One general approach to random number generation is to take a uniformly distributed (0,1) random variable and then invert the cumulative distribution function (CDF) to generate samples from another distribution.  This talk follows this approach, approximating the inverse CDF for the Poisson distribution in a way which is particularly efficient for vector execution on NVIDIA GPUs.

I shall show that for every conjugacy class O in a connected semisimple algebraic group G over an algebraically closed field of characteristic good for G one can find a special transversal slice S to the set of conjugacy classes in G such that O intersects S and dim O=codim S. The construction of the slice utilizes some new combinatorics related to invariant planes for the action of Weyl group elements in the reflection representation. The condition dim O=codim S is checked using some new mysterious results by Lusztig on intersection of conjugacy classes in algebraic groups with Bruhat cells.

In 1983 Kerckhoff settled a long standing conjecture by Nielsen proving that every finite subgroup of the mapping class group of a compact surface can be realized as a group of diffeomorphisms. An important consequence of this theorem is that one can now try to study subgroups of the mapping class group taking the quotient of the surface by these groups of diffeomorphisms. In this talk we will study quotients of surfaces under the action of a finite group to find bounds on the cardinality of such a group.

The Dehn function of a group measures the complexity of the group's word problem, being the upper bound on the number of relations from a group presentation required to prove that a word in the generators represents the identity element. The Filling Theorem which was first stated by Gromov connects this to the isoperimetric functions of Riemannian manifolds. In this talk, we will see the classification of hyperbolic groups as those with a linear Dehn function, and give Bowditch's proof that a subquadratic isoperimetric inequality implies a linear one (which gives the only gap in the "isoperimetric spectrum" of exponents of polynomial Dehn functions).