Algebraic geometry and its connections to number theory, representation theory, mathematical physics, or other fields. More specifically, the birational and enumerative geometry of moduli spaces in algebraic geometry. See my website (https://deopurkar.github.io) for more.

Positions

• 2018–now, Lecturer, Australian National University, Canberra, Australia
• 2016–2017, Assistant Professor (Limited Term), University of Georgia, Athens, GA.
• 2012–2016, J. F. Ritt Assistant Professor, Columbia University, New York, NY.

Education

• 2008–2012, Ph.D., Harvard University, Cambridge, MA. Advisor: Joseph Harris.
• 2004–2008, S.B., Massachusetts Institute of Technology (MIT), Cambridge, MA. (Mathematics with Computer Science).

These projects are suitable for advanced undergraduate or masters projects. For PhD projects, please read my research page and list of publications (https://deopurkar.github.io/research/).

1. Readings in commutative algebra and algebraic geometry: Reading some classics in commutative algebra and algebraic geometry based on the student's interest and level of preparation. Readings can range from introductory to more advanced (intersection theory, toric varieties, birational geometry). Expected background: MATH2322 (Algebra 1) and MATH3345 (Algebra 2).
2. Topology of singularities: A complex algebraic/analytic singularity is an amazingly intricate object from the point of view of topology. For example, intersecting a plane curve singularity with a unit sphere yields a knot that encodes important invariants of the singularity. The goal is to explore this connection between algebra, geometry, and topology. Expected background: MATH2322 (Algebra 1) and MATH 4204 (Algebraic topology).
3. Group actions and invariants: Why are the trace and the determinant of a matrix special, as opposed to some other random function of the matrix entries? Why should the discriminant b²-4ac play a fundamental role, and not, say a²+4bc? One answer (among many) is that the trace, the determinant, and the discriminant are distinguished among other functions by the fact that they are invariant functions under a natural group action. We will explore group actions and their invariants in algebra, representation theory, and geometry. Projects can be expository or include original research. Expected background: MATH2322 (Algebra 1) and MATH3345 (Algebra 2).
4. Computability in algebra and geometry: Can one write a computer program to decide whether two given rings are isomorphic? Or whether a given group is the trivial group? Or whether a given CW complex is a sphere? These are questions on the inteface of logic, theoretical computer science, and mathematics with fascinating results and open problems. Projects will involve reading (and explaining to me) some of this work, or surveying the current state of the art. Expected background: MATH2322 (Algebra 1) and COMP3630 (Theory of Computation).
5. Arithmetic and geometry over finite fields: How many solutions does the equation x²+y² = z² have if x, y, z are taken from Z/pZ? How many square-free polynomials of degree n are there with coefficients in Z/pZ? Questions of this kind have deep and seemingly unexpected connections with the arithmetic and geometry of algebraic varieties. Projects will involve an exploration of this connection. They can range from heavily theoretical to mostly  experimental, for example, involving computer programming to do these counts, and using the data to make some conjectures. Expected background: MATH2322 (Algebra 1) and MATH3345 (Algebra 2).