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Mathematical writing

Everything below is expository. For research, please take a look at my CV. If you find a mistake or are interested in something written here, please don’t hesitate to email me.

• Simultaneously diagonalizable operators exercise. One of my favorite linear algebra exercises. The statement is that if $\varphi$ and $\psi$ are a pair of linear maps from a finite-dimensional vector space $V$ to itself such that $\varphi$ and $\psi$ are diagonalizable, then $\varphi$ and $\psi$ commute if and only if they are simultaneously diagonalizable. Simultaneously diagonalizable means that there is a basis for $V$ consisting of eigenvectors for $\varphi$ and $\psi$.
• The surgery exact triangle in Heegaard Floer homology. My undergraduate thesis is an expository account of Heegaard Floer homology and a proof of a key computational tool in the theory, the surgery exact triangle. Heegaard Floer homology is a package of invariants for closed and oriented three-manifolds $Y$, defined by Peter Ozsváth (my advisor) and Zoltán Szabó. The simplest version associates to $Y$ a finitely generated Abelian group $\widehat{\mathrm{HF}}(Y)$. It is defined with the help of Heegaard diagrams and Lagrangian Floer homology. Roughly half of the thesis constructs Heegaard Floer homology with relevant background and the other half proves the surgery exact triangle. I intend to make this thesis readable for anybody who has taken first courses in algebraic topology and differential geometry.
• On Morse-Smale and Lagrangian Floer homology. A brief account and comparison of Morse homology and Lagrangian Floer homology. Morse theory allows one to understand the topology of a manifold by studying differentiable real-valued functions on the manifold. Morse homology is a particularly illuminating way of understanding the homology of smooth manifolds and is isomorphic to the singular homology of the manifold with integer coefficients. Lagrangian Floer homology is the “infinite-dimensional” analogue of Morse theory and it grew out of Andreas Floer’s solution to Arnold’s conjecture for a large class of symplectic manifolds. I tell this story here.
• The proof of Steinitz’s Theorem. A proof of a characterization of graphs of polyhedra. The statement is that $G$ is the graph of the polyhedron, that is, the graph obtained by only looking at the polyhedron’s vertices and edges, if and only if $G$ is $3$-connected, meaning that whenever one removes two of its vertices and all of the edges incident to those vertices, the graph remains connected.

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