### Complex Algebraic Geometry

Algebraic geometry is a field of mathematics that studies so-called
*varieties*, i.e. the sets of solutions of systems of polynomial
equations. The main goal of the theory is to figure out how various geometric
properties of these varieties can be read off algebraically from the
equations describing them. Algebraic geometry is a very broad field of
mathematics that has relations to many other areas of research such as e.g.
complex analysis, topology, differential geometry, singularity theory,
computer algebra, commutative algebra, and number theory.

More introductory information on complex algebraic geometry can e.g. be found in chapter 0 of my Algebraic Geometry class notes.

### Enumerative Geometry and Gromov-Witten Invariants

The goal of enumerative geometry is to count curves in a given space $X$ that satisfy some given conditions. These conditions can be of various types: we can require that the curves have specified genus, specified degree, intersect given subvarieties of $X$, are tangent to a given subvariety of $X$, have certain singularities, and so on. Although enumerative geometry is a very classical subject it has received new attention and made enormous progress in the last 25 years due to ideas and results from theoretical physics and the invention of Gromov-Witten theory that provides a very general modern framework for attacking enumerative problems.

More introductory information on enumerative geometry can e.g. be found in chapter 0 of my Enumerative Geometry class notes.

### Tropical Algebraic Geometry

Tropical algebraic geometry is a recent and active field of mathematics that tries to attack complicated algebraic or geometric problems (e.g. in enumerative geometry) by combinatorial methods. Ideally, every construction in algebraic geometry should have a combinatorial counterpart in tropical geometry. If this tropical set-up is easier to understand one can then try to transfer the tropical results back to the original algebro-geometric setting. Moreover, tropical geometry has many relations to other areas of research, e.g. algebraic statistics and even bioinformatics.

More introductory information on tropical geometry can e.g. be found in my expository paper on tropical geometry.