# Algebra, Number Theory & Representations

Listed on this page are current research projects being offered for the Vacation Scholarship Program.

For more information on this research group see: Algebra, Number Theory & Representations

## Fusion coefficients

(posted for 2015-2016)

Fusion is one of those mysterious product operations from mathematical physics for which we need a better understanding. The goal of this project is to give a way of computing fusion products by generalizing the path model technique used for computing ordinary products of group representations.

**Contact:** Arun Ram aram@unimelb.edu.au

## Explaining Mochizuki’s Concept of Species

(posted for 2015-2016)

In Shinichi Mochizuki’s Inter-Universal Teichmuller Theory IV: Log-Volume Computations and Set- Theoretic Foundations, §3, “Inter-Universal Formalism: the Language of Species,” Mochizuki introduces the concept of species to formalise the intuitive notion of a “type of mathematical object.” The goal of this project is to explain this concept, and how this concept resolves Russell’s Paradox.

**Contact:** Arun Ram aram@unimelb.edu.au

## The Glass Bead Game

(posted for 2014-2015)

The glass bead game is a toy model for the algebraic structure of the quiver Hecke algebra. This project explores whether this model is also useful for modelling the preprojective variety and the loop Grassmannian. The goal is to understand what these objects are by studying the possible plays that lead to each configuration of glass beads.

**Contact:** Arun Ram aram@unimelb.edu.au

## Grading with strip matrices

(posted for 2014-2015)

A graded algebra is a more rigid and more controllable version of the original structure. The goal of this project is to represent the structure (the affine BMW algebra) as matrices and then compute the relations satisfied by the diagonal strips in these matrices.

**Contact:** Arun Ram aram@unimelb.edu.au

## Elliptic reflections

(posted for 2014-2015)

This project aims to complete the analogy between Euclidean (flat) reflections, circular reflections, and elliptic reflections. The goal is to copy the standard proof that the polynomials have a basis (over the reflection invariants) to the elliptic setting. Here the word basis is in the sense of linear algebra: basis of a vector space.

**Contact:** Arun Ram aram@unimelb.edu.au