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Michael Chou
Norbert Wiener Assistant Professor
Michael Chou
Contact Info:
Tufts University
Department of Mathematics
Bromfield-Pearson Hall
503 Boston Avenue
Medford, MA 02155

Email @tufts.edu:
Phone: 617-627-3235

Number theory

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!