# Work

## Presentations

Introduction to Growth in Groups (Winter 2017; 30 minutes)

Transcript on a talk about group counting problems I gave at the University of Waterloo.

I want to introduce you to the particularly beautiful group counting problem of how many elements can be spelled with $\leq n$ letters?

The growth function of a group is a large-scale geometric property that is directly connected to two out of three of Dehn’s decision problems: the word problem and the isomorphism problem.

Computation in the Completion of the Free Group Algebra (Winter 2015; 19 slides).

Abstract:

It is known (e.g., due to independent results of Malcev and B.H. Neumann) that $\mathbb{Q}[F_n]$, the (rational) group algebra of the free group of rank $n$, can be embedded in a division algebra $D$. We consider the problem of making this embedding algorithmic. More precisely, if $D_0$ is the smallest sub-division algebra of $D$ containing $\mathbb{Q}[F_n]$, we consider the problem of representing elements of $D_0$ by a finite data structure with which the algebra operations (addition, multiplication, inversion) can be calculated algorithmically. We present a solution to this problem on a particular subalgebra of $D_0$, using noncommutative formal power series with recurrence relations, and discuss how to generalize this idea to all of $D_0$. We also discuss applications of our results to problems in algebraic topology.

## Selected Blog Posts

Beautiful Link (Fall 2017; 3,745 words)

An intuitive introduction to mapping class groups, Teichmüller spaces their link to hyperbolic geometry using one example: the torus. Written with the help of Prof. Kasra Rafi at the University of Toronto.

Introducing Growth of Groups (Winter 2017; 2,984 words)

The growth of a group is an aesthetically pleasing large-scale property that is borderline geometric and algorithmic. I will give an overview on the research on growth by historical and intuitive examples.

The Social Role of Mathematical Proofs (Fall 2016; 1,220 words)

Although proofs are often thought as a way to show beyond reasonable doubt that a mathematical statement is true, this is not always the case in practice, even for mathematicians at the top of the field. For example, Andrew Wiles proof of Fermat’s Last Theorem originally contained a gap, and Fields medalist Vladimir Voevodsky found one of his papers to be very wrong seven years after publication. […] Using today’s computing power, there are three main ways that mathematical proofs can be going:

## Expository Papers

Introduction to $\ell^2$-Betti numbers (Spring 2016; 8 pages).

The goal of this presentation is to set up the framework for $\ell^2$ - Betti Numbers from the point of view of von Neumann Algebras.

Overview of the classification of tripartite entanglement under SLOCC (Summer 2014; 23 pages).

The goal of this document is to give the undergraduate reader an overview of tripartite quantum entanglement under SLOCC, with no background assumed. Warning: juvenilia!

Adventures in $C^*$-algebra (Summer 2016; 49 pages)

Notes for Jane Panangaden’s excellent $C^*$-algebra reading course.

Keywords: time evolution dynamics, infinite dimensional spaces, trace class operators, bounded operators on a Hilbert space, Banach Algebra, Riesz representation theorem for Hilbert spaces, spectral theory, functional calculus, Hille-Yoshida, structure theorem for $C^*$-algebra, Tomita-Takesaki, KMS states.

## Research Logs

Warning: by the nature of these documents, some stuff in there may be blatantly wrong. Let me know.

Growth in the Heisenberg Group (Fall 2016; 32 pages)

Collection of definition and concepts, insights and breakthroughs to formulating a conjecture on the period of the quasi-polynomial growth function of the three-dimensional Heisenberg group with arbitrary generating set. Updated weekly.

Keywords: growth in groups, nilpotent groups, Malcev normal form, Carnot groups, Gromov, growth in the polynomial range, rational growth, Duchin, Shapiro, geodesics, Cayley graph, CC metric, complexity, polynomial-time algorithm.