Algebraic geometry studies systems of algebraic equations and their solutions.  It is a central field in modern mathematics.  I am happy to supervise a project in algebraic geometry on a topic based on your (and my) interests and background.

An undergraduate, honours, or masters project can range from supervised reading of a book or a paper to independent research.  It can be about core algebraic geometry or about its connections with other fields.  I am most interested in the connections with number theory, representation theory, logic, and computer science, but I am open to other ideas.  Here is a sample of projects students have done with me in the past:

- The algebraic geometry of rigid frameworks
- Determinantal cubic surfaces and 3-by-3-by-4 tensors
- The Borel-Weil-Bott theorem
- The Hilbert scheme of points
- Elliptic curves over finite fields
- Topology of plane curve singularities

An MPhil/PhD project will align more closely with my research interests and expertise.  My interests include the geometry of algebraic curves and surfaces, construction and properties of moduli spaces and stacks, enumerative algebraic geometry, cohomology and Chow rings, and connections to homological and commutative algebra.  Please look at the papers/pre-prints on my webpage (https://ananddeopurkar.org) to get a more accurate picture of my research.