In its most basic form, number theory studies properties of the integers.
Examples include studying prime numbers and the solubility of polynomial equations in integers. Looking at these and related questions from a geometrical point of view leads to arithmetic algebraic geometry. An everyday application of early number theory is the RSA cryptography scheme used by banks, etc. More modern (and potentially more secure) cryptography uses curves over finite fields, which are studied by combining techniques from both number theory and arithmetic algebraic geometry.
Algebraic K-theory can be viewed as a far-reaching generalization of the notion of dimension of a vector space. Although originally defined because it satisfied some convenient properties, it is now related to many fields in mathematics, including number theory, algebraic geometry, hyperbolic geometry, and even theoretical physics.
At our department we are also interested in computer algebra, including formally verifying mathematics.