Students

Asmita Sodhi (Dalhousie University)

Integer-valued polynomials and a game called $p$-ordering

In this talk we will visit the world of integer-valued polynomials, and also introduce the ring of polynomials that are integer-valued over a subset of $\mathbb{Z}$. We will explore Bhargava’s “game called $p$-ordering”, and see how $p$-orderings and $p$-sequences allow us to find a $\mathbb{Z}$-module basis for the ring of integer-valued polynomials for a subset of the integers. Finally, we will briefly see how Bhargava’s tools may be extended to the noncommutative case of integer-valued polynomials over the ring $M_n(\mathbb{Z})$ of $n\times n$ integer matrices.


Nicole Kitt (University of Calgary)

How to calculate perverse sheaves on quiver representation varieties of type A

In their 1997 paper, Geometric construction of crystal bases, Masaki Kashiwara and Yoshihisa Kashiwa Saito described a singularity in a quiver representation variety of type $A_5$ with the property that the characteristic cycles of the singularity is reducible, thus providing a counterexample to a conjecture of Kazhdan and Lusztig. This singularity is now commonly known as the Kashiwara-Saito singularity. While the 1997 paper showed that the characteristic cycles of the Kashiwara-Saito singularity decomposes into at least two irreducible cycles, they promised, but did not prove, that it decomposes into exactly two irreducible cycles.\\

The goal of this project is to complete this calculation using geometric techniques developed in the example part of the Voganish paper. The first step in this calculation is to compute perverse sheaves on the quiver representation variety of type $A_5$. In this talk, I will illustrate the methods used to make such a calculation by calculating perverse sheaves for a specific quiver representation variety of type $A$. In doing so, I will show how to construct a proper smooth cover for any quiver variety of type $A$.


Anne Dranowski (Univeristy of Toronto)

MV cycles from generalized orbital varieties

Representations constructed from the geometry of homogeneous spaces involve many choices, so we would like to parametrize coarse invariants, like dimensions of weight spaces of irreducible representations, by combinatorial objects. A classical example is the Grothendieck–Springer resolution of the variety of nilpotent elements $\mathcal{N}$ in a semi-simple Lie algebra: the top Borel-Moore homology of a fibre of this resolution is an irreducible representation of the associated Weyl group. In type A, a canonical basis is parametrized by Young tableaux. This talk will review a more modern example: the torus-equivariant cohomology of upper-triangular Slodowy slices. We explain the representation theory and combinatorics of this example: using the geometric Satake correspondence and a Spaltenstein decomposition, we show that orbital varieties in Slodowy slices define bases in representations. Under the magnifying glass of a finer geometric invariant — the Duistermaat-Heckmann measure — we show that not all bases are created equal.

Reila Zheng (Univeristy of Toronto)

Random walks on Gromov hyperbolic spaces

I will describe the geometry of Gromov hyperbolic spaces, Gromov boundary, Poisson boundary, shadow sets, horofunctions, and other useful constructs. I will also introduce some tools from probability theory used to analyze random walks of isometries on such spaces. If time allows I will state some recent results in literature and sketch their proofs.


Peter Bradshaw (Simon Fraser University)

Cops and robbers on Cayley graphs

We discuss the pursuit-evasion game “Cops and Robbers” with regard to Cayley graphs. We show that Meyniel’s Conjecture holds for several classes of Cayley graphs, including abelian Cayley graphs and dihedral Cayley graphs. We also extend several known Cops and Robbers results for Cayley graphs to other pursuit-evasion variants.