I do mathematical research in a topic called Algebraic Geometry.
The main goal of Algebraic Geometry is to classify Algebraic Varieties, which in a wide sense are all geometric objects which can be defined by polynomial equations. Obviously, such goal is impossible because of the immense size of the object of study.
So, in order to do progress in this area we often focus in a smaller set of algebraic varieties.
Such smaller sets comes in different flavours, for example we can impose conditions on the number of polynomials,
degree of the polynomials, or number of terms of the polynomials, on the other hand we may also try to focus
in the outcome of such polynomial equations instead, meaning that we can require conditions in the geometric object itself
to make it easier to study.
Calabi-Yau Algebraic Variety.
From this point of view, there are two main ways to proceed:
- studying a small set of algebraic varieties and give really precise characterizations,
- studying a big set of algebraic varieties and give a more qualitative description.
Right now, I am more interested in this second approach, I am trying to understand the moduli space (parametrizing space) of varieties of general type, which are morally speaking, the algebraic varieties which one would expect to get when we choose the coefficients of the defining polynomials randomly. The moduli space of smooth curves is one of the most studied geometric objects, understanding the analogous for higher dimensional algebraic varieties is a problem in which many questions remain open.
Smooth Algebraic Curve or Riemann Surface.