The Bloch-Kato conjecture is the statement that the Galois cohomology of the absolute Galois group of a field which contains a primitive pth root of unity in mod p coefficients is isomorphic to Milnor K-theory reduced modulo p. This statement is now a theorem proved by Rost and Voevodsky. In particular it says that the cohomology ring of the absolute Galois group is generated by one dimensional classes. It is a natural question to find intermediate Galois extensions of the base field where every element in the cohomology ring decomposes into a sum of products of one dimensional classes. In degree two we answer this question by providing a tower of Galois extensions where indecomposable elements decompose in the next level of the tower. We also illustrate this refinement by directly computing the cohomology rings of superpythagorean fields and p-rigid fields. This is a joint work with J. Minac and S. Chebolu.

The topics we continue are as follows: ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weizenbok formula, ---Heat kernel and its asymptotic expansion, Gilkey's formula, Mackean-Singer formula.

Estimates involving symmetric geometric means of mixed norms have appeared since at least Littlewood's $4/3$ inequality, and remain relevant. New theorems provide a simple general framework, replacing ad-hoc methods.

More flexible generalizations of H{\"o}lder's inequality, both in one variable and for mixed norms, are crucial. These reformulate the exponent condition in terms of harmonic means, and the conclusion in terms of geometric means. I will also describe a generalization to weighted means by Albuquerque, Araujo, Pellegrino, and Seoane-Sepulveda.

The key results follow from generalized H{\"o}lder, after a combinatorial argument. The basic techniques used are just the Holder and Minkowski integral inequalities, but the final results easily produce generalizations of Littlewood's $4/3$ inequality, with applications to multilinearity, Sobolev embeddings, and other topics.

Given a group of 100 men and 100 women, can we always arrange 100 (heterosexual) marriages, which would be *stable* in that no man and no woman simultaneously prefer each other over their assigned partners (which may lead to them leaving their partners and running away)? The positive answer to this question was given in 1962 by mathematicians David Gale and Lloyd Shapley and it was one of the results for which in 2012, Shapley was awarded the Nobel Prize in economics.

I will present the mathematics behind the Gale--Shapley algorithm and some of its interesting applications. Afterwards, I will discuss a few variations on the original problem, such as: the stable roommate problem and the college admission problem.

Pizza and pops to follow!