For an algebraic group G over an arbitrary field F, the geometry of projective homogeneous G-varieties has yet to be fully classified. A effective tool used towards such a classification is the cohomological invariant given by the set of Tits algebras of G. A result of Panin provides a connection from the Tits algebras of G to the Grothendieck group of G, and in particular to its associated gamma-filtration. In this talk, we use the Tits algebras of G to construct a torsion element in the gamma-filtration of a flag variety twisted by means of a PGO-torsor. This generalizes a construction in the HSpin case previously obtained by Zainoulline.

I will explain the construction of Kontsevich-Vishik canonical trace on non-integer order classical pseudodifferential operators. This construction has it roots in the old methods of extracting a finite part from a divergent sum or integral (infra-red and ultra-violet divergence), used by mathematicians and physicists.

If time permits I will explain some of the results on generalizations of this construction to noncommutative setting.

We survey the real normed division algebras (the real numbers, the complex numbers, the quaternions, and the octonions) and discuss how a result of M. Franz over the complex numbers was effectively generalized to all four of these number systems in the summer of 2014. The recommended background for this talk is second year linear algebra.

Algebraic groups are a well studied object that arise when one has an algebraic variety with a group structure compatible with the variety. In the same vein algebraic monoids are varieties with a monoid structure imposed on them. An interesting thing happens to certain algebraic groups and algebraic monoids called the Bruhat deomposition, which provides a wealth of knowledge about the groups/monoids in terms of double cosets. We'll take a look at two collections of algebraic monoids and their Bruhat decompositions, and determine generating functions for the "sizes" of their Bruhat decompositions.

We will use (primary and) secondary cohomology operations to describe the structure of the stable stems in low dimensions and compute a few Toda brackets in the stable stems.

In this talk I will define spin structure and spin manifolds and give some examples. I will also quickly review some notions of Riemannian geometry like connections, curvature; in particular, the Levi-Civita connection. Then I will define the spin connection and using that I will introduce the Dirac operator.

We will outline a certain program for Nakajima quiver varieties, in the cyclic quiver example. The picture includes two algebras that act on the K-theory of these varieties: one is the original picture by Nakajima, rephrased in terms of shuffle algebras, and the other one is the Maulik-Okounkov quantum toroidal algebra. The connection between the two is provided by the action of certain operators in the so-called "stable basis", and we will present formulas for this action. These formulas can be perceived as a generalization of Lascoux-Leclerc-Thibon ribbon tableau Pieri rules.