Norbert Wiener Assistant Professor
Department of Mathematics
503 Boston Avenue
Medford, MA 02155
My research is in arithmetic number theory. I'm interested in
studying rational solutions to Diophantine equations. A Diophantine
equation is any polynomial equation $f(x_1, x_2, \ldots, x_n)=0$
with integer coefficients. While generally quite easy to solve over
the real numbers, finding rational solutions can prove to be quite
difficult. My research focuses on a particular type of Diophantine
equation called an elliptic curve. These are equations of the form
$y^2 = x^3 + ax + b$ for some coefficients $a$ and $b$. These are
genuine curves that can be plotted on your favorite graphing tool
and can be thought of as geometric objects. However, there is a
rigorous way in which two points on the curve can be "added"
together to obtain a third point on the curve. We call this the
group law of the elliptic curve. Thus, elliptic curves have both
algebraic and geometric properties which can both be leveraged to
help understand their rational points.
My research is focused on how the solutions of these Diophantine
equations can change when you allow solutions in what are called
"number fields". For instance, $y^2= x(x^2 + 1)$ has only one
solution over the rationals but if we allow ourselves the use of $i$,
the square root of $-1$, then we obtain two additional solutions!
Elliptic curves are beautiful mathematical objects that have a
myriad of applications. For instance the proof of Fermat's Last
Theorem translated the original problem into a question about
elliptic curves. In cryptography, the group law of elliptic curves
can be used to safely encrypt information sent via the world wide
web. If you're interested in learning more about them, stop by!