RESEARCH
The three sphere famously admits a foiliation by circles: the Hopf fibration.  However, if we loosen our requirements a little bit and allow nearby circles to twist about one another, it admits infinitely many more.  Here we will construct pretty pictures of these Seifert fibrations by filling the complement of a Hopf link with torus knots.
(UNDER CONSTRUCTION)
It is natural to want to view the space of lattices in the plane as a topological space: just looking at them theres an obvious notion of when two lattices are "close".  It turns out the space of unimodular lattices is homeomorphic to the trefoil knot complement, which allows for some really pretty mathematics.  Both the modular surface and the hyperbolic plane's unit tangent bundle make surprise appearances in this story!
(UNDER CONSTRUCTION)
​The three strand braid group is the fundamental group of the trefoil complement - and an enlightening isomorphism can be constructed by thinking about cubic polynomials with distinct roots.  In fact this picture generalizes - there's a connection between the n strand braid group and degree n polynomials!
(UNDER CONSTRUCTION)
​Take a tetrahedron and inscribe it in a sphere.  The space of ways to do this has a natural topology on it, but what does it look like?  We can play the same game with dodecahedra, cubes, or even regular tilings of other shapes such as the Euclidean or Hyperbolic planes.  A beautiful story involving the unit tangent bundles of orbifolds can be told to help us understand these spaces.
(COMING SOON)
​The modular group $\textrm{SL}(2,\mathbb{R})$ has the kernel of the "reduce mod $N$" map as a subgroup for each $N$.  In the world of orbifolds, the modular group corresponds to the $(2,3,\infty)$ triangular pillowcase, and these subgroups correspond to covers.  But what do these covers look like?  Punctured platonic solids, honeycomb patters on tori and the Klein quartic, in fact!
We are often accustomed to viewing the three sphere through stereographic projection - but this is not what it would look like to an inhabitant.  Here, we will try to understand what it means to "look around" inside a Riemannian manifold using $\mathbb{S}^3$ as a guiding example.​

UNDERGRADUATE
Experience tutoring various levels of mathematics over the years has led me to think a lot about the different ways new ideas can be presented.  As I myself am rather geometrically inclined, I've collected some of the more interesting geometric perspectives that I have come across or thought up below; the topics covered range from grade school through undergraduate mathematics.
Taking a geometric viewpoint with respect to basic arithmetic and algebra allows us to visualize factoring, foiling and the proof of the quadratic formula.
A very visual argument for the pythagorean theorem, one you can replicate yourself with a scissors and some paper!

​Geometric series are the simplest introduction to the idea of adding up infinitely many things.  Here we will discuss the potential pitfalls of infinite addition, and work through a couple of really clever examples which will help us expose a general pattern for what happens when we want to compute sums of the form $1+x+x^2+\cdots$.
Exponential functions are everywhere, and with them comes the mysterious number $e$.  Here we will discuss what it means to raise numbers to powers, and how we extend that process to a function on the real line which has fundamental applications to the study of growth.
(COMING SOON)
The gradient from multivariable calculus is a generalization of the derivative for functions from $\mathbb{R}^n\to\mathbb{R}.  And along with this notion of derivative there is a corresponding version of the fundamental theorem of calculus.  Here we discuss one way to concretely understand the gradient through the analogy of slopes on a mountain range.
Divergence and curl are two generalizations of the idea of derivative to higher dimensions: but instead of measuring changes of functions they measure changes in vector fields.  The curl of a vector field measures the "infintesimal circulation" and the divergence the "infintesimal expansion": these ideas will be made precise here.
Fourier series (and the associated Fourier transform) are extremely powerful tools for solving problems in mathematics and the sciences - here we discuss the linear algebra behind the scary looking formulas involved.​
Convolution is a process by which two functions are combined by integrating against each other in an (seemingly) complicated way.  This operation is actually rather natural however - here we will talk through heuristic arguments that lead us to construct this operation for ourselves.
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