 copies of the K-theory of the ground ring. - The combinatorial approach is flexible enough to include the case of schemes defined over an arbitrary (possibly non-commutative) ring.

A knot is a smooth embedding of a circle in \(R^3\). A major problem in knot theory is to classify knots. Two knots are identified if one can transform one knot to the other one without tearing.

The discovery of the Jones polynomial in 1980's has been considered as a great achievement in the classification of knots. The Jones polynomial is a sensitive knot invariant which is easy to compute. On the other hand, theories including geometry of low dimensional spaces, quantum groups and statistical mechanics meet each other because of this polynomial.

In this talk we introduce the Jones polynomial in an elementary way from two different point of views. This talk will be accessible to undergraduate students.

I am going to explain gently how one can define link invariants using braid groups and their representations as matrices. Then I am going to explain how representations endowed with a compatible symplectic form give rise to link invariants with values in the Witt ring of the field considered; of course I will define the Witt ring as well. The construction makes use of Maslov indices. In the end, using the Burau representation, we get one invariant which "contains" many others: signatures, Jones metaplectic nvariants, and a polynomial which is almost the one by Alexander-Conway.

There is a general cohomology defined by Sweedler for co-commutative Hopf algebras, generalizing the usual cohomology of a group or a Lie algebra. Recently it was discovered that low-dimensional groups could be defined without the co-commutativity requirement. In joint work with Christian Kassel, we have given the first few examples of computations with these, in the case of algebras of functions on groups. These turn out to be related to torsors in algebraic geometry, and Drinfeld twists in quantum groups theory.