## Department of Mathematics and Philosophy

### Undergraduate Research - Number Theory and Arithmetic Geometry

##### Seyfi Turkelli

**Areas of Interest:** Number Theory, Arithmetic Algebraic Geometry

**Description:** I welcome all students who enjoy learning mathematics and solving problems. My areas of interest are number theory and arithmetic algebraic geometry.

In short, number theory is the study of integers and the (algebraic or analytic) objects that are made out of them. An example of a well-known problem in number theory is the following: Are there infinitely many pairs of prime numbers that differ by 2? Very recently, it has been proven that there are infinitely many pairs of prime whose difference is less than 70,000,000. One wants to bring this 70-million bound all the way down to 2!

Arithmetic algebraic geometry is the study of polynomial equations and their solutions in integers. For example, we know from high school math classes that we can find all non-zero integers x, y, z such that x^2 + y^2 = z^2-- that's to say this polynomial equation has infinitely many nontrivial solutions in integers--. Can you think of one? In 1637, Pierre de Fermat conjectured one cannot find ANY triple of all non-zero integers x, y, z such that x^n + y^n = z^n for n>2. This problem was solved by Andrew Wiles in 1993-- almost 400 years later!

If these problems sound interesting to you, or if you'd simply like to talk about mathematics, please stop by my office. Maybe we can find a problem for you!

## Connect with us: